# New explicit CMC cylinders and same–lobed CMC multibubbletons

Joseph Cho  Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner
Hauptstrasse 8-10/104, 1040 Wien, Austria<EMAIL_ADDRESS>,
Katrin Leschke  Department of Mathematics, University of Leicester, University
Road, Leicester LE1 7RH, United Kingdom
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
<EMAIL_ADDRESS>and  Yuta Ogata  Department of Mathematics, Faculty
of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto-City,
603-8555, Japan<EMAIL_ADDRESS>
###### Abstract.

We provide explicit parametrisations of all Darboux transforms of Delaunay
surfaces. Using the Darboux transformation on a multiple cover, we obtain this
way new closed CMC surfaces with dihedral symmetry. These can be used to
construct closed same-lobed CMC multibubbletons by applying Bianchi
permutability.

###### Key words and phrases:

CMC surfaces, Darboux transformations, Delaunay surfaces

###### 2020 Mathematics Subject Classification:

Primary 53A10; Secondary 37K35, 58E20.

##  1\. Introduction

In this paper, we investigate Darboux transforms of Delaunay surfaces, that
is, of constant mean curvature surfaces (CMC) of revolution with non–vanishing
mean curvature. Recall that the _classical Darboux transformation_ [darboux]
is defined for isothermic surfaces, those surfaces which allow a conformal
curvature line parameterisation. Geometrically, a Darboux pair is given by a
pair of isothermic surfaces, both conformally enveloping a sphere congruence.
Since a CMC surface $f:M\to\mathbb{R}^{3}$ is isothermic, one can apply the
classical Darboux transformation; however, a classical Darboux transform
$\hat{f}$ is in general just isothermic, and has constant mean curvature only
if $\hat{f}$ has pointwise constant distance from the parallel constant mean
curvature surface of $f$, see [darboux_isothermic]. We will denote these
Darboux transforms for short by _CMC Darboux transforms_. CMC Darboux
transforms are equivalent to the Bianchi-Bäcklund transforms
[bianchi_lezioni_1903] and simple-factor dressings of the extended frame
[kobayashi_characterizations_2005, burstall_isothermic_2006,
simple_factor_dressing, cho_simple_2019].

We will construct CMC Darboux transforms via the $\mu$–Darboux transformation
for CMC surfaces: this transformation is a special case of the (generalised)
Darboux transformation for conformal immersions, [conformal_tori]. We briefly
recall the $\mu$–Darboux transformation (see also Section 2): By the Ruh–Vilms
Theorem [ruh_vilms] a surface has constant mean curvature if and only if its
Gauss map is harmonic. The harmonicity of the Gauss map allows for an
introduction of an associated family of flat complex connections
$d_{\lambda}$, $\lambda\in\mathbb{C}_{*}=\mathbb{C}\setminus\\{0\\}$. Then the
$\mu$–Darboux transforms of a CMC surface are exactly given by the
$d_{\mu}$–parallel sections for some fixed _spectral parameter_
$\mu\in\mathbb{C}_{*}$. In the case when
$\mu\in\mathbb{R}_{*}=\mathbb{R}\setminus\\{0\\}$, this construction gives
exactly all (classical) CMC Darboux transforms, [cmc].

In the case of CMC Darboux transforms of Delaunay surfaces, for a generic
spectral parameter $\mu\in\mathbb{R}_{*}$ the only periodic CMC Darboux
transforms are trivial, that is the Darboux transforms are again
reparametrisations, rotations and translations of the original Delaunay
surface.

In case of a round cylinder however Sterling and Wente [wente_sterling]
discovered, while using the Bianchi–Bäcklund transform, that there are special
spectral parameters, the so–called _resonance points_ , so that there are
closed, non–trivial CMC Darboux transforms; these surfaces are called (CMC)
_bubbletons_ and are not embedded [Kilian_bubbletons]. For each
$n\in\mathbb{N},n>1,$ there is a corresponding resonance point $\mu_{n}$, and
$n$ gives the number of lobes on the bubble of the bubbleton in case of a
single cover of the surface. This process has been extended
[KobayashiBubbletons, KobayashiBubbletons2] to obtain bubbletons from any
Delaunay surface (and in all space forms).

From our point of view (Section 3), closed CMC Darboux transforms are
characterised by parallel sections _with multiplier_ : going around the circle
direction, the parallel section only changes by a complex multiple. Resonance
points are then spectral parameter $\mu\in\mathbb{R}\setminus\\{0,\pm 1\\}$
for which all parallel sections are sections with multiplier. In the case of
Delaunay surfaces we first provide an explicit parameterisation of all Darboux
transforms in terms of Jacobi elliptic functions and the incomplete elliptic
integral of the third kind by computing parallel sections in a quaternionic
formalism (see Theorem 3.1). We then investigate the parallel sections with
multipliers, and in particular obtain resonance points $\mu_{n}$ which depend
on the necksize $r$ of the Delaunay surface. For necksize $r=\frac{1}{2}$ we
have the round cylinder, for $0<r<\frac{1}{2}$ an unduloid and $r<0$ a nodoid.
As before $n\in\mathbb{N},n>1$, gives the number of lobes if the Delaunay
surface is embedded. In the case of nodoids, that is, non–embedded Delaunay
surfaces, the necksize $r<0$ imposes the condition $n>1-2r$ on the number of
lobes on a bubbleton.

Given two Darboux transforms $f_{1}$ and $f_{2}$ one can algebraically
construct a common Darboux transform $\hat{f}$ via Bianchi permutability,
[bianchi_ricerche_1905, conformal_tori]. Applying this construction in our
setting, we obtain a _doublebubbleton_ for two resonance points
$\mu_{n},\mu_{l}\in\mathbb{R}_{*}$, $n\not=l,n,l\in\mathbb{N}$, and
corresponding bubbletons $f_{1}$ and $f_{2}$ with $n$ and $l$ lobes
respectively, that is, a common closed Darboux transform $\hat{f}$ of $f_{1}$
and $f_{2}$ which has constant mean curvature. Such a doublebubbleton has two
bubbles, one with $n$ lobes, the other with $l$ lobes. For this construction
the assumption $n\not=l$ is critical: for $\mu_{n}=\mu_{l}$ the common CMC
Darboux transform obtained by Bianchi permutability is the original Delaunay
surface. Repeating the procedure we can obtain multibubbletons with an
arbitrary number of bubbles but each two bubbles have different numbers of
lobes. Put differently, with this approach we cannot obtain closed,
same–lobbed CMC multibubbletons.

On the other hand, we showed in [sym-darboux] by using a Sym–type argument
that any closed classical double Darboux transform of a Delaunay surface with
the same spectral parameter can only have constant mean curvature if it is the
original surface. Therefore, if we restrict to a single cover of the Delaunay
surface, no same–lobed closed CMC multibubbletons can occur.

We now consider multiple covers of Delaunay surfaces (Section 4). This is
motivated by results on Darboux transforms of isothermic tori [holly_tori],
circletons [kilian_circletons] and periodic smooth and discrete Darboux
transforms [periodic_discrete]: in these cases (additional) resonance points
can be found when considering multiple covers of the domain.

