The six dimensional sphere is a complex manifold
G\'abor Etesi
{Egry J. u. 1, H \'ep.,}
Existence of a complex structure on the six dimensional sphere is proved in this paper. The proof is based on re-interpreting a hypothetical complex structure as a classical ground state of a non-linear scalar field theory formulated on $S^6$. This classical vacuum solution is then constructed via Fourier expansion from the obvious one of a similar theory on the exceptional compact Lie group ${G}_2$.
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\begin{document}

\title{The six dimensional sphere is a complex manifold}

\author{G\'abor Etesi
\\ {\it Department of Geometry, Mathematical Institute, Faculty of
Science,}
\\ {\it Budapest University of Technology and Economics,}
\\{\it Egry J. u. 1, H \'ep.,}
\\{\it H-1111 Budapest, Hungary}
\footnote{e-mail: {\tt etesi@math.bme.hu}}}

\maketitle

\pagestyle{myheadings}
\markright{G. Etesi: The six-sphere is a complex manifold}

\thispagestyle{empty}

\begin{abstract}
Existence of a complex structure on the six dimensional sphere is 
proved in this paper. The proof is based on re-interpreting a hypothetical 
complex structure as a classical ground state of a non-linear scalar 
field theory formulated on $S^6$. This 
classical vacuum solution is then constructed via Fourier expansion 
from the obvious one of a similar theory on the exceptional compact Lie group 
${\rm G}_2$. 
\end{abstract}
\centerline{AMS Classification: Primary: 32J17, 32J25; Secondary: 43A30, 
53C56, 81T13}
\centerline{Keywords: {\it Six-sphere; Complex structure; 
Fourier expansion; Yang--Mills--Higgs theory}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
\label{one}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A classical result of Borel and Serre from 1951 states that among 
spheres the two and the six dimensional are the only ones which can 
carry almost complex structures \cite{bor-ser1, bor-ser2}. In the case 
of the two-sphere the almost complex structure is {\it unique} and 
stems from the embedding $S^2\subset {\rm Im}\HH$ as the unit 
sphere by taking $J_x$ to be simply the multiplication by $x\in S^2$. 
Equivalently it can be constructed through the identification $S^2\cong\C P^1$ 
consequently it is {\it integrable} in the sense that it comes from a 
complex manifold structure. On the contrary the six-sphere admits a 
{\it plethora} of almost complex structures which are 
typically {\it non-integrable}. For example the analogouos six 
dimensional almost complex structure can be constructed \cite[pp. 
163-164]{wil} from the inclusion $S^6\subset {\rm Im}\OO$ where $\OO$ 
refers to the octonions or Cayley numbers; however it is not 
integrable \cite{fro}. This can be directly proved by a lengthy 
calculation of its Nijenhuis tensor: since it is not zero this Cayley almost 
complex structure is not integrable by the Newlander--Nirenberg 
theorem \cite{new-nir}. Putting an orientation 
and a Riemannian metric $g$ onto $S^6$ an abundance of other almost complex 
structures compatible with the orientation and the metric emerges as smooth 
sections of the projectivized positive chirality spinor bundle $P\Sigma^+$, or 
equivalently, of the positive twistor space $Z^+(S^6,[g])$ where $[g]$ 
denotes the conformal class of $g$, cf. \cite[Chapter IV, \S 9]{law-mic}. 
Homotopy theory also can be used to construct almost 
complex structures \cite{pen-tan}. In spite of the two dimensional case 
however, it has been unknown whether or not any of these or other 
almost complex structures are integrable \cite[pp. 16-18]{che1}---or 
it has rather been believed that all of them are not-integrable hence 
the six-sphere cannot be a complex manifold at all \cite[p. 424]{gre-sch-wit}.

One source of this dismissive attitude might 
be an \ae sthetic aversion: if the most canonical and natural 
Cayley almost complex structure is not integrable then what else could be? 
Another one might be the feeling that the existence of a complex structure 
on $S^6$ would be somehow awkward: for example \cite{huc-keb-pet} 
it would allow one to perform non-K\"ahlerian deformations of the standard 
complex structure on $\C P^3$; a ``minor disaster'' as LeBrun says 
in \cite{leb}. 

During the past six decades several works appeared 
which claimed to prove the non-existence however they turned 
out to be erroneous. Examples are \cite{adl, hsi} while the latest 
trial was Chern's attack \cite{che2} in 2004 based on the link between 
$S^6$ and the exceptional group ${\rm G}_2$. Although it also seems to 
have a gap, Chern's approach has introduced interesting new techniques 
into gauge theories \cite{mar-nie}.

These failures are not surprising because this complex structure, if 
exists, cannot be grasped by conventional means. Indeed, let $X$ 
denote this hypothetical compact complex $3$-manifold i.e., an 
integrable almost complex structure $J$ on $S^6$. Regarding
classical geometry, $X$ cannot have anything to do with $S^6\subset\R^7$
as the usual sphere with round metric $g_0$ since $J$ cannot even be 
orthogonal with respect to the standard metric \cite{leb} that is, $X$ 
cannot be constructed from a section of $Z^+(S^6, [g_0])$ in the sense above 
(but for some properties of this standard twistor space cf. 
\cite{mus, slu}). Since $\pi_i(X)=0$ ($i=1,\dots,5$) $X$ cannot be 
fibered over any lower dimensional manifold therefore 
algebraic topology does not help to cut down the problem to a 
simpler one. It also follows that $H^2(X;\Z )\cong 0$ consequently $X$ 
cannot be K\"ahler implying it cannot be approached by 
traditional analysis or algebraic geometry. In fact any meromorphic function 
on $X$ must be constant \cite{cam-dem-pet} demonstrating on the one hand 
the inadequacy of powerful methods of complex analysis and on the 
other hand that the algebraic dimension of $X$ is zero consequently it is 
as far from being algebraic as possible. We also know that $X$ 
cannot be a homogeneous complex manifold in the sense that ${\rm 
Aut}_{\co}X$ cannot act transitively and none of its orbits can be 
open \cite{huc-keb-pet} (actually this property would permit the existence 
of the aforementioned exotic complex projective spaces). 

Nevertheless the Hodge numbers of $X$ are known in some 
extent \cite{gra2, uga}. We also mention that there is an 
extensive literature about various submanifolds of $S^6$ equipped with 
various structures. Far from being complete, for instance the almost 
complex submanifolds of the Cayley almost complex $S^6$ are studied in 
\cite{gra1} and an excellent survey about the Lagrangian submanifolds of the 
Cayley nearly K\"ahler $S^6$ is \cite{chen}. 

The long resistance of the problem against proving non-existence may 
indicate that one should rather try to seek a complex structure on $S^6$.
However the irregular features of this hypothetical space $X$ convince 
us that {\it asymmetry, transcendental} 
(i.e., non-algebraic) {\it methods} and {\it inhomogeneity} should play a 
key role in finding it. 

Indeed, by a result of Wood \cite{woo} from a ``physical'' viewpoint the 
\ae sthetic Cayley almost complex structure is energetically remarkable 
unstable (however for a contradictory result cf. \cite{bor-lam-sal}). 
Therefore it is not surprising that if integrable almost complex 
structures on $S^6$ exist then they would appear ``far'' from the Cayley one. 
For instance Peng and Tang \cite{pen-tan} recently have constructed a novel 
almost complex structure by twisting and extending the standard complex 
structure on $\C^3\subset S^6$. It is a vast deformation of the Cayley 
almost complex structure and is integrable except a narrow equatorial 
belt in $S^6$. However it is still orthognal with respect to the standard 
round metric hence LeBrun's theorem \cite{leb} forbids its full integrability.

As we have indicated above, the central difficulty with this hypothetical 
complex structure is that it should be compact and non-K\"ahler at the same 
time. However in fact plenty of such non-algebraic compact complex manifolds 
are known to exist for a long time: classical examples are the 
Hopf-manifolds; or the Calabi--Eckmann manifolds 
\cite{cal-eck} and the compact, even dimensional, simply connected 
Lie groups---as it was observed by Samelson \cite{sam}---both from 1953. 

Therefore one may try to seek $X$ by minimizing an appropriate energy 
functional over one of these spaces somehow related to $S^6$. 
{\it A fortiori} the candidate is the compact, simply connected, 14 
dimensional exceptional Lie group ${\rm G}_2$ which arises as the total space 
of a non-trivial ${\rm SU}(3)$-bundle over $S^6$. But how the 
energy functional should look like on it? A physisist is experienced that 
there is a close analogy between the existence of a geometric structure on a 
manifold on the mathematical side (cf. e.g. \cite{joy}) and the existence of 
a spontaneous symmetry breaking in Yang--Mills--Higgs-like systems on the 
physical side. This later process is familiar in quantum field theory 
\cite{che-li, kak, wei}. Consequently we are
tempted to consider an appropriate (non-linear) Higgs scalar field theory 
on ${\rm G}_2$ whose Higgs field in the spontaneously broken vacuum 
state represents the complex structure on ${\rm G}_2$. Performing a 
dimensional reduction in this theory from ${\rm G}_2$ down to $S^6$ 
and dropping unimportant fields we obtain a similar Higgs scalar field 
theory on $S^6$. Then we consider what happens to 
the original smooth vacuum solution in the limit 
${\rm Vol}({\rm SU}(3))\rightarrow 0$. One would expect that it develops a 
nasty singularity. To see that this is not the case, following \cite{wit} we 
expand the vacuum on ${\rm G}_2$ into Fourier modes with respect to 
${\rm SU}(3)$ and then prove by the aid of the stationary phase method that 
all the higher modes (corresponding to the non-trivial irreducible 
representations of the collapsing group) vanish in this limit thereby 
leaving us with the ground mode only. This is independent of the 
${\rm SU}(3)$ directions hence it smoothly descends to $S^6$ providing us 
with a smooth vacuum solution of the associated theory on $S^6$. We finally 
check that this vacuum gives rise to a complex structure on the six-sphere. 

From a physical viewpoint the passage ${\rm Vol}({\rm SU}(3))\rightarrow 0$ 
can be regarded as replacing the original theory formulated on ${\rm G}_2$ 
with an effective theory on $S^6$ by integrating out all degrees of 
freedom along the fibers. This also suggests that the vacuum of the 
original theory should survive not only classically but even 
at the quantum level in this low energy limit. 

It also should become clear in the course of the construction why this 
approach breaks down in apparently 
similar situations like ${\rm SO}(4n+1)/{\rm SO}(4n)\cong S^{4n}$. It 
will turn out that in the case of ${\rm G}_2/{\rm SU}(3)\cong S^6$ there 
is an exceptional topological coincidence related with homotopy groups (cf. 
Eq. (\ref{veletlen}) here) which makes it possible to reduce the 
original vacuum solution on ${\rm G}_2$ in a smooth and global way to $S^6$.

Guided by these ideas in what follows we hope to prove the 
existence of a complex structure on the six-sphere. 
The paper is organized as follows. In Sect. \ref{two} we formulate a 
hierarchy of classical field theories describing various spontaneous symmetry 
breakings related to the existence of various geometric structures on the 
underlying manifold. Apparently these sort of spontaneous symmetry 
breakings are not distinguished by physicists. In particular we 
identify a non-linear Higgs scalar theory on the tangent bundle of 
an even dimensional oriented Riemannian manifold describing a 
so-called ``weak spontaneous symmetry breaking'' which is the physical 
reformulation of the existence of a non-K\"ahlerian integrable almost 
complex structure on the underlying manifold. Unfortunately this 
theory is highly non-linear hence solving its field 
equations directly or even the detailed analytical study of them is far 
beyond our technical limits. 

Instead of this in Sect. \ref{three} we develop a sort of global Fourier 
expansion which makes it possible to understand the behaviour of sections of 
trivial vector bundles over ${\rm G}_2$ in our limit.

This technique enables us in Sect. \ref{four} to prove 
{\it ad hoc} the existence of a smooth vacuum solution of the field 
equations of this Higgs scalar theory over $S^6$ by 
starting with an obvious vacuum solution of a similar theory on 
${\rm G}_2$ and then taking its ground Fourier mode in the limit 
${\rm Vol}({\rm SU}(3))\rightarrow 0$. This section contains our main result, 
namely that there exists an integrable almost complex structure on the 
six-sphere (cf. Theorem \ref{Xtetel}). 
\vspace{0.1in}

\noindent{\bf Acknowledgement.} The author was partially supported by 
OTKA grant No. NK81203 (Hungary).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Geometric structures and symmetry breaking}
\label{two}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $M$ be an $n$ dimensional manifold and pick an abstract Lie group 
$G\subset{\rm GL}(n,\R )$. Recall that \cite[Section 2.6]{joy} a {\it 
$G$-structure} on $M$ is a principal sub-bundle of the frame bundle of 
$M$ whose fibers are $G$ and the structure group of this sub-bundle is also 
$G$. A $G$-structure is called {\it integrable} (or {\it torsion-free}) if 
there is a torsion-free connection $\nabla$ on $TM$ with ${\rm Hol}\nabla$ 
being (a subgroup of) $G$. An integrable $G$-structure is 
interesting because the individual $G$-structures on the tangent spaces 
stem from an underlying global non-linear structure on $M$.

As a starting setup suppose that a $G$-structure on $M$ is given 
together with a connection on $TM$ whose holonomy group satisfies 
${\rm Hol}\nabla \subseteq G$. Note that $G$ plays a 
double role here. Pick a Lie subgroup $H\subset G$. We are going 
to describe two versions of a ``spontaneous symmetry breaking'' over $M$ 
as follows. Firstly, one can speak about a {\it strong spontaneous symmetry 
breaking $G\rightarrow H$} over $M$ in which {\it $G$ 
as the holonomy group of $\nabla$} is cut down to (a subgroup of) $H$. 
That is, the connection satisfies that ${\rm Hol}\nabla\subseteq H$.

To clarify this concept, an example can be constructed as follows. Take 
a $2m$ real dimensional oriented Riemannian manifold $(M,g)$. 
Then $TM$ is an ${\rm SO}(2m)$ vector bundle associated with the 
standard representation of ${\rm SO}(2m)$ to its frame bundle and 
has an automorphism bundle providing a gauge group 
$\cg :=C^\infty (M; {\rm Aut}_{{\rm SO}(2m)}TM)$. Let $\nabla$ be 
an arbitrary SO$(2m)$ connection on $TM$ and let $F_\nabla$ and 
$T_\nabla$ denote its curvature and torsion tensor respectively.

Consider moreover the adjoint representation of SO$(2m)$ on its
Lie algebra and the associated adjoint bundle ${\rm ad}M\subset{\rm End}TM$ 
whose fibers are ${\mathfrak s}{\mathfrak o}(2m)\subset {\rm End}\:\R^{2m}$. 
Its generic section is $\Phi\in C^\infty (M;{\rm ad}M)$. A
geometric example of a section of this kind is an orthogonal almost
complex structure whose induced orientation agrees with the given one.
Indeed, since for a real oriented vector space with a 
scalar product $(V, \langle\cdot\:,\:\cdot\rangle )$ the
moduli space of orthogonal complex structures $J$ compatible with the
orienation is precisely ${\rm SO}(V)\cap{\mathfrak s}{\mathfrak o}(V)
\subset{\rm End}\:V$, it follows that we can look at any orthogonal 
almost complex structure with respect to $g$ and compatible 
induced orientation as a section $J=\Phi\in C^\infty (M;{\rm ad}M)$. By a 
little abuse of notation let us also denote by $\nabla$ the connection on 
${\rm ad}M$, associated to the SO$(2m)$ connection on $TM$ introduced above. 
Also fix a real number $e\not= 0$.

Then for example an ${\rm SO}(2m)\rightarrow{\rm 
U}(m)$ strong spontaneous symmetry breaking arises by seeking the 
hypothetical vacuum solutions (see below) of the energy functional
\[\ce (\nabla ,\Phi ,g ):=
\frac{1}{2}\int\limits_M\left(\frac{1}{e^2}\vert 
F_\nabla\vert_g^2+\vert\nabla\Phi\vert_g^2+e^2\vert\Phi\Phi^*-{\rm 
Id}_{TM}\vert^2_g\right)\dd V.\]
Various pointwise norms induced by the metric on various  SO$(2m)$ 
vector bundles and the Killing form on
${\mathfrak s}{\mathfrak o}(2m)$ are used here to calculate norms of 
tensor fields in the integral. We note that $\Phi^*$ is the pointwise 
adjoint (i.e., transpose) of the bundle map $\Phi :TM\rightarrow TM$ with 
respect to $g$. 

$\ce (\nabla ,\Phi ,g)\geqq 0$ and it is 
invariant under ${\rm SO}(2m)$ acting as gauge transformations $\gamma\in \cg$ 
given by $(\nabla ,\Phi ,g)\mapsto (\gamma^{-1}\nabla\gamma ,
\gamma^{-1}\Phi\gamma ,\gamma^*g )$. In fact $\ce (\nabla ,\Phi ,g)$ is the 
very usual Euclidean action of a Yang--Mills--Higgs theory on (the 
tangent bundle of) $M$ from the physics literature. Hence, using physicists' 
terminology \cite{che-li,kak,wei} the above functional can be regarded as the 
action functional of a {\it sponataneously broken $2m$
dimensional Euclidean ${\rm SO}(2m)$ Yang--Mills--Higgs system} on $TM$.
In this context $\nabla$ is the ``${\rm SO}(2m)$ gauge 
field'' and $\Phi$ is the ``Higgs field in the adjoint representation'', 
$e$ represents the ``coupling constant'' and the quartic polynomial 
$\vert\Phi\Phi^*-{\rm Id}_{TM}\vert^2_g$ is the ``Higgs potential''. 
If a smooth vacuum solution $\ce (\nabla ,\Phi ,g)=0$ exists then 
\[\left\{\begin{array}{ll}
0= & F_{\nabla}\\
0= & \nabla\Phi\\
0= & \Phi\Phi^*-{\rm Id}_{TM}.
        \end{array}\right.\]
(Note that these are {\it not} the Euler--Lagrange equations of the 
system!) The first equation says that $\nabla$ is flat. The third 
equation dictates $\Phi^2=-{\rm Id}_{TM}$ hence restricts the Higgs 
field to be an orthogonal almost complex structure $\Phi =J$ with 
respect to $g$. The second equation reduces the holonomy of 
$\nabla$ in the usual way \cite[Proposition 2.5.2]{joy} to 
(by flatness necessarily a discrete subgroup of) ${\rm U}(m)$. 
Therefore this vacuum---if exists---describes a (not 
necessarily integrable) orthogonal almost complex structure and a compatible 
flat connection on $M$. The fact that for this vacuum solution ${\rm 
Hol}\nabla\subset{\rm U}(m)\subset{\rm SO}(2m)$ justifies 
that this is indeed an example for our first concept given above. 

The resulting structure is non-integrable and flat hence mathematically 
less interesting. This illness of the vacuum is cured if the curvature is 
replaced by the torsion. Consequently we put the metric and the Higgs 
field to be the fundamental fields only to get a new theory
\[\ce (\Phi ,g):=
\frac{1}{2}\int\limits_M\left(\vert\nabla\Phi\vert_g^2+
e^2\vert\Phi\Phi^*-{\rm Id}_{TM}\vert^2_g\right)\dd V\]
where $\nabla$ is the Levi--Civita connection. This functional is again 
non-negative and acted upon by $\cg$ as 
$(\Phi ,g)\mapsto (\gamma^{-1}\Phi\gamma ,\gamma^*g)$ 
leaving it invariant. Although this theory looks physically less obvious 
this departure from physics yields a more familiar structure on the 
mathematical side. Namely, it is clear that a hypothetical smooth 
minimizing configuration $\ce (\Phi ,g)=0$ gives 
rise to a K\"ahler structure on $M$. Note that 
${\rm Hol}\nabla\subseteq{\rm U}(m)\subset{\rm SO}(2m)$ hence this is 
again a strong spontaneous symmetry breaking 
${\rm SO}(2m)\rightarrow{\rm U}(m)$.

Returning to the original setup, our second concept is a {\it weak 
spontaneous symmetry breaking $G\rightarrow H$} in 
which $G$ {\it as the structure group of $TM$} is reduced to (a subgroup 
of) $H$. That is, simply there is a $H$-structure on $M$ but probably 
the connection is not compatible with this structure. The strong 
symmetry breaking above if exists always implies the corresponding 
weak one cf. e.g. \cite[Theorem 2.3.6]{joy} however not the other way 
round. 

A weak symmetry breaking ${\rm SO}(2m)\rightarrow{\rm U}(m)$ is provided by 
the following example which goes along the lines of the previous two cases. 
Consider the first order non-linear differential operator
\[N_\nabla : C^\infty (M;{\rm End}TM)\longrightarrow
C^\infty (M;{\rm End}TM\otimes\Lambda^1M)\]
whose local shape on a chart $(U, x^1,\dots,x^{2m})$ of $M$ looks like
\cite[Eqn. 15.2.5]{gre-sch-wit}
\begin{equation}
\left(\left(N_\nabla\Phi\right)\vert_U\right)^k_{\:\:\:ij}=
\sum\limits_{l=1}^{2m}
\Phi^l_{\:\:i}(\nabla_l\Phi^k_{\:\:j}-\nabla_j\Phi^k_{\:\:l})-
\sum\limits_{l=1}^{2m}\Phi^l_{\:\:j}(\nabla_l\Phi^k_{\:\:i}-  
\nabla_i\Phi^k_{\:\:l}),\:\:\:\:\:i,j,k=1,\dots,2m.
\label{lokalis-nijenhuis}
\end{equation}
Note that $N_\nabla\Phi$ transforms as a tensor under ${\rm SO}(2m)$. 
This time set
\begin{equation} 
\ce (\Phi ,g):=\frac{1}{2}\int\limits_M\left(\vert N_\nabla\Phi\vert^2_g
+e^2\vert\Phi\Phi^*-{\rm Id}_{TM}\vert^2_g\right)\dd V.
\label{YMH}
\end{equation}
This functional is again invariant under $\cg$ acting as before moreover it 
is non-negative.

Therefore it describes a sort of gauge theory on $M$. The two peculiarities 
here compared to the usual Yang--Mills--Higgs Lagrangian is that the 
curvature has again been dropped and the coupling 
$\nabla\Phi$ has been replaced with $N_\nabla\Phi$ regarded as a 
non-minimal quadratic coupling in (\ref{YMH}). This further departure from 
the physical side gives rise again to a familiar structure on $M$ on the 
mathematical side as follows.

If for a fixed coupling constant $e\not=0$ the smooth vacuum 
$\ce (\Phi ,g)=0$ is achieved in (\ref{YMH}) over $M$ then it satisfies
\begin{equation}
\left\{\begin{array}{ll}
0= & N_\nabla\Phi\\
0= & \Phi\Phi^*-{\rm Id}_{TM}.
        \end{array}\right.
\label{vakuum}
\end{equation}
The second equation ensures us that $\Phi^2=-{\rm Id}_{TM}$ i.e., the
Higgs field reduces to an orthogonal almost complex structure $\Phi =J$ 
with respect to the metric and is compatible with the orientation. 
Exploiting this together with the fact that $\nabla$ is torsion-free 
the operator (\ref{lokalis-nijenhuis}) in the gauge 
induced by the coordinate system on $(U, x^1,\dots, x^{2m})$ cuts down to
\[\left(\left(N_{\nabla}J\right)\vert_U\right)^k_{\:\:\:ij}=\sum\limits_{l=1}^{2m}
J^l_{\:\:i}\left(\partial_lJ^k_{\:\:j}-\partial_jJ^k_{\:\:l}\right)
-\sum\limits_{l=1}^{2m}J^l_{\:\:j}\left(\partial_lJ^k_{\:\:i}-
\partial_iJ^k_{\:\:l}\right),\:\:\:\:\:i,j,k=1,\dots,2m\]
hence $N_{\nabla} J=N_J$ is the 
{\it Nijenhuis tensor} of the almost complex structure $J$ given by
\[N_J(u,v)= [u,v]-[Ju, Jv]+J[u, Jv]+J[Ju, v]\]
for two vector fields $u,v$ on $M$. Consequently the first equation of 
(\ref{vakuum}) says that $N_J=0$ that is, this almost 
complex structure is integrable on $M$ and is orthogonal with respect to 
$g$. Therefore if a vacuum solution to
(\ref{YMH}) exists it makes $(M, J)$ into 
a complex analytic $m$-space (together with a compatible metric $g$ however 
this is not important to us) in light of the Newlander--Nirenberg theorem 
\cite{new-nir}. Note that this time the structure group of $TM$ 
has been cut down to (a subgroup of) ${\rm U}(m)$ but probably for the 
connection ${\rm Hol}\nabla\not\subseteq{\rm U}(m)$ hence this example is 
indeed an example for our second concept. Accordingly the resulting 
complex structure on $M$ is weaker than the K\"ahler one in the previous 
example.

We have described two different symmetry breakings ${\rm 
SO}(2m)\rightarrow{\rm U}(m)$ over a Riemannian manifold. However for 
physicists these concepts coincide because for them a ``spontaneous 
symmetry breaking'' has a slightly different meaning. It simply means that 
although one starts with an ${\rm SO}(2m)$-invariant theory over $M$, 
the vacuum $(\Phi ,g)$ is stabilized by a subgroup of ${\rm SO}(2m)$ 
isomorphic to ${\rm U}(m)$ only. Of course this criteria holds 
in all of our cases. 

We know that all integrable solutions to the previous strong 
problem---i.e., the K\"ahler structures---provide integrable solutions 
to the weaker one, too. But we rather raise the question whether or not 
there are solutions to the weaker problem which do not stem 
from strong solutions. 
Approaching the problem directly with analysis is certainly very hard.

Nevertheless the answer is yes, even in the compact case. In what 
follows we take $M:=G$, a connected, simply connected, compact, even 
dimensional Lie group to be the base manifold in (\ref{YMH}). 
Left-invariant complex structures on $G$ can be constructed as follows 
\cite{sam}. Write ${\mathfrak g}:=T_eG$ for the Lie algebra of $G$ and 
let ${\mathfrak g}^\C$ be its complexification. A complex Lie subalgebra 
${\mathfrak s}\subset{\mathfrak g}^\C$ is a {\it Samelson subalgebra} if 
$\dim_\C{\mathfrak s}= \frac{1}{2}\dim_\C({\mathfrak g}^\C )$ and 
${\mathfrak s}\cap{\mathfrak g}=\{ 0\}$. Then an almost complex structure 
$J:{\mathfrak g}\rightarrow {\mathfrak g}$ is determined by ${\mathfrak s}$ 
such that its complex linear extension $J^\C :{\mathfrak g}^\C
\rightarrow {\mathfrak g}^\C$ has the property that ${\mathfrak s}$ is 
the $-\ii$ and ${\mathfrak g}$ is the $+\ii$ eigenspace of $J^\C$ 
respectively. Let $G^\C$ be the complexified Lie group and $S\subset G^\C$ 
the complex Lie subgroup for ${\mathfrak s}\subset{\mathfrak g}^\C$; then 
there exists a diffeomorphism $G^\C\cong G\times S$ and a homogeneous complex 
structure on $G$ induced by $J$ (i.e., by ${\mathfrak s}$) arises by taking 
$G\cong G^\C /S$. If $Y$ denotes this compact complex manifold then, 
since $H^2(Y;\R )\cong \{ 0\}$, it 
{\it cannot} be algebraic (cf. e.g. \cite[Lemma 4.1]{cam-dem-pet}) 
or even K\"ahler hence $\nabla J\not =0$ with respect to any metric. 

As a $T$-module with respect to the adjoint action of a maximal torus 
$T\subseteq G$ we have the usual decomposition
\begin{equation}
{\mathfrak g}={\mathfrak t}\oplus\bigoplus\limits_{\alpha\in R^+}
{\mathfrak g}_\alpha
\label{gyokok}
\end{equation}
into the Cartan subalgebra ${\mathfrak t}$ corresponding to $T$ and the 
positive root spaces ${\mathfrak g}_\alpha$. Since this action commutes 
with the almost complex structure, $J$ also decomposes as follows. As a 
real vector space ${\mathfrak t}\cong\R^l$ where $l=\dim_\R{\mathfrak t}$ 
is the rank of $G$. Since $\dim_\R G$ is even, $l$ is even hence $J_e$ 
restricted to ${\mathfrak t} $ gives a complex structure $J_0$ on 
${\mathfrak t}$ providing an isomorphism ${\mathfrak t}\cong\C^{\frac{l}{2}}$. 
In addition, on any ${\mathfrak g}_\alpha\cong\R^2$ we obtain a complex 
structure $J_\alpha$ on $\R^2$ providing ${\mathfrak g}_\alpha\cong\C$. 
Consequently at $e\in G$ the integrable almost complex operator $J$ 
representing the complex structure splits up:
\begin{equation}
J_e=J_0\oplus\bigoplus\limits_{\alpha\in R^+}J_\alpha .
\label{J-szetszedes}
\end{equation}

An example is ${\rm SU}(2)\times{\rm SU}(2)$ as a complex manifold. Note 
that this space carries another kind of complex structure too and 
is also known as a Calabi--Eckmann manifold \cite{cal-eck}. However in 
principle this later space differs from our example here. Another 
example is the exceptional Lie group ${\rm G}_2$ as a complex manifold.

Picking a representative $J$ of the complex structure on $G$, we put an 
induced orientation onto $G$ as well as a metric by defining it to be 
$g(u,v):=\frac{1}{2}g'(u,v)+\frac{1}{2}g'(Ju, Jv)$ for some metric $g'$ 
and any vector fields $u,v$ on $G$. Then the almost complex 
operator $J$ is orthogonal with respect to $g$. Consequently taking $\Phi$ 
to be $J$ and the metric to be $g$ just constructed we obtain that 
$(\Phi ,g)$ solves (\ref{vakuum}) on the compact space $G$ i.e., gives rise 
to a strictly weak spontaneous symmetry breaking 
${\rm SO}(2m)\rightarrow{\rm U}(m)$ where $\dim_\R G=2m$. The spontaneously 
broken vacuum represents the complex structure on $G$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Fourier series expansion over principal bundles}
\label{three}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this technical section we collect some useful facts about 
expansions of functions into Fourier series over a compact Lie group $G$. 
A general reference is for instance \cite{hel}
but we shall use the language of \cite{may}. Then we generalize this 
procedure to expand sections of trivial vector bundles over 
principal $G$-bundles. 

Let $G$ be a compact Lie group and let us denote by ${\rm Irr}(G;\C )$ the 
set of isomorphism classes of its finite dimensional complex irreducible 
representations. That is, if $\rho\in {\rm Irr}(G;\C )$ then there exists 
a finite dimensional complex vector space $V_\rho$ and a homomorphism 
$\rho : G\rightarrow{\rm Aut}V_\rho$. For a fixed (isomorphism 
class of) $\rho$ pick a basis in $V_\rho$ to write $(V_\rho 
,e_1,\dots,e_{\dim_\C V_\rho})\cong\C^{\dim_\C V_\rho}$ and denote the 
corresponding matrix elements of $\gamma\in G$ as $\rho^{ij}(\gamma )$. 
Then $\rho^{ij}:G\rightarrow\C$ is a continuous function and the 
Peter--Weyl theorem \cite[Theorem 3.1]{bro-tom} implies that
\[\left\{\sqrt{\dim_\C V_\rho}\:\rho^{ij}:G\rightarrow\C\:\vert\:\rho\in 
{\rm Irr}(G;\C );\:\: i,j=1,\dots,\dim_\C V_\rho\right\}\]
forms a complete set of orthonormal functions in the Hilbert space 
$L^2(G;\C )$ of square-integrable complex-valued functions on $G$ with 
the usual scalar product $(f_1,f_2)_{L^2}:=\lambda\int_Gf_1
\overline{f}_2\dd V$ where $\dd V$ is a Haar measure on $G$ 
defined by the volume form of a left-invariant metric on $G$ and 
$\lambda^{-1}:={\rm Vol}\:G$ is the volume of $G$ with respect to this 
measure i.e., $\lambda^{-1}=\int_G\dd V$. For any element $f\in L^2(G;\C )$ 
the formal sum
\begin{equation}
f(\gamma )\sim\sum\limits_{\rho\in {\rm Irr}(G;\C )}
\sqrt{\dim_\C V_\rho}\sum\limits_{i,j=1}^{\dim_\C V_\rho}a_\rho^{ij}
\rho^{ij}(\gamma )
\label{fourier}
\end{equation}
is called the {\it Fourier expansion of $f$ with respect to $G$ of volume 
$\lambda^{-1}$} if the constants $a_\rho^{ij}\in\C$ are defined by 
\begin{equation}
a_\rho^{ij}:=\sqrt{\dim_\C V_\rho}\:\lambda\int\limits_G 
f(\gamma )\rho^{ij}(\gamma^{-1})\dd V.
\label{egyutthatok}
\end{equation}
The number $a_\rho^{ij}$ is called the {\it $ij^{\it th}$ Fourier 
coefficient (of $f$) belonging to $\rho$}. We also call the collection 
$\{ a_\rho^{ij}\:\vert\: i,j=1,\dots,\dim_\C V_\rho\}$ of
Fourier coefficients as the {\it Fourier mode (of $f$) belonging to
$\rho$} and denote it simply by $a_\rho$. 

We need the following lemma about the asymptotics of the Fourier 
coefficients in the small volume regime:
\begin{lemma}
Let $G$ be a compact matrix Lie group of 
positive dimension with a Haar measure on it such that the volume of $G$ 
is $\lambda^{-1}={\rm Vol}\:G>0$. Then for any non-trivial complex 
irreducible representation $\rho\in {\rm Irr}(G;\C )$ one has
\[\lim\limits_{\lambda\rightarrow +\infty}a_\rho^{ij} (\lambda )=0\]
where $a_\rho^{ij}(\lambda )=a_\rho^{ij}\in\C$, regarded as a function 
of $\lambda$, is the Fourier coefficient (\ref{egyutthatok}) of an 
arbitrary smooth function $f\in C^\infty (G;\C )\subset L^2(G;\C )$.
\label{eltuneslemma1}
\end{lemma}
\noindent{\it Proof.} We give a proof if $G={\rm SU}(3)$ because 
this case will be relevant to us. The proof of the general case is 
very similar but its formalization would be tedious hence we omit.