In the case of a multiple–cover of the round cylinder already Sterling and
Wente [wente_sterling] observed that additional resonance points $\mu_{n}^{m}$
can occur: one obtains again a closed CMC surface with $n$ lobes on the
$m$–fold cover of the round cylinder where $n$ must be bigger than $m$. This
is a natural generalisation of the situation on a single cover. Indeed, in
this paper we show that for nodoids and unduloids, one can also obtain
additional resonance points $\mu_{n}^{m}$ for $n>m$. Geometrically, these
bubbletons close on an $m$–fold cover and have non–embedded ends but look
similar to the “normal” bubbletons with $n$ lobes on a single bubble (if $m$
and $n$ are co–prime). As before, in the case of nodoids there is an
obstruction on the number of lobes $n>m(1-2r)$.

However, the situation is more intriguing for unduloids and nodoids: in this
situation the case $n<m$ can occur (Theorem 4.1). Geometrically, now the
“bubbles” are not localised anymore but the number $n$ still gives the
dihedral symmetry of the surface. In contrast to the case of a single cover,
there is a restriction on the order of the symmetry for unduloids $n<m(1-2r)$
whereas for any choice of $n<m$ we obtain a bubbleton of a nodoid. In
particular, for all nodoids (and unduloids with $0<r<\frac{m-1}{2m}$) it is
possible to obtain a CMC surface with $n=1$, which just has a reflectional
symmetry and one “lobe”.

To obtain closed, same–lobed CMC multibubbletons, we change our point of view
(Section 5). Limiting to the case when $n>m$ (to obtain localised bubbles with
lobes on our bubbletons), we investigate multibubbletons: if there are two
co–prime pairs $(m_{1},n)$ and $(m_{2},n)$, $m_{1}\not=m_{2}$, so that the
corresponding Darboux transforms $f_{1}$ and $f_{2}$ have $n$ bubbles on the
$m_{1}$ and $m_{2}$–fold cover respectively, then the common Darboux transform
of $f_{1}$ and $f_{2}$ given by Bianchi permutability has two bubbles, each
with $n$ lobes — we obtain a CMC same–lobed doublebubbleton which closes on an
$L$–fold cover, where $L$ is the least common multiple of $m_{1}$ and $m_{2}$.

Finally, we observe that the number $1\leq l_{0}\leq m$ of resonance points
given by $n$, that is the number of $n$–lobed bubbles we can put on a Delaunay
surface, depends on the necksize of the Delaunay surface and co–prime pairs
$(m,n)$: for any $l\leq l_{0}$ we show that there exists a multibubbleton with
$l$ bubbles with $n$ lobes each (Theorem 5.1).

Acknowledgements. The second author would like to thank the members of the
Research Institute for Mathematical Sciences at Kyoto University for their
hospitality during her stay as Visiting Professor at the Institute. The third
author gratefully acknowledges the support from Grants-in-Aid of JSPS Research
Fellowships for Young Scientist 21K13799.

##  2\. Background

In this section we will give a short summary of results and methods used in
this paper. For details on the quaternionic formalism and CMC surfaces we
refer to [coimbra, klassiker, cmc, simple_factor_dressing,
darboux_isothermic].

###  2.1. CMC surfaces in the quaternionic model

In this paper, we investigate the Darboux transformation on conformal
immersions $f:M\to\mathbb{R}^{3}$ from a Riemann surface into 3–space with
constant mean curvature, where we assume without loss of generality that the
mean curvature is $H=1$. By the Ruh–Vilms theorem [ruh_vilms] the Gauss map
$N$ of a CMC surface $f$ is harmonic. This allows to introduce a spectral
parameter and to define, for example, the associated family of flat
connections, the associated family of CMC surfaces, and the spectral curve of
a CMC torus, [hitchin-harmonic, pin&ster, Bob].

Here we use a quaternionic model and identify 3–space with the imaginary
quaternions $\mathbb{R}^{3}=\operatorname{Im}\mathbb{H}$ where
$\mathbb{H}=\operatorname{span}_{\mathbb{R}}\\{1,i,j,k\\}$ and
$i^{2}=j^{2}=k^{2}=ijk=-1$. In the following we will identify

$\mathbb{H}={\rm
Re}\,\mathbb{H}\oplus\operatorname{Im}\mathbb{H}=\mathbb{R}\oplus\mathbb{R}^{3}.$

For imaginary quaternions the product in the quaternions is related to the
inner product $\langle\cdot,\cdot\rangle$ and the cross product in
$\mathbb{R}^{3}$ by

$ab=-\langle a,b\rangle+a\times b,\quad a,b\in\operatorname{Im}\mathbb{H}.$

In particular, we see

$S^{2}=\\{n\in\operatorname{Im}\mathbb{H}\mid n^{2}=-1\\}.$

Thus, if $f:M\to\mathbb{R}^{3}$ is an immersion then its Gauss map $N:M\to
S^{2}$ is a complex structure $N^{2}=-1$ on $\mathbb{R}^{4}=\mathbb{H}$.
Moreover, if $(M,J_{TM})$ is a Riemann surface, where $J_{TM}$ is the complex
structure on the tangent space, then $f:M\to\mathbb{R}^{3}$ is conformal if
and only if

$*df=Ndf=-dfN,$

where $*$ denotes the negative Hodge star operator, that is,
$*\omega(X)=\omega(J_{TM}X)$ for $X\in TM$, $\omega\in\Omega^{1}(M)$.

Denoting the $(1,0)$–part ($(0,1)$-part, respectively) of a 1–form $\omega$
with respect to the complex structure $N$ by

$\omega^{\prime}=\frac{1}{2}(\omega-N*\omega)$

(and $\omega^{\prime\prime}=\frac{1}{2}(\omega+N*\omega)$, respectively), we
have for any conformal immersion $f:M\to\mathbb{R}^{3}$ that

(2.1) $dfH=-(dN)^{\prime}.$

In the case when $H=1$, the decomposition of $dN$ into $(0,1)$– and
$(1,0)$–part is thus given by

$dN=dg-df$

where $dg=(dN)^{\prime\prime}$ and $df=-(dN)^{\prime}$. Put differently,
$g=f+N$ is the parallel CMC surface of $f$ since $N_{g}=-N$ and thus
$dg=-(dN_{g})^{\prime}$.

In particular, we obtain the following reformulation of the Ruh–Vilms theorem:
a conformal immersion $f$ has constant mean curvature $H$ if and only if $N$
is harmonic, that is,

$d((dN)^{\prime})=0.$

Harmonicity allows for the introduction of a complex spectral parameter (for
details in our setting, see [cmc]) to obtain the family of connections
$d_{\lambda}$, $\lambda\in\mathbb{C}_{*}=\mathbb{C}\setminus\\{0\\}$, given by

(2.2) $d_{\lambda}\alpha=d\alpha-\frac{1}{2}(dN)^{\prime}(N\alpha(a-1)+\alpha
b)$

where $a=\frac{\lambda+\lambda^{-1}}{2},b=i\frac{\lambda^{-1}-\lambda}{2}$.