Recalling \cite[Eq. 2.10]{bro} we parameterize $\gamma\in{\rm SU}(3)$ as 
follows:
\[\begin{array}{ll}
\gamma^{11}=\cos\theta^1\cos\theta^2\:
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}\phi^1},\hspace{2cm} 
\gamma^{12}=\sin\theta^1\:{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}\phi^3}, 
& \gamma^{13}=\cos\theta^1\sin\theta^2\:
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}\phi^4},\\
        & \\
\gamma^{21}=\sin\theta^2\sin\theta^3\:
{\rm e}^{-{\bf i}\lambda^{\frac{1}{5}}(\phi^4+\phi^5)}-
\sin\theta^1\cos\theta^2\cos\theta^3\:{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}
(\phi^1+\phi^2-\phi^3)}, &\gamma^{22}=
\cos\theta^1\cos\theta^3\:{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}\phi^2},\\
             & \\
\gamma^{23}=-\cos\theta^2\sin\theta^3\:{\rm e}^{-{\bf i}
\lambda^{\frac{1}{5}}(\phi^1+\phi^5)}-\sin\theta^1\sin\theta^2\cos\theta^3\:
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}(\phi^2-\phi^3+\phi^4)}, & \\
           & \\
\gamma^{31}=-\sin\theta^1\cos\theta^2\sin\theta^3\:
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}(\phi^1-\phi^3+\phi^5)}-
\sin\theta^2\cos\theta^3\:{\rm e}^{-{\bf i}
\lambda^{\frac{1}{5}}(\phi^2+\phi^4)}, & 
\gamma^{32}=\cos\theta^1\sin\theta^3\:{\rm e}^{{\bf i}
\lambda^{\frac{1}{5}}\phi^5},\\
          & \\
\gamma^{33}=\cos\theta^2\cos\theta^3\:{\rm e}^{-{\bf i}
\lambda^{\frac{1}{5}}(\phi^1+\phi^2)}-\sin\theta^1\sin\theta^2\sin\theta^3\:
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}(-\phi^3+\phi^4+\phi^5)} & \\
         \end{array}\]
where $0\leqq\theta^1,\theta^2,\theta^3<\pi /2,\:\:\:
0\leqq\phi^1,\dots,\phi^5<2\pi/\lambda^{\frac{1}{5}}$. The volume form 
looks like \cite[Eq. 2.11]{bro}
\[\dd V=\frac{1}{2\pi^5}\sin\theta^1\cos^3\theta^1\dd\theta^1\wedge
\sin\theta^2\cos\theta^2\dd\theta^2\wedge\sin\theta^3\cos\theta^3\dd\theta^3
\wedge\dd\phi^1\wedge\dots\wedge\dd\phi^5\]
normalized so that the group volume is $\lambda^{-1}$.

The representation ring of ${\rm SU}(3)$ is well-known (\cite[p. 
265]{bro-tom}) to be generated by two fundamental representations 
$\rho_1,\rho_2\in{\rm Irr}({\rm SU}(3);\C )$ acting on 
$V_{\rho_1}:=\Lambda^1V$ and $V_{\rho_2}:=\Lambda^2V$ where $V\cong\C^3$ 
acted upon by ${\rm SU}(3)$ in the standard way. 
Moreover (cf. e.g. \cite[Claim 13.4]{ful-har}) a generic irreducible 
representation arises as (an irreducible subspace of) the induced 
representation space $S^{m_1}V_{\rho_1}\otimes S^{m_2}V_{\rho_2}$ where 
$S^kW$ denotes the $k^{\rm th}$ symmetric power of $W$. In other 
words all irreducible representations of ${\rm SU}(3)$ are 
algebraic. Consequently any matrix element $\rho^{ij}:{\rm SU}(3)
\rightarrow\C$ can be written as a finite sum of terms like
\begin{equation}
Q^{ij}(\cos\theta^1,\sin\theta^1,\dots,\cos\theta^3,\sin\theta^3)
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}P^{ij}(\phi^1,\dots,\phi^5)}
\label{polar}
\end{equation}
where both $Q^{ij},P^{ij}$ are real-valued polynomials in 
their variables and are independent of $\lambda$. Note that all the 
$P^{ij}$ are of first order.

Therefore at least in the case of ${\rm SU}(3)$ we cast (\ref{egyutthatok}) 
in the form of a finite sum of integrals like
\[\sqrt{\dim_\C V_\rho}\:
\lambda\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}\dots
\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}
\int\limits_0^{\frac{\pi}{2}}\int\limits_0^{\frac{\pi}{2}}
\int\limits_0^{\frac{\pi}{2}}
f(\theta^1,\theta^2,\theta^3,\phi^1,\dots,\phi^5)
Q^{ij}(\cos\theta^1,\dots,\sin\theta^3)
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}P^{ij}(\phi^1,\dots,\phi^5)}\dd V.\]
Integrating with respect to the $\theta^i$ we end up with 
oscillatory integrals
\[\lambda\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}\dots
\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}F^{ij}(\phi^1,\dots,\phi^5)
{\rm e}^{{\bf i}\lambda^{\frac{1}{5}}P^{ij}(\phi^1,\dots,\phi^5)}
\dd\phi^1\dots\dd\phi^5\]
where $F^{ij}$ are smooth functions. 
Taking into account that $P^{ij}$ is of first order hence has no 
critical value on the semi-open $5$-cube 
$\left[ 0,2\pi/\lambda^{\frac{1}{5}}\right)^5$ parameterized by 
$(\phi^1,\dots,\phi^5)$ it then follows for example from the stationary 
phase formula (cf. e.g. \cite[Proposition 6.B.2]{tay}) that 
these oscillatory integrals decay fast if $\lambda\rightarrow +\infty$. 
The result then follows from this. $\Diamond$
\vspace{0.1in} 

\noindent Now we proceed further and construct more general Fourier 
expansions. Let $M$ be an arbitrary connected and compact smooth manifold. 
As usual let $G$ be a compact Lie group and consider a trivial principal 
$G$-bundle $\pi : P\rightarrow M$. Suppose that there is a Riemannian 
metric on $P$ such that all fibers (isomorphic to $G$) are isometric to 
each other and the restricted metric is left-invariant along them. 
Consequently there is a fixed real number $0<\lambda <+\infty$ such that all 
the fibers have the same fixed volume $\lambda^{-1}$. Also let $E$ be an 
arbitrary {\it trivial} real or complex vector bundle  $p:E\rightarrow P$ 
equipped with some privilaged trivialization 
$\tau :E\rightarrow P\times\C^{{\rm rk}E}$ 
(for simplicity we assume that $E$ is a complex vector bundle). We emphasize 
that we do not suppose that $E$ has anything to do with $G$. Write 
$\tau E\cong P\times\C^{{\rm rk}E}$ for the bundle in this fixed 
trivialization. Let the Hilbert space $L^2(P; \tau E)$ be the direct 
product of ${\rm rk}E$ copies of Hilbert spaces $L^2(P;\C )$ where this later 
space is defined by the metric on $P$. Pick a section $\Phi$ of $E$ whose 
trivialization satisfies $\tau\Phi\in L^2(P; \tau E)$. 

Given a global trivialization $\sigma :P\rightarrow M\times G$ write 
$y\in P$ in the form $\sigma (y)=(x,\gamma )\in M\times G$. The 
left-invariant metric on the fibers gives Haar measures $\dd V_x$ of 
volume $\lambda^{-1}$ on $\{x\}\times G$. Also write 
$\tau\Phi =(\Phi^1,\dots,\Phi^{{\rm rk}E})$ for 
the trivialized section. In analogy with (\ref{fourier}) define the 
{\it Fourier expansion of $\Phi$ with respect to $G$ of volume 
$\lambda^{-1}$ and the trivializations $\tau ,\sigma$} by the formal sum
\[\Phi^s (x,\gamma )\sim\sum\limits_{\rho\in {\rm Irr}(G;\C )}
\sqrt{\dim_\C V_\rho}\sum\limits_{i,j=1}^{\dim_\C V_\rho}a_\rho ^{s,ij}(x)
\rho^{ij}(\gamma ),\:\:\:\:\:s=1,\dots ,{\rm rk}E\]
where the {\it $ij^{\it th}$ Fourier coefficient $a_\rho ^{s,ij}:
M\rightarrow\C$ (of $\Phi$) belonging to $\rho$} given by
\[a_\rho^{ s,ij}(x):=\sqrt{\dim_\C V_\rho}\:\lambda\int\limits_G\Phi^s 
(x,\gamma )\rho^{ij}(\gamma^{-1})\dd V_x\] 
is now a globally defined function over $M$ for all $s=1,\dots,{\rm rk}E$. 

Next suppose that $\pi :P\rightarrow M$ is a 
smooth, not necessarily trivial principal $G$-bundle over 
$M$ and there is a Riemannian metric on $P$ with the previous 
properties. Let $\{ (U_\alpha ,\sigma_\alpha )\}$ be local 
trivializations for $P$ i.e., $U_\alpha\subset M$ are open with 
$\cup_{\alpha\in I} U_\alpha =M$ where $I$ is a finite indexing set and 
$\sigma_\alpha :P\vert_{U_\alpha}\rightarrow U_\alpha\times G$ are local 
trivializations. Again take a {\it trivial} and globally trivialized 
real or complex vector bundle $\tau E$ over $P$. Then 
for any trivialized section $\tau\Phi\in L^2(P;\tau E)$ we 
define its {\it local Fourier expansion with respect to $G$ of volume 
$\lambda^{-1}$ and trivializations $\tau ,\sigma_\alpha$} to be the formal sum 
\begin{equation}
\Phi^s(x_\alpha ,\gamma_\alpha )\sim\sum\limits_{\rho\in 
{\rm Irr}(G;\C )}\sqrt{\dim_\C V_\rho}\sum\limits_{i,j=1}^{\dim_\C V_\rho}
a_\rho^{s ,ij}(x_\alpha )
\rho^{ij}(\gamma_\alpha ),\:\:\:\:\:s=1,\dots ,{\rm rk}E
\label{lokalisfourier}
\end{equation}
where the {\it $ij^{\it th}$ local Fourier coefficient $a_\rho^{s ,ij}: 
U_\alpha\rightarrow\C$ (of $\Phi$) belonging 
to $\rho$} is now only a local function given by
\begin{equation}
a_\rho^{s ,ij}(x_\alpha ):=\sqrt{\dim_\C V_\rho}\:\lambda
\int\limits_G\Phi^s(x_\alpha,\gamma_\alpha)\rho^{ij}(\gamma^{-1}_\alpha )
\dd V_{x_\alpha }
\label{lokalisegyutthatok}
\end{equation}
with $\sigma_\alpha (y)=(x_\alpha ,\gamma_\alpha )$ and $\tau\Phi 
=(\Phi^1,\dots,\Phi^{{\rm rk}E})$. 

As a last step we build up something global from these local creatures. 
The natural way to do this appears by adopting the more invariant language of 
\cite{may} (also cf. \cite[Theorem III.5.10]{bro-tom}). An irreducible 
complex representation $\rho$ of $G$ can be identified with a minimal 
closed two-sided ideal $I_\rho$ in the convolution algebra $\cC (G;\C )$ 
introduced on the Hilbert space $L^2(G;\C )$. Let 
$\Pi_\rho :\cC (G;\C )\rightarrow I_\rho$ be the projection; then for a 
function $f\in\cC (G;\C )$ its Fourier expansion (\ref{fourier}) can be 
regarded as a formal sum $f\sim\sum_\rho\Pi_\rho f$. Now if 
$\pi :P\rightarrow M$ is a principal bundle as before then there is no 
preferred identification of the fibers with $G$. With some particular
$\pi^{-1}(x)\cong G$ we can make $L^2(\pi^{-1}(x);\C )$ into a 
convolution algebra $\cC (\pi^{-1}(x);\C )$ by declaring it to be 
$\cC (G;\C )$ as above. If $\pi^{-1}(x)\cong G'$ with $G'\cong Ga$ and 
$a\in G$ is another identification then it easily follows from the 
invariance of integration over $G$ that 
as an algebra $\cC (\pi^{-1}(x);\C )=\cC (G';\C )$. 
Consequently we can talk about {\it the} convolution algebra 
$\cC (\pi^{-1}(x);\C )$ over every fiber $\pi^{-1}(x)$ correctly. 
This suggests that if we have a section $\Phi\in L^2(P;E)$ 
more precisely a trivialized section $\tau\Phi\in L^2(P\:;\:\tau E)$ we 
can define its {\it global Fourier expansion with respect to $G$ of volume 
$\lambda^{-1}$ and trivialization $\tau$} as the formal sum
\[\tau\Phi\sim\sum\limits_{\rho\in {\rm Irr}(G;\C )}(\tau A)_\rho\]
with $(\tau A)_\rho \in L^2(P\:;\:\tau E)$ given by
\[\mbox{$(\tau A)_\rho (y):=({\bf \Pi}_{\rho ,x}\tau\Phi )(y)$
\:\:\:\:\:if\:\:\:\:\:$x\in M$ and $y\in\pi^{-1}(x)\subset P$.}\]
Putting $\cC (\pi^{-1}(x)\:;\:\tau(E\vert_{\pi^{-1}(x)})):=\cC
(\pi^{-1}(x);\C )\times\dots\times\cC (\pi^{-1}(x);\C )$ and
${\bf I}_{\rho ,x}:=I_{\rho ,x}\times\dots\times I_{\rho ,x}$ the map 
${\bf \Pi}_{\rho ,x}:\cC (\pi^{-1}(x)\:;\:\tau 
(E\vert_{\pi^{-1}(x)}))\rightarrow {\bf I}_{\rho ,x}$ is 
the obvious fiberwise projection. That is, ${\bf \Pi}_{\rho ,x}:=
\Pi_{\rho ,x}\times\dots\times\Pi_{\rho ,x}$ is a direct product of ${\rm rk}E$
copies of projections $\Pi_{\rho ,x}:\cC (\pi^{-1}(x);\C )
\rightarrow I_{\rho ,x}$ onto pairwise isomorphic minimal two-sided ideals.

This projection is supposed to be continuous in its $x$ variable in the 
following sense. Let $\tau\cC$ denote the total space of the topological 
convolution-algebra-bundle over $M$ whose fibers for all $x\in M$ are 
the direct product convolution algebras $\cC (\pi^{-1}(x)\:;\:\tau 
(E\vert_{\pi^{-1}(x)}))$. Our considerations yield that this 
is a product bundle: $\tau\cC\cong M\times
\cC (G;\C )\times\dots\times\cC (G;\C )$. (Since the fibers are also Hilbert 
spaces this accords with Kuiper's theorem.) By Fubini's theorem there is a 
canonical linear isomorphism as vector spaces 
$L^2(P;\tau E)\cong L^2(M;\tau\cC )$. Hence regarding an element as 
$\tau\Phi\in  L^2(M;\tau\cC )$ we suppose that the collection 
${\bf \Pi}_\rho :=\{ {\bf \Pi}_{\rho ,x}\:\vert\:x\in M\}$ gives rise to a 
continuous section ${\bf \Pi}_\rho\in C^0(M; {\rm End}(\tau\cC ))$. Note 
that connectedness of $M$, the continuity of the 
projection and compactness of $G$ ensure us that 
the type of the representation cannot jump i.e., labeling the individual 
terms $(\tau A)_\rho\in L^2(M;\tau\cC )$ with $\rho$'s is globally correct. 

If $\Phi$ and $\tau$ are smooth i.e., $\tau\Phi\in C^\infty (P;\tau E)$ then 
it follows from the general theory of Fourier series
\cite[Theorems 7.1 or 8.6]{may} that $(\tau A)_\rho\in C^\infty (P ;\tau E)$ 
and the global Fouries series constructed above converges 
uniformly over $P$ and pointwise equality holds. Writing $(\tau A)_\rho 
=(A^1_\rho ,\dots, A^{{\rm rk}E}_\rho )\in C^\infty (P;\C )\times\dots 
\times C^\infty (P;\C )$ and using some Hermitian metric $g$ on $E$ set in 
this case
\[\Vert (\tau A)_\rho\Vert_{L^\infty (P)}
:=\sup\limits_{y\in P}\vert (\tau A)_\rho (y)\vert_g 
=\sup\limits_{y\in P}\left(\sum\limits_{r,s=1}^{{\rm rk}E}
g^{rs}(y)A^r_\rho (y)\overline{A^s_\rho (y)}\right)^{\frac{1}{2}}\]
(this is the usual sup, not any ess-sup!). Suppose that the volume of our 
compact Lie group goes to zero in $\pi :P\rightarrow M$ i.e., the 
volume $\lambda^{-1}$ of the fibers uniformly approaches zero with respect 
to a $1$-parameter family of degenerating metrics on $P$. 
\begin{lemma} Assume that a trivialized section $\tau\Phi\in C^\infty 
(P; \tau E)$ is given hence for its Fourier modes 
$\Vert (\tau A)_\rho\Vert_{L^\infty (P)} <+\infty$. 

{\rm (i)} Then for any non-trivial representation 
$\rho$ of $G$ one finds $(\tau A)_\rho\rightarrow 0$ in $L^\infty$ 
as $\lambda\rightarrow +\infty$.

{\rm (ii)} Put a norm onto $\Lambda^1 P$ and also take a smooh connection 
$\nabla$ on $E$. Let $\tau\Phi$ be as before. 

Then for all non-trivial representations $\rho$ of $G$ we find 
$(\tau\nabla )(\tau A)_\rho\rightarrow 0$ in $L^\infty$ as 
$\lambda\rightarrow +\infty$.
\label{eltuneslemma2}
\end{lemma}

\noindent{\it Proof.} Again we present the proof for $G={\rm SU}(3)$; 
the general case is more complicated only in notations.

(i) In the trivializations $\tau ,\sigma$ around a given point 
$\sigma (y)=(x,\gamma )$ the pointwise norm of a Fourier mode looks like
\[\vert (\tau A)_\rho (y)\vert^2_g 
=\sum_{r,s}\sum_{i,j}g^{rs} (x,\gamma ) 
a_\rho^{r,ij}(x)\rho^{ij}(\gamma )
\overline{ a_\rho^{s,ij}(x)\rho^{ij}(\gamma )}.\]
Let $y_0$ be a point where $\vert (\tau A)_\rho\vert_g$ attains 
its supremum. Then taking into account (\ref{lokalisfourier}), 
(\ref{lokalisegyutthatok}) 
and (\ref{polar}) any term on the right hand side can be estimated 
from above with the finite sum of terms like
\[\vert g^{rs}(x_0,\gamma_0)\vert \vert 
a_\rho^{r,ij}(x_0)\vert\vert a_\rho^{s,ij}(x_0)\vert 
\left( Q^{ij}(\cos\theta^1_0,\dots,\sin\theta^3_0)\right)^2\]
with $\sigma (y_0)=(x_0,\gamma_0)$. Lemma \ref{eltuneslemma1} for 
$a_\rho^{r ,ij}(x_0)$ and $a_\rho^{s ,ij}(x_0)$ with this fixed $x_0\in M$ 
applies: we know that for any non-trivial representation 
$\rho\in{\rm Irr}({\rm SU}(3);\C )$ they decay fast while 
$\vert g^{rs} (x_0,\gamma_0)\vert
\left( Q^{ij}(\cos\theta^1_0,\dots,\sin\theta^3_0)\right)^2$ 
is constant as $\lambda\rightarrow +\infty$ hence the 
claim follows. 

(ii) Write the derivative of one component as $(\tau\nabla) A^s_\rho 
=\dd A^s_\rho +\sum_r\omega^s_rA^r_\rho$. The algebraic term 
$\sum_r\omega^s_rA^r_\rho$ brings nothing new. Take a point 
$\sigma (y)=(x,\gamma (\psi ))$ then the 
non-algebraic term $\dd A^s_\rho$ is 
\[\sum\limits_m\frac{\partial a_\rho^{s,ij}(x)}{\partial x^m}\rho^{ij}
(\gamma (\psi ) )\dd x^m+\sum\limits_n a_\rho^{s,ij}(x)
\frac{\partial\rho^{ij}(\gamma (\psi ))}{\partial\psi^n}\dd\psi^n\]
where
\[\frac{\partial a_\rho^{s,ij}(x)}{\partial x^m} =\sqrt{\dim_\C V_\rho}\:
\lambda\int\limits_{{\rm SU}(3)}\frac{\partial\Phi^s 
(x,\gamma (\psi ))}{\partial x^m}\rho^{ij}(\gamma^{-1}(\psi))\dd V_x(\psi )\]
and with (\ref{polar}) we see that all $\partial\rho^{ij}(\gamma 
(\psi ))/\partial\psi^n$ is a finite sum of terms like 
\[\left(\frac{\partial Q^{ij}(\psi )}{\partial\psi^n}+{\bf 
i}\lambda^{\frac{1}{5}}Q^{ij}(\psi )\: 
\frac{\partial P^{ij} (\psi )}{\partial\psi^n}\right){\rm 
e}^{{\bf i}\lambda^{\frac{1}{5}} P^{ij}(\psi )} =
\left(\frac{\partial Q^{ij}(\psi )}{\partial\psi^n}+ 
{\bf i}\lambda^{\frac{1}{5}}c^{ij}_nQ^{ij}(\psi )\right){\rm e}^{{\bf i}
\lambda^{\frac{1}{5}} P^{ij}(\psi )}.\]
Since by assumption $(\tau A)_\rho$ hence all its components $A^s_\rho$ 
are smooth, all the derivatives above are smooth consequently for a fixed 
$x_0\in M$ Lemma \ref{eltuneslemma1} with the stationary phase formula works 
again. Taking into account the fast decay in $\lambda$ we proceed as 
in part (i). $\Diamond$
\vspace{0.1in}

\noindent When $\lambda\rightarrow +\infty$ i.e., the fibers of 
$P$ uniformly collapse, and both $\Phi$ and $\tau$ are smooth then 
the pointwise equality in the global Fourier series and Lemma 
\ref{eltuneslemma2} implies that in the $L^\infty_1$ norm on $E$ given by 
$g$ and $\nabla$ we have a strong convergence 
\[\lim\limits_{\lambda\rightarrow +\infty}\left\Vert 
(\tau\Phi )-(\tau A)_{\rho_0}\right\Vert_{L^\infty_1(P)}\leqq
\lim\limits_{\lambda\rightarrow +\infty}\sum\limits_\stack{\rho\in{\rm 
Irr}(G;\C )}{\rho\not=\rho_0}\left\Vert (\tau 
A)_\rho\right\Vert_{L^\infty_1(P)} =0\]
which means that $\tau\Phi$ gets dominated by its ground mode $(\tau 
A)_{\rho_0}\in C^\infty (P;\tau E)$.

The geometric way to grasp this phenomenon is as follows. The limiting 
space can be naturally identified with $P/G\cong M$. We wish to cut 
the bundle $E$ on $P$ down to $M$ in this limit. This can be done 
with the help of an extra piece of data namely the global trivialization 
$\tau :E\rightarrow P\times\C^{{\rm rk} E}$. For a fixed finite $\lambda$ we 
define an equivalence relation $\sim$ on $E$ as follows. If $p:E\rightarrow P$ 
and ${\rm pr}:P\times\C^{{\rm rk} E}\rightarrow\C^{{\rm rk} E}$ are 
the projections then for $e,e'\in E$ we say that 
\[\mbox{$e\sim e'$\:\:\:\:\: if and 
only if\:\:\:\:\: }\left\{\begin{array}{ll}
                      \mbox{$\pi (p(e))=\pi (p(e'))$ and}\\
                      {\rm pr}(\tau (e))={\rm pr}(\tau (e'))
                          \end{array}\right.\]
that is, we collapse $E$ along the fibers of $P$. We obtain a 
vector bundle on $M\cong P/G$ that we denote by $\widetilde{\pi\circ p}: 
E/\tau\rightarrow M$. This is just the familiar collapsing procedure in 
$K$-theory. The collapsed bundle has the same rank as $E$ and 
its isomorphism class in principle depends on the homotopy class of 
$\tau$. However observe that $\tau$ induces a global trivialization 
$\tilde{\tau}:E/\tau\rightarrow M\times\C^{{\rm rk} E}$ given by
\[\mbox{$\tilde{\tau}([e]):=(\pi (p(e))\:,\:{\rm pr}(\tau (e)))$
\:\:\:\:\:with any $e\in [e]$}\]
hence $E/\tau$ will be also globally trivial over $M$ 
whatever $\tau$ is. That is, the isomorphism class of $E/\tau$ is 
actually independent of $\tau$ consequently from now on we shall denote 
this collapsed bundle by $\tilde{E}$. Moreover taking into 
account the triviality of both bundles, the pullback induced by $\pi$ is 
the inverse of the collapsing, that is $\pi^*\tilde{E}\cong E$. 

Finally we can get rid of referring to any trivialization as follows.
Notice from (\ref{lokalisegyutthatok}) that under a gauge
transformation of $\Phi$ by a generic automorphism of the bundle $E$ i.e.,
which locally depends also on $\gamma\in G$, the coefficients
$a^{s,ij}_\rho$ hence the modes $(\tau A)_\rho$ do {\it not} transform well. 
However if the gauge transformation is the pullback from $M$ that is, 
locally independent of $\gamma\in G$ then the coefficients in 
(\ref{lokalisegyutthatok}) transform as the original $\Phi$. Consequently 
consider the restricted general linear automorphism bundle
\[{\rm Aut}_rE:=\pi^*({\rm Aut}\tilde{E})\]
containing ``collapsable'' automorphisms of $E$. Declaring from now on 
that only ``collapsable'' automorphisms are used to change an initially 
given trivialization $\tau$ the objects $(\tau A)_\rho$ behave like a tensor 
consequently the trivialization can be dropped from our notations. 

Thus to summarize: given a section $\Phi\in L^2(P;E)$ there exists a formal sum
\begin{equation}
\Phi\sim\sum\limits_{\rho\in{\rm Irr}(G;\C )}A_\rho,
\:\:\:\:\:A_\rho\in L^2(P;E)
\label{globalis.fi.fourier} 
\end{equation}
called the {\it global Fourier expansion of $\Phi$ over $P$ with respect 
to $G$ of volume $\lambda^{-1}$}. The terms $A_\rho$ are the {\it global 
Fourier modes} of the expansion and locally are given by 
(\ref{lokalisfourier}) and (\ref{lokalisegyutthatok}). This expansion is 
well-defined in the sense that its terms transform correctly under 
gauge transformations taking values in ${\rm Aut}_rE$. Moreover if $\Phi\in 
C^\infty (P;E)$ then in (\ref{globalis.fi.fourier}) pointwise equality holds. 
If $A_{\rho_0}\in C^\infty (P;E)$ denotes the ground Fourier 
mode of the expansion then, being independent of $G$, there exists a unique 
$\tilde{A}_{\rho_0}\in C^\infty (M;\tilde{E})$ such that 
$A_{\rho_0}=\pi^*\tilde{A}_{\rho_0}$. Let us denote these objects in the 
limiting case $\lambda\rightarrow +\infty$ as $\Phi_{+\infty}$ and 
$\tilde{\Phi}_{+\infty}$ respectively (by boundedness these limiting 
objects exist). Clearly $\Phi_{+\infty}=\pi^*\tilde{\Phi}_{+\infty}$. 
This ground mode has an even stronger equivariance property. First 
notice that $\tilde{\Phi}_{+\infty}$ up to an overall constant is 
independent of the metric on $P$ used for Fourier expansion. Secondly, 
given {\it arbitrary} two trivializations $\tau '$ and $\tau ''$ of $E$
such that $\tau ''=\gamma\tau '$ with a {\it generic} $\gamma\in
C^\infty (P;{\rm Aut}E)$ there exists a unique $\tilde{\gamma}\in 
C^\infty (M;{\rm Aut}\tilde{E})$ such that 
$\tilde{\tau}''=\tilde{\gamma}\tilde{\tau}'$
consequently $\tilde{\Phi}_{+\infty}$ transforms well even 
under a generic gauge transformation of $E$.

Hence the original 
field decays to this ground Fourier mode that we write symbolically as
\begin{equation}
\Phi\in C^\infty (P; E) ~~ 
\stackrel{\lim\limits_{\lambda\rightarrow 
+\infty}\Phi}\Longrightarrow ~~
\tilde{\Phi}_{+\infty}\in C^\infty (M ; \tilde{E})
\label{limesz}
\end{equation}
and call both the procedure itself and the resulting 
$\tilde{\Phi}_{+\infty}$ as the {\it reduction of $\Phi$}. This procedure 
is well-defined in the sense that $\tilde{\Phi}_{+\infty}$ depends 
only on $\Phi$ and the trivial representation $\rho_0$ of $G$.

The reduction has the following key property. Assume 
that a section $s :M\rightarrow P$ is given with its image 
${\rm Im}\:s\subset P$. We shall denote the pullback 
$s^*(\Phi\vert_{{\rm Im}s})$, a section of $\tilde{E}$ over $M$, 
simply as $s^*\Phi$. Since in general $\pi :P\rightarrow M$ is not a 
trivial principal bundle $s$ cannot even be continuous over the 
whole $M$ consequently $s^*\Phi$ is in general singular for a finite 
$\lambda$. However 

\begin{lemma} Let $s :M\rightarrow P$ together with its singularity set 
$S\subset M$ be given. Suppose that 
$s\in C^\infty (M-\overline{S}\:;\: P\vert_{M-\overline{S}})$ and 
that its derivative $s_*$ is bounded on $M-\overline{S}$. 
Moreover assume that some fixed $L^\infty_1$ norm is given on $\tilde{E}$.

Then for any smooth section $\Phi\in C^\infty (P;E)$ the reduction 
(\ref{limesz}) satisfies
\begin{itemize}

\item[{\rm (i)}]
$\lim\limits_{\lambda\rightarrow +\infty}\Vert s^*\Phi 
-\tilde{\Phi}_{+\infty}\Vert_{L^\infty_1(M-\overline{S})} =0,$

\item[{\rm (ii)}]
$\lim\limits_{\lambda\rightarrow 
+\infty}\Vert s^*\Phi -\tilde{\Phi}_{+\infty}\Vert_{L^\infty (M)} =0.$
\end{itemize}
\label{szingularitas.lemma}
That is, $\tilde{\Phi}_{+\infty}$ extends $(s^*\Phi )\vert_{M-\overline{S}}$ 
at least continuously over the whole $M$ when $\lambda\rightarrow +\infty$.
\end{lemma}
\noindent{\it Proof.} (i) Put $C:=\sup\limits_{x\in
M-\overline{S}}\vert s_* (x)\vert <+\infty$. Since 
$\pi\circ s={\rm Id}_M$ implies $s^*\circ\pi^*={\rm Id}_{\tilde{E}}$ 
we obtain that the left hand side of part (i) can be estimated from above by
\[\lim\limits_{\lambda\rightarrow +\infty}\left\Vert s^*(\Phi
-\Phi_{+\infty})\right\Vert_{L^\infty (M-\overline{S})}+C
\lim\limits_{\lambda\rightarrow +\infty}\left\Vert s^*((\pi^*\nabla )
(\Phi-\Phi_{+\infty}))\right\Vert_{L^\infty(M-\overline{S})}.\]
However these terms vanish taking into account Lemma \ref{eltuneslemma2}.  
The proof of (ii) is similar. $\Diamond$
\vspace{0.1in}

\noindent {\it Example.} We work out a case which is simple but 
certainly irrelevant otherwise. Consider the Hopf fibration 
$\pi :S^3\rightarrow \C P^1$ i.e., assume that the 
group is $G:={\rm U}(1)$ of volume $2\pi\lambda^{-1}$, the base space is 
$M:=\C P^1\cong S^2$ and the total space is $P:=S^3\subset\C^2$ given 
by $\vert z\vert^2+\vert v\vert^2=\lambda^{-2}$ with its ${\rm 
U}(1)$-invariant standard metric. Moreover let 
$\tau E:=S^3\times\C$ be the trivialized auxiliary bundle; its sections 
$\tau\Phi$ are just complex-valued smooth functions on $S^3$ 
consequently they can be identified with smooth functions 
$F:\C^2\rightarrow\C$ restricted to $S^3$. Fixing a constant 
$a\in\C\cup\{\infty\}\cong\C P^1$ we parameterize 
a complex line in $\C^2$ as $\ell_a=\{ (z,az)\:\vert\:z\in\C\}$ and 
$\ell_\infty =\{ (0,v)\:\vert\:v\in\C\}$. Its intersection with a fiber is 
\[\ell_a\cap\pi^{-1}(a)=
\left\{\left(\frac{{\rm e}^{\ii\lambda t}}{\lambda\sqrt{1+\vert a\vert^2}}\:,\:
\frac{a\:{\rm e}^{\ii\lambda t}}{\lambda\sqrt{1+\vert 
a\vert^2}}\right)\:\left\vert\:t\in \left[ 0,\frac{2\pi}{\lambda}\right )
\right.\right\}\subset S^3,\]
\[\ell_\infty\cap\pi^{-1}(\infty )=
\left\{ \left( 0\:,\:
\frac{{\rm e}^{\ii\lambda t}}{\lambda}\right)\:\left\vert\: t\in 
\left[ 0,\frac{2\pi}{\lambda}\right )\right.\right\}\subset S^3.\]
Take for instance $F(z,v):={\rm e}^z+{\rm e}^v$ providing 
$\tau\Phi :=F\vert_{S^3}$. It easily follows that its Fourier expansion along 
a fiber $\pi^{-1}(a)$ looks like
\[(\tau\Phi )(a,t)=\sum\limits_{k=0}^{+\infty}\frac{1}{k!}
\frac{1+a^k}{\lambda^k(1+\vert a\vert^2)^{\frac{k}{2}}}\:
{\rm e}^{\ii\lambda kt},\:\:\:\:\:(\tau\Phi )(\infty ,t)=1+
\sum\limits_{k=0}^{+\infty}\frac{1}{k!\lambda^k}\:{\rm e}^{\ii\lambda kt}.\] 
Note that as $a\rightarrow\infty$ the ambiguity in the Fourier 
coefficients can be absorbed into $t$ hence the expansion is well-defined over 
the whole $\C P^1$. We obtain that $\tilde{\tau}\tilde{\Phi}_{+\infty}:\C 
P^1\rightarrow\C$ is simply the constant function 
$\tilde{\tau}\tilde{\Phi}_{+\infty}(a)=2$. Moreover for any section 
$s:\C P^1\rightarrow S^3$ whose shape is
\[s(a):=\left(\frac{{\rm e}^{\ii\lambda t_0}}
{\lambda\sqrt{1+\vert a\vert^2}}\:,\:\frac{a\:{\rm e}^{\ii\lambda t_0}}
{\lambda\sqrt{1+\vert a\vert^2}}\right)\]
for $a\not=\infty$, we find
\[\Vert s^*(\tau\Phi )-\tilde{\tau}\tilde{\Phi}_{+\infty}
\Vert_{L^\infty _1(\C P^1-\{\infty\} )}\leqq
C\sum\limits_{k=1}^{+\infty}\frac{1}{\lambda^k}
=C\left(\frac{1}{1-\lambda^{-1}}-1\right)\]
with 
\[C=\sup\limits_\stack{k\in\N}{a\in\C P^1-\{\infty\}}\frac{1}{k!}
\left(\left\vert\frac{1+a^k}{(1+\vert a\vert^2)^{\frac{k}{2}}}\right\vert 
+\left\vert\frac{\dd}{\dd a}\left(\frac{1+a^k}{(1+\vert 
a\vert^2)^{\frac{k}{2}}}\right)\right\vert\right) <+\infty .\]
This demonstrates the rapid decay to the reduction as 
$\lambda\rightarrow +\infty$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The six dimensional sphere}
\label{four}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

After these preliminaries providing the general framework, let us 
focus our attention to the case of the six-sphere. First we quickly 
demonstrate that there is no integrable strong spontaneous symmetry 
breaking ${\rm SO}(6)\rightarrow{\rm U}(3)$ over the $S^6$. 