The associated family can now be used to characterise CMC surfaces: the
connections $d_{\lambda}$ of the associated family (2.2) of a map $N:M\to
S^{2}$ are flat for all $\lambda\in\mathbb{C}_{*}$ if and only if $N$ is
harmonic.

######  Remark 2.1.

In case when $\lambda=a+ib\in S^{1}$ we have $a,b\in\mathbb{R}$ and the family
of flat connections is quaternionic. In general, the family of flat
connections is defined on the trivial $\mathbb{C}^{2}$–bundle
$M\times\mathbb{C}^{2}=\underline{\mathbb{C}}^{2}$ over $M$ where we consider
$\mathbb{C}^{2}=(\mathbb{H},I)$ with the complex structure $I$ given by right
multiplication by $i$. In this case, we have the reality condition

(2.3) $d_{\lambda}(\alpha j)=(d_{\bar{\lambda}^{-1}}\alpha)j.$

###  2.2. CMC surfaces and Darboux transforms

We are now in the position to recall the Darboux transformation of CMC
surfaces. In [conformal_tori] a Darboux transformation on conformal immersions
$f:M\to\mathbb{R}^{3}$ was defined, generalising the classical Darboux
transformation in [darboux]. In this paper we consider the so–called
$\mu$–Darboux transformation on CMC surfaces, which gives the $\mu$–Darboux
transforms by the choice of a spectral parameter $\mu\in\mathbb{C}_{*}$ and a
$d_{\mu}$–parallel section. The $\mu$–Darboux transformation is a special case
of the generalised Darboux transformation and preserves the CMC property, up
to translation. Additionally, the $\mu$–Darboux transforms with
$\mu\in\mathbb{R}\setminus\\{0,1\\}$ are exactly the classical Darboux
transforms of a CMC surface which are CMC. In this paper, we investigate these
_(classical) CMC Darboux transforms_ in the case of Delaunay surfaces. For a
detailed study of the link between these different Darboux transformations, we
refer to [isothermic_paper].

We recall:

######  Theorem 2.2 ([cmc]).

Let $f:M\to\mathbb{R}^{3}$ be a CMC surface with $H=1$, and let $d_{\lambda}$
be its associated family of flat connections. For
$\mu\in\mathbb{R}\setminus\\{0,1\\}$, let
$\alpha\in\Gamma(\underline{\widetilde{\mathbb{H}}})$ be a $d_{\mu}$–parallel
section of the trivial $\mathbb{C}^{2}$–bundle
$\tilde{M}\times\mathbb{C}^{2}=\tilde{M}\times\mathbb{H}$ over the universal
cover $\tilde{M}$ of $M$.

Then $\hat{f}=f+T$ is a CMC Darboux transform of $f$ where $T$ is given
algebraically by

$T^{-1}=\frac{1}{2}(N(a-1)+\alpha b\alpha^{-1}),$

and $a=\frac{\mu+\mu^{-1}}{2},b=i\frac{\mu^{-1}-\mu}{2}$. All classical CMC
Darboux transforms of $f$ arise this way.

######  Remark 2.3.

Note that $\mu=a+ib$ and therefore $\mu\in\mathbb{R}$ if and only if
$a\in\mathbb{R},b\in i\mathbb{R}$. In this case, the condition $a^{2}+b^{2}=1$
shows that $\mu^{-1}=a-ib$. Therefore, by the reality condition (2.3) $\alpha$
is $d_{a+ib}$–parallel if and only if $\alpha j$ is $d_{a-ib}$–parallel. In
particular, in the case of real parameter, the two parameter $\mu=a+ib$ and
$\mu^{-1}=a-ib$ give rise to the same Darboux transforms.

Moreover, for $\mu=-1$, that is, $a=-1,b=0$, the Darboux transform is
independent of the parallel section $\alpha$ and we obtain the parallel CMC
surface $\hat{f}=f+N$ as the only Darboux transform.

We now recall the Bianchi permutability theorem [bianchi_ricerche_1905,
conformal_tori] in our setting which gives two–step Darboux transforms
algebraically, that is, without further integration, from the parallel
sections of the first CMC surface.

######  Theorem 2.4 ([isothermic_paper]).

Let $f:M\to\mathbb{R}^{3}$ be a CMC surface and
$\alpha_{l}\in\Gamma(\underline{\widetilde{\mathbb{H}}})$ parallel sections
with respect to the flat connections $d_{\mu_{l}}$ where
$\mu_{l}\in\mathbb{R}\setminus\\{0,1\\}$, $l=1,2$, $\mu_{1}\not=\mu_{2}$. Let
$f_{l}=f+\alpha_{l}\beta_{l}^{-1}$ be the CMC Darboux transforms given by
$\alpha_{l}$ and

(2.4) $\beta_{l}=\frac{1}{2}(N\alpha_{l}(a_{l}-1)+\alpha_{l}b_{l})$

where
$a_{l}=\frac{\mu_{l}+\mu_{l}^{-1}}{2},b_{l}=i\frac{\mu_{l}^{-1}-\mu_{l}}{2}$.

Then the common $\mu$–Darboux transform $\hat{f}$ of $f_{1}$ and $f_{2}$ is a
CMC surface. Moreover, the common $\mu$–Darboux transform is given by
$\hat{f}=f_{1}+\alpha\beta^{-1}$ where

(2.5)
$\alpha=\alpha_{2}-\alpha_{1}\beta_{1}^{-1}\beta_{2},\qquad\beta=\beta_{2}-\beta_{1}\alpha_{1}^{-1}\alpha_{2}\frac{a_{2}-1}{a_{1}-1}.$

Note that one could also allow $\mu_{1}=\mu_{2}$ and consider two independent
$d_{\mu_{1}}$–parallel sections, but in this case the resulting common CMC
Darboux transform is $\hat{f}=f$.

To investigate closed surfaces we recall:

######  Definition 2.5.

Let $f:M\to\mathbb{R}^{3}$ be a CMC surface and $d_{\lambda}$ its associated
family of flat connections. For $\mu\in\mathbb{C}_{*}$ a $d_{\mu}$–parallel
section $\alpha$ is called a _section with multiplier_ if
$\gamma^{*}\alpha=\alpha h_{\gamma}$ for all $\gamma\in\pi_{1}(M)$. Here
$h_{\gamma}:\pi_{1}(M)\to\mathbb{C}_{*}$ is a group homomorphism, and
$\gamma^{*}\alpha=\alpha\circ\gamma_{\sharp}$ where $\gamma_{\sharp}$ is the
associated deck transformation of $\gamma\in\pi_{1}(M)$.

A parameter $\mu\in\mathbb{C}_{*}$ is called a _resonance point_ if all
$d_{\mu}$–parallel sections have a multiplier.

With this definition we have in the case of a CMC surface (for the general
case of a conformal immersion, see [conformal_tori]):

######  Proposition 2.6.

Let $f:M\to\mathbb{R}^{3}$ be a CMC surface and $\hat{f}=f+T$ a $\mu$–Darboux
transform given by a $d_{\mu}$–parallel section $\alpha$,
$\mu\in\mathbb{R}\setminus\\{0,1\\}$. Then $\hat{f}$ is defined on $M$, rather
than the universal cover $\tilde{M}$, if and only if $\alpha$ is a section
with multiplier.