To begin with, let us recall the topological origin why almost 
complex structures exist on the six-sphere. Consider the standard 
representation of SO$(6)$ 
on $\R^6$ and let $E$ be an SO$(6)$ vector bundle on $S^6$ associated to 
this representation. Up to isomorphism, these bundles are classified 
by the homotopy equivalence classes of maps from the equator of $S^6$ 
into SO$(6)$ that is, by the group $\pi_5({\rm SO}(6))\cong\Z$. In 
particular picking an isomorphism $E\cong TS^6$ is equivalent to the 
existence of an orientation and a Riemannian metric on $S^6$. 
However a continuous map $f: S^5\rightarrow {\rm SO}(6)$ is homotopic to a 
continuous map $f': S^5\rightarrow {\rm U}(3)\subset{\rm SO}(6)$ because any 
particular embedding $i:{\rm U}(3)\subset {\rm SO}(6)$ induces a 
homomorphism $i_*:\pi_5({\rm U}(3))\rightarrow\pi_5({\rm SO}(6))$
which is an isomorphism: since ${\rm SO}(6)/{\rm U}(3)\cong\C P^3$ and 
$\pi_j(\C P^3)\cong 0$ if $j=5,6$ the desired isomorphism is provided by the 
associated homotopy exact sequence of the ${\rm U}(3)$-fibration 
${\rm SO}(6)\rightarrow\C P^3$. Therefore in fact any ${\rm SO}(6)$ vector 
bundle isomorphism $E\cong TS^6$ descends in a non-unique way to an U$(3)$ 
vector bundle isomorphism $E'\cong TS^6$ which is equivalent to saying that 
in addition to an orientation and a Riemannian metric $g$ on $S^6$ there 
exists a further (non-unique) compatible almost complex structure 
$J:TS^6\rightarrow TS^6$. That is $J^2=-{\rm Id}_{TS^6}$ in a manner such 
that it is orthogonal with respect to $g$ which means that 
$g(Ju, Jv)=g(u,v)$ for all vector fields $u,v$ on $S^6$. One can assume 
that these structures are smooth.

A fixed $g$ and $J$ uniquely determines a smooth non-degenerate 
$2$-form by $\omega (u,v):=g(Ju, v)$ which is not closed in general. 
In fact it cannot be closed at all: 
$\int_{S^6}\omega\wedge\omega\wedge\omega\sim{\rm Vol}(S^6)>0$ on 
the one hand but since $H^2(S^6;\Z )=0$ if $\dd\omega =0$ then $\omega 
=\dd\phi$ for a $1$-form $\phi$ hence 
one would find $\int_{S^6}\omega\wedge\omega\wedge\omega 
=\int_{S^6}\dd(\phi\wedge\dd\phi\wedge\dd\phi )=0$ via Stokes' theorem 
on the other hand, a contradiction. Consequently 
with any choice of $g$ and $J$ we cannot expect the metric to be K\"ahler or 
even almost K\"ahler. The quadruple $(S^6, g,J,\omega )$ is 
called an {\it almost Hermitian structure} on $S^6$. 

The non-existence of a K\"ahler structure on $S^6$ can be 
reformulated as saying that for any $(S^6, g,J,\omega )$ the almost 
complex tensor is not parallel with respect to the connection $\nabla$ on the 
endomorphism bundle, associated to the Levi--Civita connection i.e., 
$\nabla J\not=0$. Or using our physical language: {\it no integrable 
strong spontaneous symmetry breaking ${\rm SO}(6)\rightarrow{\rm U}(3)$ 
exists over the six-sphere.} However this does not exclude the 
possibility that a weak one may exist. 

Our aim is to construct an example here. Consider the 14 dimensional 
compact, simply connected exceptional Lie group ${\rm G}_2$. As it is 
well-known (for a complete proof cf. e.g. \cite[pp. 306-311]{pos}) 
this group arises as the total space of a non-trivial 
${\rm SU}(3)$-fibration $\pi :{\rm G}_2\rightarrow S^6$ over the six-sphere. 
Put an orientation and a Riemannian metric $g$ onto ${\rm G}_2$. Furthermore 
pick a Higgs field $\Phi\in C^\infty ({\rm G}_2\:;\:{\rm ad}{\rm G}_2)$. Then 
consider the theory (\ref{YMH}) on ${\rm G}_2$ as follows:
\begin{equation}
\ce (\Phi ,g):=
\frac{1}{2}\int\limits_{{\rm G}_2}\left(\vert N_\nabla\Phi\vert^2_g+e^2
\vert\Phi\Phi^*-{\rm Id}_{T{\rm G}_2}\vert^2_g\right)\dd V .
\label{G2YMH}
\end{equation}
Recall that this theory has a smooth solution $(\Phi ,g_\lambda )$ to its 
vacuum equations (\ref{vakuum}) as follows. 
Let the Higgs field $\Phi$ be an integrable almost complex structure of 
${\rm G}_2$ constructed in Sect. \ref{two}. Fix the orientation on 
${\rm G}_2$ to be compatible with $\Phi$. Next recalling \cite{gib-lu-pop} we 
put a smooth left-invariant ``squashed'' Riemannian metric $g$ onto 
${\rm G}_2$ as follows \cite[pp. 12-13 and 21]{gib-lu-pop}:
\begin{equation}
\begin{array}{ll}
      \dd s^2 := & \\
             & \\
(\sigma_{2\hat 3} -\sigma_{3\hat 2})^2 +
                 \frac{1}{3}(\sigma_{23} -2 \sigma_1)^2+
        (\sigma_{3\hat 1} -\sigma_{1\hat 3})^2 +
          \frac{1}{3}(\sigma_{31} -2 \sigma_2)^2
         +(\sigma_{1\hat 2} -\sigma_{2\hat 1})^2 +
                 \frac{1}{3}(\sigma_{12} -2 \sigma_3)^2 &\\
                        & \\
+\lambda^{-\frac{1}{4}}\left( (\sigma_{2\hat3}+\sigma_{3\hat2})^2+
                           3\sigma_{23}^2\right)
+\lambda^{-\frac{1}{4}}\left( (\sigma_{3\hat1}+\sigma_{1\hat3})^2+
                             3\sigma_{31}^2\right) 
+\lambda^{-\frac{1}{4}}\left( (\sigma_{1\hat2}+\sigma_{2\hat1})^2+
                           3\sigma_{12}^2\right) & \\   
                        & \\
+ \lambda^{-\frac{1}{4}}(\sigma_{1\hat1} -\sigma_{2\hat2})^2 +
3\lambda^{-\frac{1}{4}}(\sigma_{1\hat1} + \sigma_{2\hat2})^2 &
\end{array}
\label{metrika}
\end{equation}
with a positive real parameter $0<\lambda <+\infty$ and $\sigma_i, 
\sigma_{ij},\sigma_{i\hat{j}}$ are certain natural 
left-invariant $1$-forms on ${\rm G}_2$ (cf. \cite{gib-lu-pop}). All 
fibers have the same induced volume 
$0<{\rm Vol}({\rm SU}(3))\sim\lambda^{-1}<+\infty$ with respect to $g$.
With this metric define a new one by 
\[g_\lambda (u,v):=\frac{1}{2}g(u,v)+\frac{1}{2}g(\Phi u,\Phi v)\] 
for all vector fields hence $\Phi$ is $g_\lambda$-orthogonal. 
Moreover the ${\rm SU}(3)$-fibers continue to have a constant volume 
proportional to $\lambda^{-1}$ since $\Phi$ is also left-invariant 
consequently so is $g_\lambda$. 

Inserting these into (\ref{G2YMH}) we summarize the situation ahead as 
\begin{equation}
\mbox{$\ce (\Phi ,g_\lambda )=0$\:\:\:\:\:for all 
$0<\lambda <+\infty$.}
\label{G2vakuum}
\end{equation}
We shall analyze the limit $\lambda\rightarrow +\infty$ in our theory i.e., 
when ${\rm G}_2$ cuts down to ${\rm G}_2/{\rm SU}(3)\cong S^6$. 

First of all let us take a look at the topology underlying our setup. 
Fix a finite $\lambda$. Taking account that $T{\rm G}_2$ as a 
plain real oriented vector bundle is trivial there exists a global 
trivialization $\tau : T{\rm G}_2\rightarrow{\rm G}_2\times\R^{14}$. 
In addition to its triviality we also have a splitting 
\begin{equation}
T{\rm G}_2=W\oplus H
\label{hasitas}
\end{equation}
where $W$ is the vertical part with respect to the fibration 
$\pi :{\rm G}_2\rightarrow S^6$ i.e., $W\vert_{\pi^{-1}(x)}:=
T\pi^{-1}(x)\cong T{\rm SU}(3)$ for all $x\in S^6$ and the 
horizontal part $H\cong T{\rm G}_2/W$ is isomorphic to the normal bundle of 
the fibers. The actual choice of $H$ in (\ref{hasitas}) will be fixed shortly.

Let us collapse (\ref{hasitas}) term by term. First its left hand side 
collapses as in Sect. \ref{three} and the resulting bundle 
$\widetilde{T{\rm G}}_2$ is a trivial, oriented real rank $14$ bundle over 
$S^6$. Next we collapse $W$. Since of course as a real vector bundle 
$T{\rm SU}(3)\cong{\rm SU}(3)\times\R^8$, for all $x\in S^6$ we can find a 
``fiberwise trivialization'' $\sigma_x: W\vert_{\pi^{-1}(x)}
\rightarrow\pi^{-1}(x)\times\R^8$ which is local in its $x$ variable that 
is, $W\subset T{\rm G}_2$ is not necessarily a trivial bundle over ${\rm G}_2$. 
The collection of these fiberwise trivializations of $W$ will be denoted by 
$\sigma$. We can carry out a fiberwise collapsing of the bundle 
$W$ as in Sect. \ref{three}: if $w,w'\in W$ then we say that
\[\mbox{$w\sim w'$\:\:\:\:\: if and
only if\:\:\:\:\: }\left\{\begin{array}{ll}
                      \mbox{$\pi (p(w))=\pi (p(w'))=x\in S^6$ and}\\
                      {\rm pr}_x(\sigma_x(w))={\rm pr}_x(\sigma_x(w'))
                          \end{array}\right.\]
where again $p:W\rightarrow{\rm G}_2$ and ${\rm pr}_x:
\pi^{-1}(x)\times\R^8\rightarrow\R^8$ are 
the projections. The resulting bundle 
$\widetilde{\pi\circ p}:W/\sigma\rightarrow S^6$ is a 
smooth real vector bundle over $S^6$ whose rank is equal to 
$\dim_\R {\rm SU}(3)=8$. Its isomorphism class, which is not necessarily 
the trivial one any more, depends on the homotopy class of $\sigma$. However by 
an accidental coincidence (at least from our present viewpoint)
\begin{equation}
\pi_5 ({\rm GL}(n;\R ))\cong\pi_5({\rm O}(n))\cong 0,\:\:\:\:\:n\geqq 7
\label{veletlen}
\end{equation}
via real Bott periodicity (cf. e.g. \cite[p. 142]{mil}) implying that all 
real vector bundles of this kind over $S^6$ are trivial hence in 
particular orientable. We eventually find that whatever the
original $W$ on ${\rm G}_2$ and the fiberwise trivialization $\sigma$ was, 
{\it the collapsed bundle $W/\sigma$ is trivial on $S^6$}. Consequently 
from now on we shall denote it as $\widetilde{W}$.

Finally consider $H$. If $h,h'\in H$ then we say that
\[\mbox{$h\sim h'$\:\:\:\:\: if and
only if\:\:\:\:\: }\left\{\begin{array}{ll}
                      \mbox{$\pi (p(h))=\pi (p(h'))$ and}\\
                      \pi_*\vert_H(h)=\pi_*\vert_H(h')
                          \end{array}\right.\]
where $p :H\rightarrow {\rm G}_2$ is the projection and 
$\pi_*\vert_H:H\rightarrow TS^6$ is the restriction of the derivative of 
$\pi$ which is a linear isomorphism between $H_y$ and $T_xS^6$ if 
$\pi (y)=x$. We denote the quotient bundle by 
$\widetilde{\pi\circ p}:H/\pi_*\rightarrow S^6$ which can 
be understood like a ``twisted collapsing'' of $H$. Moreover there is a 
natural isomorphism $H/\pi_*\cong TS^6$. 

This term-by-term collapsing does not distroy the splitting (\ref{hasitas}): 
one can prove that any ${\rm GL}^+(6;\R )$ vector bundle $E$ over $S^6$, 
including $TS^6$, is stably trivial in the sense that 
$\Theta^{14}\cong\Theta^8\oplus E$ where $\Theta^k$ denotes the trivial rank 
$k$ real vector bundle over $S^6$. This yields that in fact we end up with a 
${\rm GL}^+(14;\R )\supset{\rm GL}^+(8;\R )\times{\rm GL}^+(6;\R )$-invariant 
induced splitting 
\begin{equation}
\widetilde{T{\rm G}}_2=\widetilde{W}\oplus H/\pi_*
\label{vegtelen.hasitas}
\end{equation}
such that $\widetilde{T{\rm G}}_2\cong\Theta^{14}$, 
$\widetilde{W}\cong\Theta^8$ and $H/\pi_*\cong TS^6$. 

Now we fix the choice of the horizontal components in (\ref{hasitas}) and 
(\ref{vegtelen.hasitas}). Observe that the things above are not only plain 
vector bundles but carry several additional structures. For instance some 
fibers are Lie algebras: $T_y{\rm G}_2\cong{\mathfrak g}_2$ and 
$W_y\cong{\mathfrak s}{\mathfrak u}(3)$ for all 
$y\in{\rm G}_2$ and this identification is independent of the base 
point. Therefore we also obtain that $(\widetilde{T{\rm G}}_2)_x
\cong {\mathfrak g}_2$ as well as $\widetilde{W}_x\cong
{\mathfrak s}{\mathfrak u}(3)$ for all $x\in S^6$ yielding that the 
global splittings (\ref{hasitas}) or (\ref{vegtelen.hasitas}) 
pointwise correspond to the Lie algebra decomposition ${\mathfrak 
g}_2={\mathfrak s}{\mathfrak u}(3)\oplus{\mathfrak m}$. Since the
ranks of ${\rm G}_2$ and ${\rm SU}(3)$ are equal we can suppose that
a given Cartan subalgebra ${\mathfrak t}\subset{\mathfrak g}_2$
is a Cartan subalgebra of ${\rm SU}(3)$ as well i.e., satisfies
${\mathfrak t}\subset{\mathfrak s}{\mathfrak u}(3)\subset{\mathfrak g}_2$.
A positive root system $P^+$ of ${\rm SU}(3)$ can be a subsystem of a
positive root system $R^+$ of ${\rm G}_2$ hence with this special Cartan
subalgebra ${\mathfrak t}$ the decomposition (\ref{gyokok}) is 
compatible with ${\mathfrak g}_2={\mathfrak s}{\mathfrak u}(3)
\oplus{\mathfrak m}$. That is, we obtain decompositions 
\begin{equation}
{\mathfrak s}{\mathfrak u}(3)={\mathfrak 
t}\oplus\bigoplus\limits_{\alpha\in P^+}{\mathfrak g}_\alpha,\:\:\:\:\:
{\mathfrak m}:=\bigoplus\limits_{\beta\in R^+-P^+}{\mathfrak g}_\beta
\label{lie}
\end{equation}
which fixes the choice of ${\mathfrak m}\subset {\mathfrak g}_2$. 
Consequently taking $H_y:={\mathfrak m}$ for all $y\in{\rm G}_2$ or 
$(H/\pi_*)_x:={\mathfrak m}$ for all $x\in S^6$ we fix the splittings 
(\ref{hasitas}) and (\ref{vegtelen.hasitas}) up to a Cartan subalgebra 
of ${\rm SU}(3)$.

After these topological considerations let us return to our fields in 
(\ref{G2YMH}). Notice that tensor bundles over ${\rm G}_2$ are also globally 
trivial. Hence taking the trivial auxiliary vector bundle $E$ 
to be any of these bundles with a distinguished trivialization induced by 
that of the tangent bundle, our global Fourier theory from Sect. \ref{three} 
works in this situation. 