In particular, we can apply this to Bianchi permutability:

######  Corollary 2.7.

Let $f:M\to\mathbb{R}^{3}$ be a CMC surface and $f_{l}:M\to\mathbb{R}^{3}$ be
the CMC Darboux transforms given by $d_{\mu_{l}}$–parallel sections
$\alpha_{l}$ with multipliers where $\mu_{l}\in\mathbb{R}\setminus\\{0,1\\}$,
$l=1,2$, $\mu_{1}\not=\mu_{2}$. Then the common CMC Darboux transform
$\hat{f}:M\to\mathbb{R}^{3}$ of $f_{1}$ and $f_{2}$ is defined on $M$ as well.

###### Proof.

Let $h_{l}$ denote the multipliers of the $d_{\mu_{l}}$–parallel sections
$\alpha_{l}$. By (2.4) we see that
$\gamma^{*}\beta_{l}=\beta_{l}(h_{l})_{\gamma}$. Then (2.5) shows that

$\displaystyle\gamma^{*}\alpha$
$\displaystyle=\gamma^{*}\alpha_{2}-(\gamma^{*}\alpha_{1})(\gamma^{*}\beta_{1})^{-1}(\gamma^{*}\beta_{2})=\alpha_{2}(h_{2})_{\gamma}-\alpha_{1}(h_{1})_{\gamma}(h_{1})_{\gamma}^{-1}\beta_{1}^{-1}\beta_{2}(h_{2})_{\gamma}$
$\displaystyle=\alpha(h_{2})_{\gamma}\,.$

Therefore, $\alpha$ is a section with multiplier, and
$\hat{f}=f_{1}+\alpha\beta^{-1}$ is defined on $M$. ∎

##  3\. Delaunay bubbletons

We will now recall the construction of (multi-) bubbletons which are CMC
Darboux transforms at a resonance point of Delaunay surfaces, that is, of CMC
surfaces of revolution.

More generally, we will give all closed CMC Darboux transforms of Delaunay
surfaces in terms of Jacobi elliptic functions and elliptic integrals. To this
end, we need to find all parallel sections of the associated family of flat
connections explicitly. Recall that surfaces of revolution can be conformally
parametrised by

$f(x,y)=ip(x)+jq(x)e^{-iy}\,,\qquad\text{ with
}\quad(p^{\prime})^{2}+(q^{\prime})^{2}=q^{2}.$

In the cases of Delaunay surfaces, $p,q$ are given by the Jacobi elliptic
functions with parameter $M=1-(1-\frac{1}{1-r})^{2}$ where
$r\in(-\infty,\frac{1}{2})$ is the necksize and

$q(x)=(1-r)\operatorname{dn}((1-r)x,M),\qquad p^{\prime}(x)=q(x)^{2}+r(1-r).$

Note that $r=\frac{1}{2}$ gives a cylinder, $r\in(0,\frac{1}{2})$ unduloids
and $r\in(-\infty,0)$ nodoids.

To find parallel sections $\alpha$ of a flat connection $d_{\mu}$ with
$\mu\in\mathbb{R}\setminus\\{0,1\\}$, in the associated family $d_{\lambda}$
of the harmonic Gauss map $N$ of $f$, we need to solve

$d\alpha=-\frac{1}{2}df(N\alpha(a-1)+\alpha b),$

where $a=\frac{\mu+\mu^{-1}}{2},b=i\frac{\mu^{-1}-\mu}{2}$. Here we used (2.1)
to express (2.2) in terms of the CMC surface $f$. Since $f$ is conformal we
have $f_{x}N=-f_{y}$ and $f_{y}N=f_{x}$ so that we need to solve the PDEs

(3.1) $\displaystyle\alpha_{x}$
$\displaystyle=\frac{1}{2}(f_{y}\alpha(a-1)-f_{x}\alpha b),$ (3.2)
$\displaystyle\alpha_{y}$
$\displaystyle=-\frac{1}{2}(f_{x}\alpha(a-1)+f_{y}\alpha b).$

We consider the second equation (3.2) first. We decompose
$\alpha=\alpha_{0}+j\alpha_{1}$ and set $\tilde{\alpha}_{1}=e^{iy}\alpha_{1}$
to obtain the linear system

$\begin{pmatrix}\alpha_{0}\\\
\tilde{\alpha}_{1}\end{pmatrix}_{y}=\frac{1}{2}\begin{pmatrix}-ip^{\prime}(a-1)&iqb+q^{\prime}(a-1)\\\
iqb-q^{\prime}(a-1)&i(2+p^{\prime}(a-1))\end{pmatrix}\begin{pmatrix}\alpha_{0}\\\
\tilde{\alpha}_{1}\end{pmatrix}=:A\begin{pmatrix}\alpha_{0}\\\
\tilde{\alpha}_{1}\end{pmatrix}.$

The matrix $A$ has eigenvalues $\frac{i}{2}(1\pm t)$ where

$t=\sqrt{1+2r(1-r)(a-1)}.$

If $t=0$ then $\alpha_{+}=\alpha_{-}$ gives only one solution. In the case
when $r=\frac{1}{2}$, we obtain $\mu=-1$, and the general solutions are given
by $\alpha=e^{\frac{iy}{2}}c$, $c\in\mathbb{H}$. By Remark 2.3 the only
Darboux transform in this case is the parallel CMC surface. For
$r<\frac{1}{2}$ and $t=0$, the spectral parameter are $\mu_{b}=\frac{r}{r-1}$
and $\mu_{b}^{-1}=\frac{r-1}{r}$, and the matrix $A$ is not diagonalisable.
Therefore, we obtain a second independent parallel section $\hat{\alpha}$ by
the generalised eigenvector, but $\hat{\alpha}$ is not a section with
multiplier. This shows that $\mu_{b},\mu_{b}^{-1}$ are not resonance points.
(Indeed they are the branch points of the multiplier spectral curve of the
unduloids and nodoids with necksize $r\not=\frac{1}{2}$.)

Therefore, we can from now on assume $t\not=0$. We obtain all
$d_{\mu}$–parallel sections, since the matrix has constant coefficients in $y$
and $p^{\prime}-q^{2}=r(1-r)$, by the following fundamental sections

$\alpha_{\pm}=e^{\frac{iy}{2}}\left(qb-
iq^{\prime}(a-1)+j\left(1+p^{\prime}(a-1)\pm
t\right)\right)e^{\pm\frac{ity}{2}}c_{\pm},$

where $c_{\pm}$ are complex valued functions in $x$.

To solve for $c_{\pm}$, we compute $(\alpha_{\pm})_{x}$ in two ways. Since
$c_{\pm}$ only depend on $x$, we can evaluate at $y=0$ and obtain

$\displaystyle(\alpha_{\pm})_{x}$
$\displaystyle=\left(q^{\prime}b-iq^{\prime\prime}(a-1)+jp^{\prime\prime}(a-1)\right)c_{\pm}$
$\displaystyle\quad+\left(qb-iq^{\prime}(a-1)+j\left(1+p^{\prime}(a-1)\pm
t\right)\right)c_{\pm}^{\prime}.$

By (3.1) we also have at $y=0$

$\displaystyle(\alpha_{\pm})_{x}$
$\displaystyle=\frac{1}{2}(f_{y}\alpha_{\pm}(a-1)-f_{x}\alpha_{\pm}b)$
$\displaystyle=\frac{1}{2}\Big{(}-iq\left(1-2p^{\prime}(a-1)\pm
t\right)(a-1)+q^{\prime}\left(1\pm t\right)b$ $\displaystyle\hskip
113.81102pt+j\big{(}p^{\prime\prime}(a-1)+ip^{\prime}(1\pm
t)b\big{)}\Big{)}c_{\pm}\,,$

where we used $a^{2}+b^{2}=1$, $(p^{\prime})^{2}+(q^{\prime})^{2}=q^{2}$ and
$p^{\prime\prime}=2qq^{\prime}$ since $p^{\prime}=q^{2}+r(1-r)$. By comparing
the $\mathbb{C}$ and $j\mathbb{C}$ parts, respectively, we obtain

$\displaystyle(bq-i(a-1)q^{\prime})c_{\pm}^{\prime}$
$\displaystyle=\frac{1}{2}(-i(a-1)q(1\pm t-2p^{\prime}(a-1))-bq^{\prime}(1\mp
t)+2i(a-1)q^{\prime\prime})c_{\pm},$ $\displaystyle(1\pm
t+(a-1)p^{\prime})c_{\pm}^{\prime}$
$\displaystyle=\frac{1}{2}\left(-(a-1)p^{\prime\prime}+i(1\pm
t)bp^{\prime}\right)c_{\pm}\,.$

Up to multiplication by the factor $-\frac{bq+i(a-1)q^{\prime}}{1\mp
t+(a-1)p^{\prime}}$ these two equations are the same. To solve for $c_{\pm}$,
we note that when $1\pm t+(a-1)p^{\prime}\neq 0$, the second equation gives

$\displaystyle\frac{c_{\pm}^{\prime}}{c_{\pm}}$
$\displaystyle=-\frac{1}{2}(\log|1\pm
t+(a-1)p^{\prime}|)^{\prime}+\frac{ib(1\pm t)}{2(a-1)}-\frac{ib(1\pm
t)^{2}}{2(a-1)}\frac{1}{1\pm t+(a-1)p^{\prime}}.$

Integrating both sides, we obtain

$\log|c_{\pm}|=-\frac{1}{2}\log|1\pm t+(a-1)p^{\prime}|+\frac{ib(1\pm
t)}{2(a-1)}x-\frac{ib(1\pm t)^{2}}{2(a-1)}\int\frac{1}{1\pm
t+(a-1)p^{\prime}}\,dx.$

Abbreviating the Jacobi elliptic functions so that, for example,
$\operatorname{dn}((1-r)x,M)=\operatorname{dn}$, we have
$p^{\prime}=q^{2}+r(1-r)$ for $q=(1-r)\operatorname{dn}$. Thus, using
$\operatorname{dn}^{2}=1-M\operatorname{sn}^{2}$, one calculates

$\displaystyle 1\pm t+(a-1)p^{\prime}$ $\displaystyle=(1\pm
t+(a-1)(1-r))(1-N\operatorname{sn}^{2})$

where $N=\frac{(a-1)(1-r)^{2}M}{1\pm t+(a-1)(1-r)}$. Therefore, it remains to
compute $\int\frac{1}{1-N\operatorname{sn}^{2}}\,dx,$ which is well-known to
be the incomplete elliptic integral of the third kind

$\displaystyle\int\frac{1}{1-N\operatorname{sn}^{2}}\,dx$
$\displaystyle=\frac{1}{1-r}\Pi(N;\operatorname{am}((1-r)x,M)\,|\,M)\,,$

where $\operatorname{am}((1-r)x,M)$ is the Jacobi amplitude.

Observing that $\alpha_{\pm}$ are parallel sections, our results extend
smoothly into the zeros of $1\pm t+(a-1)p^{\prime}$. We summarise:

######  Theorem 3.1.

Let $f(x,y)=ip+jqe^{-iy}$ be a Delaunay surface with necksize $r$. For
$\mu\in\mathbb{R}\setminus\\{0,\pm 1\\}$, let $a=\frac{\mu+\mu^{-1}}{2}$ and

$t=\sqrt{1+2r(1-r)(a-1)}\,.$

Then all $d_{\mu}$–parallel sections for $t\not=0$ are explicitly given by

$\alpha=\alpha_{+}m_{+}+\alpha_{-}m_{-}$

with constants $m_{\pm}\in\mathbb{C}$, where

$\alpha_{\pm}=e^{\frac{iy}{2}}\left(qb-
iq^{\prime}(a-1)+j\left(1+p^{\prime}(a-1)\pm
t\right)\right)e^{\pm\frac{ity}{2}}c_{\pm}$

and

$c_{\pm}=\frac{\exp\left(\frac{ib(1\pm t)}{2(a-1)}\left(x-\frac{1\pm t}{1\pm
t+(a-1)(1-r)}\frac{1}{1-r}\Pi(\operatorname{am}((1-r)x,M),M)\right)\right)}{\sqrt{|1\pm
t+(a-1)p^{\prime}|}}\,.$

Here $\Pi$ is the incomplete elliptic integral of the third kind and
$\operatorname{am}$ the Jacobi amplitude.

Since by Theorem 2.2 all CMC Darboux transforms of Delaunay surfaces are given
algebraically by $d_{\mu}$–parallel sections, we obtain also:

######  Theorem 3.2.

The CMC Darboux transforms of a Delaunay surfaces can be expressed explicitly
in terms of Jacobi elliptic functions and the incomplete elliptic integral of
the third kind.

######  Remark 3.3.

The parallel sections obtained here are special cases of the parallel sections
of a surface of revolution in [isothermic_paper, sym-darboux] (after adjusting
the spectral parameter to account for the choice of dual surface): we obtain
exactly those classical Darboux transforms which are CMC.

Now, observe that $\alpha_{\pm}$ are sections with multipliers

$h_{\pm}=-e^{\pm\pi i\sqrt{1+2r(1-r)(a-1)}},$

and therefore, since $c_{\pm}$ are independent of $y$, resonance points have
to satisfy

$1+2r(1-r)(a-1)=n^{2}\,,\quad n\in\mathbb{N}\,.$

Since $\mu\not=1$ and $t\not=0$ by assumption, the cases $n=0$ and $n=1$
cannot occur. Therefore, the (relevant) resonance points are given by

$\mu_{n}^{\pm}=\frac{1}{2r(1-r)}\left(n^{2}-1+2r(1-r)\pm\sqrt{(n^{2}-1)^{2}+4r(1-r)(n^{2}-1)}\right),$

for $n\in\mathbb{N},n>1$.

We also recall that $\mu_{n}^{\pm}$ give rise to the same Darboux transform by
Remark 2.3, allowing us to consider only one of the solutions
$\mu_{n}=\mu_{n}^{+}$.