In particular we can take $E:={\rm End}T{\rm G}_2$ and write 
$\tau ({\rm End}T{\rm G}_2)={\rm End}(\tau T{\rm G}_2)
\cong{\rm G}_2\times{\rm End}\:\R^{14}$ for its smooth trivialization induced 
by that of $T{\rm G}_2$. Note that 
$\Phi\in C^\infty ({\rm G}_2\:;\:{\rm End}\:T{\rm G}_2)$ because 
${\rm ad}{\rm G}_2\subset {\rm End}\:T{\rm G}_2$ moreover 
$C^\infty ({\rm G}_2; {\rm Aut}_rT{\rm G}_2)$ acts on ${\rm End}\:T{\rm G}_2$ 
in the usual way. Consequently the Higgs field admits a 
$C^\infty ({\rm G}_2; {\rm Aut}_rT{\rm G}_2)$-equivariant global 
Fourier expansion (\ref{globalis.fi.fourier}) of the shape
\[\Phi =\sum\limits_{\rho\in{\rm Irr}({\rm SU}(3);\C )}A_\rho\]
and here pointwise equality holds. Moreover in the trivialization 
$\sigma (y)=(x,\gamma )$ about $y\in{\rm G}_2$ the coefficients of the ground 
Fourier mode $(\tilde{\tau}\tilde{A})_{\rho_0}\in C^\infty 
(S^6;\tilde{\tau}({\rm End}(T{\rm G}_2/\tau )))$ in every $x\in S^6$ 
look like \[a^s_{\rho_0}(x)=
\lambda\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}\dots
\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}\int\limits_0^{\frac{\pi}{2}}
\int\limits_0^{\frac{\pi}{2}}\int\limits_0^{\frac{\pi}{2}}
\Phi^s(x,\theta^1,\theta^2,\theta^3,\phi^1,\dots,\phi^5)\dd V_x.\]
Performing the integrals with respect to $\theta^i$ we find for 
all $0<\lambda <+\infty$ that
\[\vert a^s_{\rho_0}(x)\vert\leqq 
\lambda\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}\dots
\int\limits_0^{\frac{2\pi}{\lambda^{\frac{1}{5}}}}
\vert\Psi^s(x,\phi^1,\dots,\phi^5)\vert\dd\phi^1\dots\dd\phi^5
\leqq (2\pi )^5\sup\limits_\stack{\phi^i\in 
\left[ 0,2\pi/\lambda^{\frac{1}{5}}\right)}{i=1,\dots,5}\vert\Psi^s 
(x,\phi^1,\dots,\phi^5)\vert .\]
Consequently $(\tilde{\tau}\tilde{A})_{\rho_0}$ exists in the limit  
$\lambda\rightarrow +\infty$ hence the (\ref{limesz}) applies. We end up 
with a reduced field as in Sect. \ref{three} and denote it in the usual way as 
\begin{equation}
\tilde{\Phi}_{+\infty}\in C^\infty\left( S^6\:;\:
{\rm End}\widetilde{T{\rm G}}_2\right)
\label{fi}
\end{equation}
where ${\rm End}\widetilde{T{\rm G}}_2\cong{\rm End}(T{\rm G}_2/\tau )$ is 
the trivial bundle for all trivializations as we already know moreover the 
field $\tilde{\Phi}_{+\infty}$ up to an automorphism of ${\rm 
End}\widetilde{T{\rm G}}_2$ is independent of the trivialization used to 
construct it.

\begin{lemma} Consider a Higgs field $\Phi\in C^\infty ({\rm G}_2;{\rm 
End}\:T{\rm G}_2)$ representing a complex structure on ${\rm G}_2$ and 
consider its ground Fourier mode (\ref{fi}). 

Then there exist 
\[\tilde{Q}_{+\infty}\in C^\infty (S^6\:;\:{\rm End}\widetilde{W}),
\:\:\:\:\: \tilde{\phi}_{+\infty}\in C^\infty (S^6\:;\:{\rm End}(H/\pi_*))\]
such that
\[\tilde{\Phi}_{+\infty} =\begin{pmatrix}
                  \tilde{Q}_{+\infty} & 0\\
                   0 & \tilde{\phi}_{+\infty}
         \end{pmatrix}\]
i.e., the ground Fourier mode admits a ${\rm GL}^+(8;\R )
\times{\rm GL}^+(6;\R )$-invariant blockdiagonal decomposition 
with respect to the splitting (\ref{vegtelen.hasitas}).
\label{hasitolemma}
\end{lemma}

\noindent{\it Proof.} Recall from Sect. \ref{two} the construction of the 
Higgs field $\Phi$ making ${\rm G}_2$ into a complex manifold $Y$. 
Taking into account (\ref{J-szetszedes}) and (\ref{lie}) and that 
$W_y={\mathfrak s}{\mathfrak u}(3)$ and $H_y={\mathfrak m}$ for all 
$y\in {\rm G}_2$, on the one hand with respect to the splitting 
(\ref{hasitas}) the Higgs field $\Phi$ admits a global blockdiagonal 
decomposition
\begin{equation}
\Phi =\begin{pmatrix}
                  Q & 0\\
                   0 & \phi
         \end{pmatrix}
\label{hasadas1}
\end{equation}
where $Q\in C^\infty ({\rm G}_2\:;\:{\rm End} W)$ and $\phi\in C^\infty 
({\rm G}_2\:;\:{\rm End}H)$.

On the other hand we can expand $\Phi$ into Fourier series making use 
of some trivialization of $T{\rm G}_2$ which is however globally 
incompatible with the splitting (\ref{hasadas1}). Therefore we use local 
gauges induced by sections to transfer (\ref{hasadas1}) to $S^6$ as follows. 
Let $s' :S^6\rightarrow{\rm G}_2$ be a section of
$\pi :{\rm G}_2\rightarrow S^6$ such that it is non-continuous in a single
point $x'_0\in S^6$ only and is smooth elsewhere. Let $\tau '$ be a 
trivialization of ${\rm End}T{\rm G}_2$ such that (i) it is smooth on 
${\rm G}_2-\pi^{-1}(x'_0)$ and (ii) it is compatible with (\ref{hasitas}) along 
the image of $s'\vert_{S^6-\{x'_0\}}$ and (iii) it trivializes
$({\rm End}W)\vert_{\pi^{-1}(x'_0)}$. It follows from (\ref{hasadas1}) and 
property (ii) of $\tau '$ that if we restrict $\Phi$ to 
${\rm Im}(s'\vert_{S^6-\{x'_0\}})\subset {\rm G}_2-\pi^{-1}(x'_0)$ and 
pull it back then
\[s'^*(\tau '\Phi )=\begin{pmatrix}
                         s'^*(\tau ' Q )& 0\\
                           0 & s'^*(\tau '\phi )
                     \end{pmatrix} =\tilde{\tau}'(s'^*\Phi).\]
By virtue of part (ii) of Lemma \ref{szingularitas.lemma} we find that as 
$\lambda\rightarrow +\infty$
\[\tilde{\tau}'(s'^*\Phi )=\tilde{\tau}'\tilde{\Phi}_{+\infty}+
\tilde{\tau}'(s'^*\Phi-\tilde{\Phi}_{+\infty})\longrightarrow 
\tilde{\tau}'\tilde{\Phi}_{+\infty}\]
holds pointwise yielding $\tilde{\tau}'\tilde{\Phi}_{+\infty}$ is 
smooth and blockdiagonal with respect to the splitting 
(\ref{vegtelen.hasitas}) over $U'=S^6-\{x'_0\}$. 

Now take another section $s'':S^6\rightarrow{\rm G}_2$ with its isolated 
singularity $x''_0\not= x'_0$, smooth elsewhere, pick a compatible 
trivialization $\tau ''$ as before and construct 
$\tilde{\tau}''\tilde{\Phi}_{+\infty}$. 
This section is smooth and blockdiagonal on $U''=S^6-\{x''_0\}$. 
We know that there exists a gauge transformation 
$\tilde{\gamma}\in C^\infty (U'\cap U''\:;\:
{\rm Aut}\widetilde{T{\rm G}}_2)$ such that 
$\tilde{\gamma}^{-1}(\tilde{\tau}'\tilde{\Phi}_{+\infty}\vert_{U'\cap 
U''})\tilde{\gamma} = \tilde{\tau}''\tilde{\Phi}_{+\infty}\vert_{U'\cap U''}$. 
However, taking into account that $\widetilde{W}$ hence 
${\rm End}\widetilde{W}$ is a trivial bundle (by virtue of 
(\ref{veletlen}) as we have seen) both of the restricted 
trivializations $\tilde{\tau}'\vert_{{\rm End}\widetilde{W}}$ 
and $\tilde{\tau}''\vert_{{\rm End}\widetilde{W}}$ extend over the 
whole $S^6$ by their property (iii) above. Consequently 
we can suppose that $\tilde{\gamma}$ takes values 
only in ${\rm Aut}\widetilde{W}\times{\rm Aut}(H/\pi_*)$. 
Therefore $\tilde{\gamma}$ obeys the blockdiagonal shapes and gluing 
$\tilde{\tau}'\tilde{\Phi}_{+\infty}\vert_{U'}$ and 
$\tilde{\tau}''\tilde{\Phi}_{+\infty}\vert_{U''}$ together along 
$U'\cap U''$ with $\tilde{\gamma}$ we obtain a smooth globally 
blockdiagonal section 
$\tilde{\Phi}_{+\infty}=\tilde{Q}_{+\infty}\oplus\tilde{\phi}_{+\infty}$. In 
principle this is a section of the endomorphism bundle of a 
twisted real rank $14$ vector bundle over $S^6$. However in fact it must 
be the trivial one i.e., ${\rm End}\widetilde{T{\rm G}}_2$ taking into 
account (\ref{veletlen}) again. $\Diamond$
\vspace{0.1in}

\noindent Next consider the metric. Let $g$ denote the limiting field on 
${\rm G}_2$ given by the metric (\ref{metrika}) at $\lambda =+\infty$. 
This limiting object is a symmetric bilinear form field on $T{\rm G}_2$ and 
is of the form $\pi^*\tilde{g}$ where $\tilde{g}$ is a similar field 
on $\widetilde{T{\rm G}}_2$. Now consider the projected field 
$\tilde{g}_{+\infty}$ with respect to (\ref{fi}) that is, 
$\tilde{g}_{+\infty}(u,v):=\frac{1}{2}\tilde{g}(u,v)+
\frac{1}{2}\tilde{g}(\tilde{\Phi}_{+\infty}u,\tilde{\Phi}_{+\infty}v)$ for 
all vector fields $u,v$ on $\widetilde{T{\rm G}}_2$. This construction is 
supposed to be gauge equivariant in the sense that if a representative 
$\tilde{\tau}\tilde{\Phi}_{+\infty}$ of (\ref{fi}) is replaced with another 
one $\tilde{\gamma}^{-1}(\tilde{\tau}\tilde{\Phi}_{+\infty})\tilde{\gamma}$ 
then $\tilde{\tau}\tilde{g}$ is also replaced with 
$\tilde{\gamma}^*(\tilde{\tau}\tilde{g})$ which yields 
$\tilde{\gamma}^*(\tilde{\tau}\tilde{g}_{+\infty})$. 

After restriction we obtain from $\tilde{g}_{+\infty}$ 
a smooth non-degenerate symmetric bilinear form field 
$\tilde{h}_{+\infty}$ on $TS^6\cong H/\pi_*\subset\widetilde{T{\rm G}}_2$ 
i.e., a Riemannian metric
\begin{equation}
\tilde{h}_{+\infty}\in C^\infty (S^6\:;\:S^2\Lambda^1S^6).
\label{h}
\end{equation}
Indeed, since $\tilde{\Phi}_{+\infty}$ of (\ref{fi}) 
is independent of any metric by perturbing slightly the original metric 
(\ref{metrika}) if necessary, we can suppose that $\tilde{h}_{+\infty}$ is
non-degenerate.\footnote{For instance if $\tilde{\Phi}_{+\infty}=0$ then 
$\tilde{h}_{+\infty}$, essentially the first line of 
(\ref{metrika}), is the round metric on $S^6$; cf. \cite[p. 21]{gib-lu-pop}.} 
We shall denote by $\tilde{D}_{+\infty}$ its Levi--Civita connection. 
Notice however that we do not expect to get a metric along the other 
component $\widetilde{W}$ of (\ref{vegtelen.hasitas}) since $g_\lambda$ 
degenerates in this direction in our limit.

\begin{lemma} Consider the sections $\tilde{\phi}_{+\infty}\in C^\infty 
(S^6;{\rm End}(H/\pi_*))$ and $\phi\in C^\infty ({\rm G}_2;{\rm End}H)$ of 
Lemma \ref{hasitolemma}. Take a section $s :S^6\rightarrow {\rm G}_2$ 
with the only singularity at $x_0\in S^6$ and bounded derivative on 
$S^6-\{x_0\}$ as in Lemma \ref{szingularitas.lemma}. Then 
\begin{itemize}

\item[{\rm (i)}]
$\lim\limits_{\lambda\rightarrow +\infty}\Vert s^*\phi
-\tilde{\phi}_{+\infty}\Vert_{L^\infty_1(S^6-\{x_0\})} =0,$

\item[{\rm (ii)}]
$\lim\limits_{\lambda\rightarrow +\infty}\Vert s^*\phi 
-\tilde{\phi}_{+\infty}\Vert_{L^\infty (S^6)} =0$
\end{itemize}
where the $L^\infty_1$-norm on $S^6$ is 
defined with respect to $\tilde{h}_{+\infty}$ and $\tilde{D}_{+\infty}$ 
constructed above.
\label{eltuneslemma3}
\end{lemma}

\noindent{\it Proof.} This is a consequence of Lemmata 
\ref{szingularitas.lemma} and \ref{hasitolemma}. We can suppose that 
the section $s$ is any of $s'$ or $s''$ used to split (\ref{fi}). 
Extend the metric $\tilde{h}_{+\infty}$ and the connection 
$\tilde{D}_{+\infty}$ on $H/\pi_*$ to some metric and connection on 
$\widetilde{W}\oplus H/\pi_*$ to calculate $L^\infty_1$ norms. Adding the 
other components and writing $\Phi =Q\oplus\phi$ which is (\ref{hasadas1}) as 
well as $\tilde{\Phi}_{+\infty}=
\tilde{Q}_{+\infty}\oplus\tilde{\phi}_{+\infty}$ with the fields of 
Lemma \ref{hasitolemma} for any finite $\lambda$ and $k=0,1$ we estimate like
$\Vert s^*\phi -\tilde{\phi}_{+\infty}\Vert_{L^\infty_k(S^6-\{x_0\})}
\leqq\Vert s^*\Phi -\tilde{\Phi}_{+\infty}\Vert_{L^\infty_k(S^6-\{x_0\})}$ 
hence the result follows. $\Diamond$ \vspace{0.1in}

\noindent The time has come to return to our field theory (\ref{G2YMH}). 
As we already noted our aim is to perform a dimensional reduction, to take 
the limit$\lambda\rightarrow +\infty$ in this theory and then to understand what
happens to its spontaneously broken vacuum solution
(\ref{G2vakuum})---representing a complex structure on ${\rm G}_2$---in 
this limit.

Consider $S^6$ with its induced orientation and define a theory 
akin to (\ref{G2YMH}) hence (\ref{YMH}) acted upon by the gauge group 
$\cg =C^\infty (S^6; {\rm Aut}_{{\rm SO}(6)}TS^6)$ as in Sect. \ref{two}. 
The energy functional is:
\begin{equation}
\ce (\phi ,h):=\frac{1}{2}\int\limits_{S^6}\left(\vert N_D\phi\vert^2_h+e^2
\vert\phi\phi^*-{\rm Id}_{TS^6}\vert^2_h\right)\dd V
\label{S6YMH}
\end{equation}
where $N_D\phi$ is again given locally by (\ref{lokalis-nijenhuis}). We 
claim that

\begin{lemma} There exists a smooth pair 
$(\tilde{\phi}_{+\infty} ,\tilde{h}_{+\infty})$ 
satisfying $\ce (\tilde{\phi}_{+\infty},\tilde{h}_{+\infty})=0$ that is, 
it provides us with a smooth solution to the associated vacuum equations 
(\ref{vakuum}) of the Higgs-like theory (\ref{S6YMH}) over $S^6$.
\label{vakuumlemma}
\end{lemma}   

\noindent{\it Proof.} Fix $0<\lambda <+\infty$. The splitting 
(\ref{hasadas1}) yields with respect to the metric $g_\lambda$ on 
$T{\rm G}_2$ that
\begin{equation}
\Phi\Phi^* =\begin{pmatrix}
                    QQ^* & 0\\
                        0 & \phi\phi^*
                 \end{pmatrix} .
\label{hasadas2}
\end{equation}
Since in the vacuum $(\Phi ,g_\lambda )$ the Levi--Civita 
connection $\nabla_\lambda$ of $g_\lambda$ is torsion-free and 
$\Phi^2=-{\rm Id}_{T{\rm G}_2}$ the tensor
$N_{\nabla_\lambda}\Phi$ reduces to the Nijenhuis tensor $N_\Phi$.
Decomposing the Higgs field via (\ref{hasadas1}) and (\ref{hasadas2}),
the vacuum (\ref{G2vakuum}) is equivalent to
\[\vert N_\phi\vert^2_{g_\lambda}+\vert N_Q\vert^2_{g_\lambda}+\vert
N(\phi ,Q)\vert^2_{g_\lambda}
+e^2\left(\vert \phi\phi^*-{\rm Id}_H\vert^2_{g_\lambda}+
\vert QQ^*-{\rm Id}_W\vert^2_{g_\lambda}\right) =0.\]  
Picking the horizontal components this implies that the pointwise equality
\begin{equation}
\vert N_\phi\vert^2_{g_\lambda}+e^2\vert \phi\phi^* -{\rm 
Id}_H\vert^2_{g_\lambda}=0
\label{nulla}
\end{equation}
holds on ${\rm G}_2$ for all $0<\lambda <+\infty$. Recalling (\ref{hasadas1}) 
we note that $N_\phi =N_{D_\lambda}\phi$ where 
$D_\lambda :=\nabla_\lambda\vert_H$ is the 
Levi--Civita connection of a metric $h_\lambda =g_\lambda\vert_H$ and of 
course $T_{D_\lambda}=0$. Consequently the tensor field $N_\phi$, yet 
depending on the vertical coordinates as well, is locally given by 
(\ref{lokalis-nijenhuis}) with $\Phi$ replaced by $\phi$.