The (non–trivial) CMC Darboux transforms of a Delaunay surface at resonance
points $\mu_{n}$ are called _bubbletons_. Geometrically, the integer
$n\in\mathbb{N}$ gives the number of lobes of the bubble on the bubbleton (see
Figures 1 and 2).

Figure 1. Bubbletons of the cylinder, that is, $r=\frac{1}{2}$, at resonance
points $\mu_{2}=7+4\sqrt{3},\mu_{3}=17+12\sqrt{2}$.

Figure 2. An unduloid with $r=\frac{1}{5}$ and its bubbletons with $n=2$ and
$n=3$.

Note that we require $\mu_{n}^{\pm}\in\mathbb{R}_{*}$ so that there is an
additional condition on $n$ in the case of a nodoid, see also
[KobayashiBubbletons]:

$\frac{1-n}{2}\leq r<0.$

The case when $n=1-2r$ gives $\mu=-1$ and the corresponding CMC Darboux
transform is the parallel CMC surface, that is, we only obtain bubbletons for
$\frac{1-n}{2}<r$ (see Figure 3). (In the case of the cylinder and unduloids,
all $\mu_{n}^{\pm}\in\mathbb{R}\setminus\\{0,\pm 1\\}$ for $n\in\mathbb{N}$,
$n>1$.)

Figure 3. A nodoid with necksize $r=-\frac{1}{2}$ and a bubbleton with $n=3$:
for $n=2$ the parameter $\mu_{2}=-1$ and there is no classical Darboux
transform with 2 lobes for necksize $r=-\frac{1}{2}$.

Finally, we can use Bianchi permutability to obtain multibubbletons. Given two
parallel sections at distinct resonance points $\mu_{n}\not=\mu_{l}$, that is,
$n\not=l$, we obtain two bubbletons with $n$ and $l$ lobes respectively. The
common Darboux transform $\hat{f}$, a _doublebubbleton_ , of $f_{n},f_{l}$ is
a surface with two bubbles on it, where one bubble has $n$ lobes and the other
one has $l$ lobes (see Figure 4).

Figure 4. Delaunay doublebubbleton as a common Darboux transform of bubbletons
with $n=2$ and $n=3$ lobes where $r=\frac{1}{5}$.

Applying Bianchi permutability one can now add an arbitrary number of bubbles
to the Delaunay surface. Since Bianchi permutability needs distinct spectral
parameter, we obtain CMC multibubbletons with any number of bubbles but every
pair of bubbles has different numbers of lobes.

On the other hand, in a recent paper [sym-darboux] we showed that the double
Darboux transformation of a Delaunay surface which uses the same spectral
parameter twice (using a Sym–type argument) does not give new closed CMC
surfaces: In particular, we cannot obtain same-lobed CMC multibubbletons this
way.

##  4\. New CMC cylinders via the Darboux transformation

In this section we will discuss CMC Darboux transforms of Delaunay surfaces
which close on a multiple cover. This is primarily motivated by our aim to
find same–lobed CMC multibubbletons but also in itself gives rise to new
interesting CMC surfaces.

In the case of a cylinder, Sterling and Wente [wente_sterling] discussed
bubbletons on an $m$–fold cover via the Bianchi–Bäcklund transform. In this
case, the number of lobes $n$ must be bigger than $m$. This naturally extends
the result for the single cover cylinder where $n>1$. We will recover this
result in this section by investigating the multipliers of parallel sections
of a Delaunay surface on a multiple cover. Surprisingly, in the case of
unduloids and nodoids, additional cases appear: in addition to the natural
generalisations of the single cover conditions $n>m$, we also obtain solutions
for $n<m$.

To this end, we will first investigate the resonance points of a multiple
cover of a Delaunay surface, that is, we will consider the surfaces

$f(x,y)=ip(x)+jq(x)e^{-iy}$

on a $m$–fold cover, that is, $x\in\mathbb{R},y\in[0,2m\pi],m\in\mathbb{N}$.
Then the parallel sections for $\mu\in\mathbb{R}\setminus\\{0,1\\}$ are still
given by Theorem 3.1

$\alpha_{\pm}^{m}=e^{\frac{iy}{2}}\left(qb-
iq^{\prime}(a-1)+j\left(1+p^{\prime}(a-1)\pm
t\right)\right)e^{\pm\frac{ity}{2}}c_{\pm}$

for $t=\sqrt{1+2r(1-r)(a-1)}\not=0$.

However, the resonance points and multipliers change since we are now
investigating the parallel sections on an $m$–fold cover. As before excluding
$t=0$, the multipliers of the parallel sections $\alpha_{\pm}^{m}$ are

$h_{\pm}^{m}=(-1)^{m}e^{\pm m\pi it}$

which shows that the resonance points $\mu_{n}^{m}$ have to satisfy

$1+2r(1-r)(a-1)=\frac{n^{2}}{m^{2}}\,,\quad n\in\mathbb{N}_{*},$

that is, the resonance points must be of the form

$\mu_{n}^{m}=\frac{1}{2r(1-r)}\left(\frac{n^{2}-m^{2}}{m^{2}}+2r(1-r)\pm\sqrt{\frac{n^{2}-m^{2}}{m^{2}}\left(\frac{n^{2}-m^{2}}{m^{2}}+4r(1-r)\right)}\right).$

Note that $\alpha_{\pm}^{m}$ are only sections with multipliers if we consider
them as sections on a $m$–fold cover. That is, we cannot construct Darboux
transforms with period $2\pi\tau$ with $\tau>0$ this way if
$\tau\not\in\mathbb{Z}$. Moreover, we can assume without loss of generality
that $(m,n)$ are co–prime since $\mu_{n}^{m}=\mu_{n^{\prime}}^{m^{\prime}}$
for $\frac{n}{m}=\frac{n^{\prime}}{m^{\prime}}$.

Additionally, $(m,n)$ must satisfy the condition

$\frac{n^{2}-m^{2}}{m^{2}}\left(\frac{n^{2}-m^{2}}{m^{2}}+4r(1-r)\right)>0,$

since we consider only real spectral parameter
$\mu\in\mathbb{R}\setminus\\{0,\pm 1\\}$.

Note that by Theorem 2.2 the resulting $\mu$–Darboux transforms for spectral
parameter $\mu_{n}^{m}\in\mathbb{R}\setminus\\{0,\pm 1\\}$ are CMC surfaces in
3–space. Thus we have extended the result for round cylinders in
[wente_sterling] (obtained via Bianchi’s Bäcklund transformation) to general
Delaunay surfaces:

######  Theorem 4.1.

Let $f(x,y)=ip+jqe^{-iy}$ be a Delaunay surface, $r\in(-\infty,\frac{1}{2})$,
$r\not=0$, $M=1-(1-\frac{1}{1-r})^{2}$,
$q(x)=(1-r)\operatorname{dn}((1-r)x,M),p^{\prime}(x)=q^{2}+r(1-r)$. Let
$(m,n)$ with co–prime $m,n\in\mathbb{N}_{*}$, be an _admissible pair_ , that
is,

$\begin{array}[]{@{}r@{\quad\quad}l@{\qquad}l@{}}\text{for
}r<0:&m<n,\,\frac{m-n}{2m}<r&\text{or}\qquad m>n\\\\[5.69046pt] \text{for
}0<r<\frac{1}{2}:&m<n&\text{or}\qquad m>n,\,r<\frac{m-n}{2m}\\\\[5.69046pt]
\text{for }r=\frac{1}{2}:&m<n\,.&\end{array}$

Then every Darboux transform with parameter $\mu_{n}^{m}$ is a closed CMC
surface on the $m$–fold cover of the Delaunay surface. Put differently, for
each admissible pair $(m,n)$ the spectral parameter

$\mu_{n}^{m}=\frac{1}{2r(1-r)}\left(\frac{n^{2}-m^{2}}{m^{2}}+2r(1-r)\pm\sqrt{\frac{n^{2}-m^{2}}{m^{2}}\left(\frac{n^{2}-m^{2}}{m^{2}}+4r(1-r)\right)}\right)$

is a resonance point and $\mu_{n}^{m}\in\mathbb{R}\setminus\\{0,\pm 1\\}$.