Now consider the smooth split field $\tilde{\phi}_{+\infty}$ of Lemma 
\ref{hasitolemma}, the smooth metric $\tilde{h}_{+\infty}$ in (\ref{h}) 
together with its Levi--Civita connection $\tilde{D}_{+\infty}$, all 
defined on the whole of $S^6$. Take again a section 
$s :S^6\rightarrow {\rm G}_2$. We want to estimate the difference between 
(\ref{nulla}) on ${\rm Im}\:s\subset{\rm G}_2$ and 
\[\vert N_{\tilde{\phi}_{+\infty}}\vert^2_{\tilde{h}_{+\infty}}+
e^2\vert \tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty} 
-{\rm Id}_{TS^6}\vert^2_{\tilde{h}_{+\infty}}\]
on ${\rm G}_2/{\rm SU}(3)\cong S^6$. We can suppose that the only singular 
point of $s :S^6\rightarrow {\rm G}_2$ is $x_0\in S^6$ and in a local 
trivialization $\sigma :{\rm G}_2\vert_{S^6-\{x_0\}}
\rightarrow (S^6-\{x_0\})\times{\rm SU}(3)$ if $\sigma (s (x))=(x,\gamma (x))$ 
then $\gamma (x)=e\in {\rm SU}(3)$. This implies that its derivative 
$s_*\vert_{S^6-\{x_0\}}=0$ is bounded. By virtue of (\ref{nulla}) 
we obtain on $S^6-\{x_0\}$ that
\[0=s^*\vert N_\phi\vert_{g_\lambda}=
\vert s^*(N_\phi)\vert_{s^*g_\lambda}=
\vert s^*(N_\phi )\vert_{\tilde{h}_{+\infty}}=
\vert N_{s^*\phi}\vert_{\tilde{h}_{+\infty}}\]
and (\ref{nulla}) also yields that on $S^6-\{x_0\}$
\[0=s^*\vert\phi\phi^*-{\rm Id}_H\vert_{g_\lambda}=
\vert (s^*\phi )(s^*\phi )^*-{\rm Id}_{H/\pi_*}\vert_{s^*g_\lambda}
=\vert (s^*\phi )B^{-1}_\lambda (s^*\phi )^*B_\lambda 
-{\rm Id}_{H/\pi_*}\vert_{\tilde{h}_{+\infty}}\] 
where the map $B_\lambda\in C^\infty (S^6-\{x_0\};
{\rm Aut}((H/\pi_*)\vert_{S^6-\{x_0\}}))$ measures the discrepancy between 
taking adjoints with respect to $s^*g_\lambda$ or $\tilde{h}_{+\infty}$ 
(in our notation both are denoted by $*$ and the subscripts for the 
metric indicate which is being used).

Therefore we can estimate the aforementioned difference on $S^6-\{x_0\}$ like
\begin{eqnarray}
& &\left\vert s^*\left(\vert
N_\phi\vert^2_{g_\lambda}+e^2\vert \phi\phi^*
-{\rm Id}_H\vert^2_{g_\lambda}\right)-\left(\vert 
N_{\tilde{\phi}_{+\infty}}
\vert^2_{\tilde{h}_{+\infty}}+
e^2\vert\tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty}-
{\rm Id}_{H/\pi_*}\vert^2_{\tilde{h}_{+\infty}}\right)\right\vert\nonumber\\
 %           &    &      \nonumber             \\
& \leqq & \left\vert\vert N_{s^*\phi}\vert^2_{\tilde{h}_{+\infty}}-
\vert N_{\tilde{\phi}_{+\infty}}\vert^2_{\tilde{h}_{+\infty}}\right\vert + e^2
\left\vert\vert (s^*\phi )B^{-1}_\lambda (s^*\phi )^*B_\lambda 
-{\rm Id}_{H/\pi_*}\vert^2_{\tilde{h}_{+\infty}}
-\vert \tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty}
-{\rm Id}_{H/\pi_*}\vert^2_{\tilde{h}_{+\infty}}\right\vert .\nonumber
\end{eqnarray}
We proceed further and make two robust estimates. We begin with the 
first term as follows:
\[\left\vert\vert N_{s^*\phi}\vert^2_{\tilde{h}_{+\infty}}-\vert
N_{\tilde{\phi}_{+\infty}}\vert^2_{\tilde{h}_{+\infty}}\right\vert\leqq 
\vert  N_{s^*\phi} 
-N_{\tilde{\phi}_{+\infty}}\vert_{\tilde{h}_{+\infty}}\left(\vert 
N_{s^*\phi} \vert_{\tilde{h}_{+\infty}} +
\vert N_{\tilde{\phi}_{+\infty}}\vert_{\tilde{h}_{+\infty}}\right) .\]
Exploiting the shape (\ref{lokalis-nijenhuis}) of $N_{s^*\phi}$ and
$N_{\tilde{\phi}_{+\infty}}$ it is not difficult to prove the pointwise 
inequality
\begin{eqnarray}
& &\vert  N_{s^*\phi} 
-N_{\tilde{\phi}_{+\infty}}\vert_{\tilde{h}_{+\infty}}\nonumber\\
%         &          &\nonumber\\
&\leqq & 2\vert s^*\phi -\tilde{\phi}_{+\infty}\vert_{\tilde{h}_{+\infty}}
\vert\tilde{D}_{+\infty} (s^*\phi 
+\tilde{\phi}_{+\infty})\vert_{\tilde{h}_{+\infty}}+
2\vert\tilde{D}_{+\infty}(s^*\phi 
-\tilde{\phi}_{+\infty})\vert_{\tilde{h}_{+\infty}}
\vert s^*\phi +\tilde{\phi}_{+\infty}\vert_{\tilde{h}_{+\infty}}\nonumber
\end{eqnarray}
consequently putting 
\[ C_1(\phi ,\lambda ):=\Vert N_{s^*\phi}\Vert_{L^\infty (S^6-\{x_0\})}
+\Vert N_{\tilde{\phi}_{+\infty}}\Vert_{L^\infty (S^6)}<+\infty\]
and
\[C_2(\phi ,\lambda ):=2\Vert s^*\phi 
+\tilde{\phi}_{+\infty}\Vert_{L^\infty _1(S^6-\{x_0\})}<+\infty\]
we obtain along $S^6-\{x_0\}$ that
\begin{equation}
\left\vert\vert N_{s^*\phi}\vert^2_{\tilde{h}_{+\infty}}-
\vert N_{\tilde{\phi}_{+\infty}}\vert^2_{\tilde{h}_{+\infty}}\right\vert\leqq 
C_1C_2\Vert s^*\phi-\tilde{\phi}_{+\infty}\Vert_{L^\infty_1(S^6-\{x_0\})}.
\label{nijenhuis.becsles}
\end{equation}
The second term can be estimated like
\begin{eqnarray}
& &\left\vert\vert (s^*\phi )B^{-1}_\lambda (s^*\phi )^*B_\lambda-
{\rm Id}_{H/\pi_*}\vert^2_{\tilde{h}_{+\infty}}-\vert 
\tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty}
-{\rm Id}_{H/\pi_*}\vert^2_{\tilde{h}_{+\infty}}\right\vert\nonumber\\
%               & & \nonumber \\
&\leqq & \vert (s^*\phi )B^{-1}_\lambda (s^*\phi )^*B_\lambda 
-\tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty}\vert_{\tilde{h}_{+\infty}}
\left( 12+\vert (s^*\phi )B^{-1}_\lambda (s^*\phi )^*B_\lambda
\vert_{\tilde{h}_{+\infty}}+\vert\tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty}
\vert_{\tilde{h}_{+\infty}}\right)\nonumber
\end{eqnarray}
since $2\vert{\rm Id}_{H/\pi_*}\vert_{\tilde{h}_{+\infty}}=12$. Again writing
\[C_3(\phi ,\lambda ):=12+\Vert s^*\phi\Vert^2_{L^\infty
(S^6-\{x_0\})}+\Vert\tilde{\phi}_{+\infty}\Vert^2_{L^\infty (S^6)}<+\infty\]
and 
\[C_4(\phi ,\lambda ):=\Vert s^*\phi\Vert_{L^\infty 
(S^6-\{x_0\})}+\Vert\tilde{\phi}_{+\infty}\Vert_{L^\infty (S^6)}<+\infty\]
we come up on $S^6-\{x_0\}$ with
\begin{eqnarray}
& &\left\vert\vert (s^*\phi )B^{-1}_\lambda (s^*\phi )^*B_\lambda 
-{\rm 
Id}_{H/\pi_*}\vert^2_{\tilde{h}_{+\infty}}-
\vert\tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty}-
{\rm Id}_{H/\pi_*}\vert^2_{\tilde{h}_{+\infty}}\right\vert\nonumber\\
&\leqq & C_3C_4\left(\Vert s^*\phi-\tilde{\phi}_{+\infty}\Vert_{L^\infty 
(S^6-\{x_0\})}+\Vert B^*_\lambda 
(s^*\phi )(B^{-1}_\lambda )^*-\tilde{\phi}_{+\infty}\Vert_{L^\infty 
(S^6-\{x_0\})}\right) .
\label{polinom.becsles}
\end{eqnarray}
Note that for a fixed $\phi$ taking $\lambda\rightarrow +\infty$ 
the coefficients $C_1,\dots,C_4$ remain bounded and we find 
$B_\lambda\rightarrow {\rm Id}_{H/\pi_*}$. Therefore in light of Lemma 
\ref{eltuneslemma3} the right 
hand sides of (\ref{nijenhuis.becsles}) and (\ref{polinom.becsles}) can be 
made as small as we please when $\lambda\rightarrow +\infty$. 
Consequently (\ref{nulla}) yields that
\[\vert N_{\tilde{\phi}_{+\infty}}\vert^2_{\tilde{h}_{+\infty}}+
e^2\vert\tilde{\phi}_{+\infty}\tilde{\phi}^*_{+\infty}
-{\rm Id}_{TS^6}\vert^2_{\tilde{h}_{+\infty}}=0\]
along $S^6-\{x_0\}$ hence by smoothness of the fields, on the whole 
$S^6\cong{\rm G}_2/{\rm SU}(3)$.

Finally taking into account that $T_{\tilde{D}_{+\infty}}=0$ because 
the connection is the Levi--Civita connection of 
$\tilde{h}_{+\infty}$ the pair 
$(\tilde{\phi}_{+\infty} ,\tilde{h}_{+\infty})$ 
is indeed a smooth vacuum solution to (\ref{S6YMH}) as we 
claimed. Moreover note that $\tilde{\phi}_{+\infty}\not= 0$ since 
$\tilde{h}_{+\infty}$ is not degenerate. $\Diamond$

\begin{remark}\rm For completeness we mention that from the physical 
viewpoint the original theory (\ref{G2YMH}) on ${\rm G}_2$ 
has been dimensionally reduced to a theory on $S^6$ whose field content is 
$(\phi , Q, h)$ where the Higgs field $\phi$ and the metric $h$ are as in 
(\ref{S6YMH}) moreover $Q\in C^\infty (S^6\:;\:{\rm End}\widetilde{W})$ is a 
new field coming from the original Higgs field. However this field theory 
in the limit $\lambda\rightarrow +\infty$ degenerates 
because (as we already noted) we do not get any metric on $\widetilde{W}$.
Consequently a non-singular Lagrangian over $S^6$ 
involving all the fields $(\phi ,Q, h)$ and 
stemming purely from (\ref{G2YMH}) cannot be written down. Instead we 
may consider a restricted theory given by (\ref{S6YMH}).
\end{remark}

\noindent Putting all of our findings 
sofar together we observe that $J:=\tilde{\phi}_{+\infty}$ is an 
everywhere smooth integrable almost complex structure on the 
six-sphere hence we have arrived at the following result:

\begin{theorem}
There exists a smooth integrable almost complex structure $J$ on $S^6$ 
given by the Higgs field $\tilde{\phi}_{+\infty}$ coming from the weakly 
spontaneosly broken vacuum $(\tilde{\phi}_{+\infty}, \tilde{h}_{+\infty})$ of 
the theory (\ref{S6YMH}) on $S^6$.
   
Consequently there exists a compact complex manifold $X$ whose underlying 
real manifold is homeomorphic to the six dimensional sphere. $\Diamond$
\label{Xtetel}
\end{theorem}


\begin{remark}\rm 1. Given a complex structure $\Phi$ on ${\rm G}_2$ its 
reduction to a complex structure $\tilde{\phi}_{+\infty}$ on $S^6$ is 
unique up to a gauge transformation in the theory (\ref{S6YMH}). In 
our approach it is natural to identify these gauge equivalent complex 
structures (note that this equivalence of complex structures is broader than 
equivalence in Kodaira--Spencer theory). Therefore in this sense assigning 
a complex structure on $S^6$ to a complex structure on ${\rm G}_2$ is unique. 

However the vacuum itself is highly degenerate: taking any metric $h$ on 
$S^6$ its usual projection along a fixed $\tilde{\phi}_{+\infty}$ 
will provide another vacuum solution. Nevertheless not all metrics on $S^6$ 
arise this way: for example the appearence of the classical round metric 
is excluded by LeBrun's theorem \cite{leb}. To get rid of 
this huge redundancy in the vacuum we could add some curvature expression 
to (\ref{YMH}) in order to control the metric as well (for instance the 
usual Einstein--Hilbert term $s_g$).

2. It is worth mentioning that a comparison of certain results 
offers an independent check of our Theorem \ref{Xtetel} here. 
Let $Y$ denote ${\rm G}_2$ equipped with a complex structre as before. 
Inserting the decomposition (\ref{hasitas}) into $\Lambda^{p,q}(TY)$ we obtain 
\[\Lambda^{p,q}(TY)\cong\Lambda^{p,q}(W)\oplus\Lambda^{p,q}(H)\oplus\dots\]
and let $\alpha^{p,q}\in [\alpha^{p,q}]\in H^{p,q}(Y;\C )$ be a 
representative of a non-zero Dolbeault cohomology class. Its 
restriction $\alpha^{p,q}\vert_H$ can be reduced as in (\ref{fi}). 
Let $X$ be a compact complex manifold diffeomorphic to $S^6$ with the 
complex structure coming from $Y$. This way one 
obtains an element $\tilde{\beta}^{p,q}\in 
C^\infty (\Lambda^{p,q}X;\C )$. It may happen that 
$[\tilde{\beta}^{p,q}]\in H^{p,q}(X;\C )$ i.e., it represents a 
Dolbeault cohomology class.

We know the following things. On the one hand \cite[Proposition 4.5]{pit} 
gives that $h^{0,1}(Y)=1$ and $h^{0,2}(Y)=0$. On the other hand it is 
proved in \cite{gra2, uga} that $h^{0,1}(X)=h^{0,2}(X)+1>0$. 
Therefore it is suggestive to expect that $h^{0,1}(X)=1$ and 
$h^{0,2}(X)=0$. Similarly, we know from \cite[Proposition 4.5]{pit} that 
$h^{2,0}(Y)=0$, $h^{1,1}(Y)=1$, $h^{1,0}(Y)=0$ and $h^{1,2}(Y)=1$ 
meanwhile \cite[Proposition 3.1]{uga} provides us that 
$h^{2,0}(X)+h^{1,1}(X)=h^{1,0}(X)+h^{1,2}(X)+1>0$. This suggests that 
$h^{2,0}(X)=0$, $h^{1,1}(X)=1$, $h^{1,0}(X)=0$ but $h^{1,2}(X)=0\not=1$. 
However the general relationship between the Hodge numbers of $Y$ and $X$ 
is certainly not straightforward.

3. Note that $Y$ is a homogeneous complex manifold since for any $y\in Y$ 
left translation $L_y$ is a biholomorphism. However this property does 
not descend to $X$ in accord with \cite{huc-keb-pet}: although for a 
finite $\lambda$ we have $[\Phi ,(L_y)_*]=0$ but taking 
the limit $\lambda\rightarrow +\infty$, while $\Phi$ remains 
meaningful and gives rise to $\Phi_{+\infty}$, the derivative 
$(L_y)_*$ of the left transition $L_y$ does not survive this limit: writing 
$y=y'\gamma$ with $y'\in {\rm G}_2$ independent of $\lambda$ and 
$\gamma\in{\rm SU}(3)$ parameterized as in \ref{eltuneslemma1} 
we find that $\Vert (L_y)_*\Vert_{L^\infty 
({\rm G}_2)}\sim\lambda^{\frac{1}{5}}$ holds.