Note that [wente_sterling] only considered the case of cylinders via the
Bianchi–Bäcklund transformation, and thus, only considered the case when
$n>m$. However, in the case of unduloids and nodoids, interesting new CMC
cylinders appear for $n<m$. We discuss the three cases separately, assuming
that $(m,n)$ is an admissible pair:

Cylinder. In the case of a cylinder, $r=\frac{1}{2}$, we have
$\mu_{n}^{m}\in\mathbb{R}\setminus\\{0,1\\}$ if and only if $n>m$. The
resulting Darboux transforms have a localised bubble with $n$ lobes and close
on the $m$–fold cover of the Delaunay surface. These are the bubbletons given
in [wente_sterling] (see Figure 5).

Figure 5. Darboux transform of the cylinder with $n=3$ on the 2–fold cover,
next to a bubbleton with $n=3$ on the single cover.

Unduloid. In the case of an unduloid we have $0<r<\frac{1}{2}$ and
$4r(1-r)\in(0,1)$. Thus, for $n>m$ we still have
$\mu_{n}^{m}\in\mathbb{R}_{*}$. In this case, we obtain Darboux transforms
with a localised bubble with $n$ lobes as in the case of cylinders (see Figure
6).

Figure 6. Darboux transform of an unduloid with necksize $r=\frac{1}{5}$,
$n=3$ on the 2–fold cover, next to a bubbleton with $n=3$ on the single cover.

However, for unduloids, we obtain additional solutions which could not be
observed in [wente_sterling]: if $n<m$ and $r<\frac{m-n}{2m}$ we also obtain
closed CMC surfaces (see Figure 7). Note that in this case, the “bubble” of
the surface is less localised, and the number $n$ gives the dihedral symmetry
of the surface rather than the number of lobes.

Figure 7. Darboux transform with $n=2$, $m=7$, of an unduloid with necksize
$r=\frac{1}{5}$.

Nodoid. In the case of a nodoid, we have $r<0$, and as in the case of the
single cover, we obtain an obstruction for the number $n$ of lobes on the
bubbleton: in this case we have a bubbleton with a localised bubble with $n>m$
lobes on the $m$–fold cover if $\frac{m-n}{2m}>r$ (see Figure 8).

Figure 8. Darboux transform of a nodoid with necksize $r=-\frac{1}{8}$, $n=3$
and $m=2$, next to a bubbleton with $n=3$ on the single cover.

In the case of a nodoid, every choice of $n<m$ gives a bubbleton on the
$m$–fold cover. In this case, the number $n$ again gives the dihedral symmetry
of the surface.

Figure 9. Darboux transform of a nodoid with necksize $r=-\frac{1}{8}$, $n=3$
and $m=4$ with two different initial conditions.

In particular, we can also use $n=1$ (see Figure 10):

Figure 10. Darboux transform of a nodoid with necksize $r=-\frac{1}{8}$, $n=1$
and $m=2$.

We will return to a more detailed analysis of these new examples of closed CMC
Darboux transforms of nodoid and unduloids in a future paper.

##  5\. Same–lobed CMC multibubbletons

Changing our point of view, we now fix the number of lobes $n$ and consider
all possible $m$ such that $\mu_{n}^{m}$ is a resonance point. To obtain
same–lobed CMC multibubbletons (with localised bubbles) we restrict here to
the case $n>m$: for such an admissible pair $(m,n)$ we then obtain a bubbleton
with $n$ lobes on an $m$–fold cover, and each $\mu_{n}^{m}\in\mathbb{R}_{*}$
is a resonance point.

Recall that Bianchi permutability requires different spectral parameters
$\mu_{1}\not=\mu_{2}$ to obtain new surfaces as common Darboux transform of
the two surfaces $f_{1}$ and $f_{2}$. Using now resonance points
$\mu_{n}^{m_{1}}\not=\mu_{n}^{m_{2}}$ given by two admissible pairs
$(m_{1},n)$ and $(m_{2},n)$ with $m_{1}\not=m_{2}$, we obtain by Bianchi
permutability a CMC doublebubbleton $\hat{f}$ which has two bubbles with $n$
lobes each. Note that $\hat{f}$ closes on the $L$–fold cover of the Delaunay
surface where $L$ is the least common multiple of $m_{1}$ and $m_{2}$.
Repeating the procedure we obtain further multibubbletons as long as there are
distinct admissible pairs $(m,n)$, $m<n$. Denoting by

$\varphi(n)=\\#\\{m\in\mathbb{N}\mid m,n\,\text{co--prime},\,1\leq
m<n,\,\frac{m-n}{2m}<r\\},$

the number of admissible pairs with $m<n$, (for cylinders and unduloids, this
is just Euler’s totient function $\varphi$ of $n$), we obtain our final result
(see also Figure 11).

######  Theorem 5.1.

Let $f$ be a Delaunay surface with necksize $r$ and denote by $\varphi(n)$ the
number of admissible pairs $(m,n)$ with $m<n$. Then for $1\leq
l\leq\varphi(n)$ there exists a CMC multibubbleton with $l$ bubbles, each with
$n$ lobes, which closes on a multiple cover of the Delaunay surface.

Figure 11. CMC same–lobed multibubbletons of a cylinder and unduloid given via
Bianchi permutability: here $n=5$, $\varphi(5)=4$ and all available resonance
points $\mu_{5}^{1},\mu_{5}^{2},\mu_{5}^{3}$ and $\mu_{5}^{4}$ are used. The
multibubbletons close on the 12–fold cover and have 4 bubbles with 5 lobes
each.

## References


# A Note on the Kaluza-Klein Theory

Li-Xin Li<EMAIL_ADDRESS><EMAIL_ADDRESS>Kavli Institute for
Astronomy and Astrophysics, Peking University, Beijing 100871, P. R. China

###### Abstract

We show that the Kaluza-Klein theory contains a fundamental problem: The four-
dimensional metric tensor and the electromagnetic potential vector assumed in
the Kaluza-Klein theory belong to four-dimensional vector spaces that are not
integrable in general, resulting that the four-dimensional physical variables
and the corresponding field equations derived from the five-dimensional
Einstein field equation (i.e., the four-dimensional Einstein field equation
and the Maxwell equations) are not defined on a four-dimensional submanifold.
That is, the four-dimensional spacetime assumed in the Kaluza-Klein theory
does not exist. No satisfactory solutions are found within the Kaluza-Klein
formalism. Perhaps the best approach to fix the problem is giving up the
Kaluza-Klein theory and looking for a new unified scheme for gravitational and
electromagnetic interactions in the framework of a spacetime with extra
dimensions, as having already been explored in some literature.