4. It is also worth pointing out why our whole construction breaks down in 
the very similar situation of ${\rm SO}(4n+1)/{\rm SO}(4n)\cong S^{4n}$. In 
our understanding the crucial difference between these quotients and 
${\rm G}_2/{\rm SU}(3)$ is that in the former cases (\ref{veletlen}) 
fails. Indeed, high rank real vector bundles over $S^{4n}$ are 
classified by the homotopy group $\pi_{4n-1}({\rm GL}(N;\R 
))\cong\pi_{4n-1}({\rm O}(N))\cong\Z$ 
(or by their $n^{{\rm th}}$ Pontryagin classes in 
$H^{4n}(S^{4n};\Z )\cong\Z$) hence they can be non-trivial. Therefore 
trivial tensor bundles over ${\rm SO}(4n+1)$ cannot be collapsed smoothly to 
$S^{4n}$ such that they contain $TS^{4n}$ as a sub-bundle.
\end{remark}


\begin{thebibliography}{99}

\bibitem{adl} Adler, A.: {\it The second fundamental forms of $S^6$ and 
$P^n(\C )$}, Amer. J. Math. {\bf 91}, 657-670 (1969);

\bibitem{bor-lam-sal} Bor, G., Hern\'andez-Lamoneda, L., Salvai, M.: 
{\it Orthogonal complex structres of minimal energy}, Geom. Dedicata {\bf 
127}, 75-85 (2007);

\bibitem{bor-ser1} Borel, A., Serre, J.-P.: {\it D\'etermination des
p-puissances r\'eduites de Steenrod dans la cohomologie des groupes
classiques. Applications.} C.R. Acad. Sci. Paris {\bf 233} 680-682
(1951);

\bibitem{bor-ser2} Borel, A., Serre, J.-P.: {\it Groupes de Lie et 
puissances r\'eduites de Steenrod}, Amer. J. Math. {\bf 75}, 
409--448 (1953);

\bibitem{bro} Bronzan, J.B.: {\it Parametrization of ${\rm SU}(3)$}, Phys. 
Rev. {\bf D38}, 1994-1999 (1988);

\bibitem{bro-tom} Br\"ocker, Th., tom Dieck, T.: {\sl Representations of 
compact Lie groups}, GTM {\bf 98}, Springer, New York (1985);

\bibitem{cal-eck} Calabi, E., Eckmann, B.: {\it A class of compact, 
complex manifolds which are not algebraic}, Ann. Math. {\bf 58}, 
494-500 (1953);

\bibitem{cam-dem-pet} Campana, F., Demailly, J.-P., Peternell, T.: {\it
The algebraic dimension of compact complex threefolds with vanishing
second Betti number}, Compos. Math. {\bf 112}, 77-91 (1998);

\bibitem{chen} Chen, B.-Y.: {\it Riemannian geometry of Lagrangian 
submanifolds}, Taiwanese Journ. Math. {\bf 5}, 681--723 (2001); 

\bibitem{che-li} Cheng, L.-P., Li, L.-F.: {\sl Gauge theory of 
elementary particle physics}, Clarendon Press, Oxford (1984);

\bibitem{che1} Chern, S.-S.: {\sl Complex manifolds without potential 
theory}, Springer, New York (1979);
 
\bibitem{che2} Chern, S.-S.: {\it On the existence of a complex 
structure on the six sphere}, Nankai Institute of Mathematics preprint 
(unpublished) (2004); 

\bibitem{fro} Fr\"olicher, A.: {\it Zur Differentialgeometrie der 
komplexen Strukturen},  Math. Ann. {\bf 129}, 50-95 (1955); 

\bibitem{ful-har} Fulton, W., Harris, J.: {\sl Representation theory}, 
GTM RIM {\bf 129}, Springer, New York (1991);

\bibitem{gib-lu-pop} Gibbons, G.W., Lu, H., Pope, C.N.: {\it Einstein 
metrics on group manifolds and cosets}, arXiv: {\tt 0903.2493 
[hep-th]}, 23 pp. (2009);
 
\bibitem{gra1} Gray, A.: {\it Almost complex submanifolds in the 
six-sphere}, Proc. Amer. Math. Soc. {\bf 20}, 277-279 (1969);

\bibitem{gra2} Gray, A.: {\it A property of a hypothetical complex 
structure on the six-sphere}, Boll. Un. Mat. Ital. {\bf B11}, 251-255 
(1997);

\bibitem{gre-sch-wit} Green, M.B., Schwarz, J.H., Witten, E.:
{\sl Superstring theory II: Loop amplitudes, anomalies $\&$ 
phenomenology}, Cambridge Univ. Press, Cambridge (1987);
 
%\bibitem{has-mas-sek} Hashimoto, H., Mashima, K., Sekigawa, K.: {\it On 
%$4$-dimensional CR-manifolds of a $6$-dimensional sphere},  in: Minimal 
%surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 
%1999),  143--154, Adv. Stud. Pure Math. {\bf 34}, Math. Soc. Japan, 
%Tokyo (2002); 

\bibitem{hel} Helgason, S.: {\sl Geometric analysis on symmetric 
spaces}, Math. Surveys and Monographs {\bf 83}, Amer. Math. Soc., 
Providence, RI (2000);
 
%\bibitem{hit} Hitchin, N.J.: {\it K\"ahlerian twistor spaces}, Proc. 
%London Math. Soc. {\bf 43}, 133-150 (1981);

\bibitem{hsi} Hsiung, Ch.-Ch.: {\it Nonexistence of a complex structure 
on the six-sphere}, Bull. Inst. Math. Acad. Sinica {\bf 14}, 231-247 (1986);

\bibitem{huc-keb-pet} Huckleberry, A.T., Kebekus, S., Peternell, T.:
{\it Group actions on $S^6$ and complex structures on ${\mathbb P}_3$},
Duke Math. Journ. {\bf 102}, 101-124 (2000);

\bibitem{joy} Joyce, D.D.: {\sl Compact manifolds with special 
holonomy}, Oxford Univ. Press, Oxford (2000);

\bibitem{kak} Kaku, M.: {\sl Quantum field theory}, Oxford Univ. Press, 
Oxford (1993);

\bibitem{law-mic} Lawson, H.B., Michelson, M.-L.: {\sl Spin geometry}, 
Princeton Univ. Press, Princeton (1990);

\bibitem{leb} LeBrun, C.: {\it Orthogonal complex structures on $S^6$},
Proc. Amer. Math. Soc. {\bf 101}, 136-138 (1987);

\bibitem{mar-nie} Marshakov, A., Niemi, A.J.: {\it Yang--Mills, complex
structures and Chern's Last Theorem}, Mod. Phys. Lett. {\bf A20}, 
2583-2600 (2005);

\bibitem{may} Mayer, R.A.: {\it Summation of Fourier series on compact 
groups}, Amer. Journ. Math. {\bf 89}, 661-692 (1967);

\bibitem{mil} Milnor, J.: {\sl Morse theory}, Annals of Math. Studies {\bf 
51}, Princeton Univ. Press (1963);

\bibitem{mus} Musso, E.: {\it On the twistor space of the six-sphere},
Bull. Austral. Math. Soc. {\bf 39}, 119--127 (1989);
 
\bibitem{new-nir} Newlander, A., Nirenberg, L.: {\it Complex analytic
co-ordinates in almost complex manifolds}, Ann. Math. {\bf 65}, 391-404
(1957);

\bibitem{pen-tan} Peng, Ch., Tang, Z.: {\it Bott generator and its 
application to an almost complex structure on $S^6$}, Chinese Ann. 
Math. {\bf B28}, 213-218 (2007);

\bibitem{pit} Pittie, H.: {\it The Dolbeault-cohomology ring of a 
compact even dimensional Lie group}, Proc. Indian Acad. Sci. Math. Sci. 
{\bf 98}, 117-152 (1988);

\bibitem{pos} Postnikov, M.: {\sl Lectures in geometry V: Lie groups and 
Lie algebras}, Mir Publishers Moscow (1986);

\bibitem{sam} Samelson, H.: {\it A class of complex-analytic manifolds}, 
Portugalie Math. {\bf 12}, 129-132 (1953);

\bibitem{slu} Slupinski, M.J.: {\it The twistor space of the conformal 
six sphere and vector bundles on quadrics}, Journ. Geom. Phys. {\bf 19}, 
246-266 (1996);

\bibitem{tay} Taylor, M.E.: {\sl Partial differential equations I: Basic 
theory}, AMS {\bf 115}, Springer, New York (1996);

\bibitem{uga} Ugarte, L.: {\it Hodge numbers of a hypothetical complex 
structure on the six sphere}, Geom. Dedicata {\bf 81}, 173-179 
(2000);

%\bibitem{yos} Yoshinari, I.: {\it An exceptional example of twistor
%spaces of four-dimensional almost Hermitian manifolds}, J. Math. Kyoto
%Univ. {\bf 46}, 179-185 (2006);

%\bibitem{wal} Wall, C.T.C.: {\it Determination of the cobordism ring}, 
%Ann. Math. {\bf 72}, 292-311 (1960);

\bibitem{wei} Weinberg, S.: {\sl The quantum theory of fields II: 
Modern applications}, Cambridge Univ. Press, Cambridge (1998);

\bibitem{wil} Willmore, T.J.: {\sl Riemannian geometry}, Clarendon 
Press, Oxford (1993);

\bibitem{wit} Witten, E.: {\it Conformal field theory in
four and six dimensions}, arXiv: {\tt 0712.0157 [math.RT]}, 16 pp. (2007);

\bibitem{woo} Wood, C.M.: {\it Instability of the nearly K\"ahler 
six-sphere}, Journ. Reine Angew. Math. {\bf 439}, 205-212 (1993).

\end{thebibliography}
\end{document}


math0505634.extracted_bib
\begin{thebibliography}{99}

\bibitem{adl} Adler, A.: {\it The second fundamental forms of $S^6$ and 
$P^n(\C )$}, Amer. J. Math. {\bf 91}, 657-670 (1969);

\bibitem{bor-lam-sal} Bor, G., Hern\'andez-Lamoneda, L., Salvai, M.: 
{\it Orthogonal complex structres of minimal energy}, Geom. Dedicata {\bf 
127}, 75-85 (2007);

\bibitem{bor-ser1} Borel, A., Serre, J.-P.: {\it D\'etermination des
p-puissances r\'eduites de Steenrod dans la cohomologie des groupes
classiques. Applications.} C.R. Acad. Sci. Paris {\bf 233} 680-682
(1951);

\bibitem{bor-ser2} Borel, A., Serre, J.-P.: {\it Groupes de Lie et 
puissances r\'eduites de Steenrod}, Amer. J. Math. {\bf 75}, 
409--448 (1953);

\bibitem{bro} Bronzan, J.B.: {\it Parametrization of ${\rm SU}(3)$}, Phys. 
Rev. {\bf D38}, 1994-1999 (1988);

\bibitem{bro-tom} Br\"ocker, Th., tom Dieck, T.: {\sl Representations of 
compact Lie groups}, GTM {\bf 98}, Springer, New York (1985);

\bibitem{cal-eck} Calabi, E., Eckmann, B.: {\it A class of compact, 
complex manifolds which are not algebraic}, Ann. Math. {\bf 58}, 
494-500 (1953);

\bibitem{cam-dem-pet} Campana, F., Demailly, J.-P., Peternell, T.: {\it
The algebraic dimension of compact complex threefolds with vanishing
second Betti number}, Compos. Math. {\bf 112}, 77-91 (1998);

\bibitem{chen} Chen, B.-Y.: {\it Riemannian geometry of Lagrangian 
submanifolds}, Taiwanese Journ. Math. {\bf 5}, 681--723 (2001); 

\bibitem{che-li} Cheng, L.-P., Li, L.-F.: {\sl Gauge theory of 
elementary particle physics}, Clarendon Press, Oxford (1984);

\bibitem{che1} Chern, S.-S.: {\sl Complex manifolds without potential 
theory}, Springer, New York (1979);
 
\bibitem{che2} Chern, S.-S.: {\it On the existence of a complex 
structure on the six sphere}, Nankai Institute of Mathematics preprint 
(unpublished) (2004); 

\bibitem{fro} Fr\"olicher, A.: {\it Zur Differentialgeometrie der 
komplexen Strukturen},  Math. Ann. {\bf 129}, 50-95 (1955); 

\bibitem{ful-har} Fulton, W., Harris, J.: {\sl Representation theory}, 
GTM RIM {\bf 129}, Springer, New York (1991);

\bibitem{gib-lu-pop} Gibbons, G.W., Lu, H., Pope, C.N.: {\it Einstein 
metrics on group manifolds and cosets}, arXiv: {\tt 0903.2493 
[hep-th]}, 23 pp. (2009);
 
\bibitem{gra1} Gray, A.: {\it Almost complex submanifolds in the 
six-sphere}, Proc. Amer. Math. Soc. {\bf 20}, 277-279 (1969);

\bibitem{gra2} Gray, A.: {\it A property of a hypothetical complex 
structure on the six-sphere}, Boll. Un. Mat. Ital. {\bf B11}, 251-255 
(1997);

\bibitem{gre-sch-wit} Green, M.B., Schwarz, J.H., Witten, E.:
{\sl Superstring theory II: Loop amplitudes, anomalies $\&$ 
phenomenology}, Cambridge Univ. Press, Cambridge (1987);
 
%\bibitem{has-mas-sek} Hashimoto, H., Mashima, K., Sekigawa, K.: {\it On 
%$4$-dimensional CR-manifolds of a $6$-dimensional sphere},  in: Minimal 
%surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 
%1999),  143--154, Adv. Stud. Pure Math. {\bf 34}, Math. Soc. Japan, 
%Tokyo (2002); 

\bibitem{hel} Helgason, S.: {\sl Geometric analysis on symmetric 
spaces}, Math. Surveys and Monographs {\bf 83}, Amer. Math. Soc., 
Providence, RI (2000);
 
%\bibitem{hit} Hitchin, N.J.: {\it K\"ahlerian twistor spaces}, Proc. 
%London Math. Soc. {\bf 43}, 133-150 (1981);

\bibitem{hsi} Hsiung, Ch.-Ch.: {\it Nonexistence of a complex structure 
on the six-sphere}, Bull. Inst. Math. Acad. Sinica {\bf 14}, 231-247 (1986);

\bibitem{huc-keb-pet} Huckleberry, A.T., Kebekus, S., Peternell, T.:
{\it Group actions on $S^6$ and complex structures on ${\mathbb P}_3$},
Duke Math. Journ. {\bf 102}, 101-124 (2000);

\bibitem{joy} Joyce, D.D.: {\sl Compact manifolds with special 
holonomy}, Oxford Univ. Press, Oxford (2000);

\bibitem{kak} Kaku, M.: {\sl Quantum field theory}, Oxford Univ. Press, 
Oxford (1993);

\bibitem{law-mic} Lawson, H.B., Michelson, M.-L.: {\sl Spin geometry}, 
Princeton Univ. Press, Princeton (1990);

\bibitem{leb} LeBrun, C.: {\it Orthogonal complex structures on $S^6$},
Proc. Amer. Math. Soc. {\bf 101}, 136-138 (1987);

\bibitem{mar-nie} Marshakov, A., Niemi, A.J.: {\it Yang--Mills, complex
structures and Chern's Last Theorem}, Mod. Phys. Lett. {\bf A20}, 
2583-2600 (2005);

\bibitem{may} Mayer, R.A.: {\it Summation of Fourier series on compact 
groups}, Amer. Journ. Math. {\bf 89}, 661-692 (1967);

\bibitem{mil} Milnor, J.: {\sl Morse theory}, Annals of Math. Studies {\bf 
51}, Princeton Univ. Press (1963);

\bibitem{mus} Musso, E.: {\it On the twistor space of the six-sphere},
Bull. Austral. Math. Soc. {\bf 39}, 119--127 (1989);
 
\bibitem{new-nir} Newlander, A., Nirenberg, L.: {\it Complex analytic
co-ordinates in almost complex manifolds}, Ann. Math. {\bf 65}, 391-404
(1957);

\bibitem{pen-tan} Peng, Ch., Tang, Z.: {\it Bott generator and its 
application to an almost complex structure on $S^6$}, Chinese Ann. 
Math. {\bf B28}, 213-218 (2007);

\bibitem{pit} Pittie, H.: {\it The Dolbeault-cohomology ring of a 
compact even dimensional Lie group}, Proc. Indian Acad. Sci. Math. Sci. 
{\bf 98}, 117-152 (1988);

\bibitem{pos} Postnikov, M.: {\sl Lectures in geometry V: Lie groups and 
Lie algebras}, Mir Publishers Moscow (1986);

\bibitem{sam} Samelson, H.: {\it A class of complex-analytic manifolds}, 
Portugalie Math. {\bf 12}, 129-132 (1953);

\bibitem{slu} Slupinski, M.J.: {\it The twistor space of the conformal 
six sphere and vector bundles on quadrics}, Journ. Geom. Phys. {\bf 19}, 
246-266 (1996);

\bibitem{tay} Taylor, M.E.: {\sl Partial differential equations I: Basic 
theory}, AMS {\bf 115}, Springer, New York (1996);

\bibitem{uga} Ugarte, L.: {\it Hodge numbers of a hypothetical complex 
structure on the six sphere}, Geom. Dedicata {\bf 81}, 173-179 
(2000);

%\bibitem{yos} Yoshinari, I.: {\it An exceptional example of twistor
%spaces of four-dimensional almost Hermitian manifolds}, J. Math. Kyoto
%Univ. {\bf 46}, 179-185 (2006);

%\bibitem{wal} Wall, C.T.C.: {\it Determination of the cobordism ring}, 
%Ann. Math. {\bf 72}, 292-311 (1960);

\bibitem{wei} Weinberg, S.: {\sl The quantum theory of fields II: 
Modern applications}, Cambridge Univ. Press, Cambridge (1998);

\bibitem{wil} Willmore, T.J.: {\sl Riemannian geometry}, Clarendon 
Press, Oxford (1993);

\bibitem{wit} Witten, E.: {\it Conformal field theory in
four and six dimensions}, arXiv: {\tt 0712.0157 [math.RT]}, 16 pp. (2007);

\bibitem{woo} Wood, C.M.: {\it Instability of the nearly K\"ahler 
six-sphere}, Journ. Reine Angew. Math. {\bf 439}, 205-212 (1993).

\end{thebibliography}