##  I Introduction

The Kaluza-Klein (KK) theory represents the first attempt to unify the
gravitational and electromagnetic interactions in the framework of general
relativity extended to a spacetime with extra dimensions (Kaluza, 1921; Klein,
1926a, b).111An even earlier attempt to unify electromagnetic and
gravitational fields in a five-dimensional spacetime before the appearance of
general relativity was given by Nordström in 1914 Nordström (1914). In the KK
theory, the bulk spacetime is assumed to be five-dimensional and described by
the five-dimensional Einstein field equation. By 4+1 decomposition of the
five-dimensional spacetime metric, a four-dimensional Einstein field equation
and the Maxwell equations are derived from the five-dimensional Einstein field
equation, which are assumed to describe the four-dimensional world where we
live. How to make the extra dimension compact, small, and static has been a
challenging problem in modern theoretical physics (Cremmer & Scherk, 1976,
1977; Luciani, 1978; Englert, 1982; Li & Gott, 1998). Nowadays, introducing
(compact or noncompact but warped) extra dimensions in addition to the four
dimensions of the spacetime where we live has been a popular strategy for
unifying all fundamental interactions in nature, e.g., the theories of
supergravity (van Nieuwenhuizen, 1981; Duff, Nilsson, & Pope, 1986),
superstring (Green, Schwarz, & Witten, 2012; Polchinski, 2003; Becker, Becker,
& Schwarz, 2007; Kiritsis, 2019), and brane gravity (Randall & Sundrum, 1999a,
1999; Maartens & Koyama, 2010). To test the existence of extra dimensions, the
KK particles arising from the excitation of fields along the compact extra
dimensions have been extensively searched by the Large Hadron Collider (Nath &
Yamaguchi, 1999; Bhattacharyya et al., 2009; Deutschmann et al., 2017).

Despite the success of the KK theory in derivation of the Maxwell equations
from the higher-dimensional Einstein field equation and its heavy influence on
modern theories of unification, in this paper we show that the KK theory has a
serious problem in its foundation: The four-dimensional metric tensor and the
electromagnetic potential vector assumed in the KK theory are defined in
vector spaces that are not integrable hence not tangent to any four-
dimensional submanifold, unless the electromagnetic field antisymmetric tensor
vanishes. That is, the four-dimensional spacetime assumed in the KK theory to
support the effective four-dimensional theory of electromagnetism and gravity
is defined only when the electromagnetic field vanishes, which conflicts the
original aim of the KK theory in unifying the gravitational and
electromagnetic interactions.

The influence of the problem just mentioned may not be limited to the KK
theory. As is well known, one of the cornerstones of string theory—extra
dimensions and compactification of extra dimensions—originated from the KK
theory with an extension from one extra dimension to multiple extra
dimensions. In string theory, a popular approach to derive gauge fields from
higher-dimensional gravity is through the KK mechanism with an extension to
spacetime of dimensions greater than five. For example, this is the case in
the eleven-dimensional supergravity when it is connected to the low-energy
limit of M-theory (Polchinski, 2003; Becker, Becker, & Schwarz, 2007;
Kiritsis, 2019).

The paper is organized as follows. In section II, we outline the KK theory and
derive the representation of the four-dimensional metric tensor and the
electromagnetic potential vector in the five-dimensional spacetime. The
representation is uniquely determined by the self-consistency requirement of
the theory. In section III, we discuss the geometric interpretation of the
above two KK variables and quantities derived from them (e.g., the
electromagnetic field antisymmetric tensor). We show that the four-dimensional
quantities are in vector spaces orthogonal to the direction of the extra
dimension. In section IV, we prove that the vector spaces containing the four-
dimensional variables are not integrable unless the electromagnetic field
tensor vanishes. Thus, in general there does not exist a four-dimensional
submanifold supporting the four-dimensional theory derived from the five-
dimensional Einstein field equation.

Section V is devoted to discussion on the action principle and
compactification of the extra dimension under the assumption of the cylinder
condition. We show that after compactification, although the four-dimensional
Einstein field equation and the Maxwell equations can be derived from the
action principle, there still does not exist a four-dimensional submanifold
supporting the four-dimensional field equations. Finally, in section VI, we
summarize the results that we have obtained in this paper and discuss their
implications.

Throughout the paper geometrized units with $G=c=1$ are adopted unless
otherwise stated, where $G$ is the four-dimensional gravitational constant and
$c$ is the speed of light. In addition, we will take $(-,+,+,+,+)$ as the
convention for the signature of the five-dimensional spacetime metric. The
abstract index notation for vectors and tensors advocated in Wald (1984) will
be used. That is, vectors and tensors are denoted by letters followed by lower
case Latin indices, e.g., $v^{a}$, $g_{ab}$, etc.

##  II The Kaluza-Klein Formalism

The success of the KK theory relies on a specific decomposition scheme of the
metric tensor of a five-dimensional bulk spacetime. Without loss of
generality, in a five-dimensional spacetime $(\tilde{\cal M},\tilde{g}_{ab})$
we take a coordinate system $\\{x^{0},x^{1},x^{2},x^{3},x^{4}=w\\}$ and write
the matrix representation of the five-dimensional metric tensor
$\tilde{g}_{ab}$ as222The form of metric decomposition in equation (3) agrees
with the general case for the KK theory generalized to a $(4+n)$-dimensional
spacetime to include non-Abelian gauge fields, where the $\phi^{2}$ is
replaced by a matrix $g_{ij}$ with the indices $i$ and $j$ running from $1$ to
$n$ for the extra dimensions (de Witt, 1964; Kerner, 1987; Cho & Freund,
1975).

$\displaystyle\tilde{g}_{AB}=\left(\begin{array}[]{cc}g_{\mu\nu}+\phi^{2}A_{\mu}A_{\nu}&\phi^{2}A_{\mu}\\\\[2.84526pt]
\phi^{2}A_{\nu}&\phi^{2}\end{array}\right)\;,$ (3)

where indices $A,B=0,1,2,3,4$, and $\mu,\nu=0,1,2,3$. Capital Latin letters
label coordinate components of five-dimensional vectors and tensors. Lower
case Greek letters label coordinate components of four-dimensional vectors and
tensors.

The $4\times 4$ matrix $g_{\mu\nu}$ is interpreted as the component of the
metric on a four-dimensional spacetime $({\cal M},g_{ab}$) associated with the
coordinate system $\\{x^{0},x^{1},x^{2},x^{3}\\}$, the $4\times 1$ matrix
$A_{\mu}$ as the component of an electromagnetic potential dual vector, and
the function $\phi$ as a scalar field in $({\cal M},g_{ab})$. With the
convention in equation (3), the five-dimensional spacetime metric tensor
$\tilde{g}_{ab}$ is represented in the coordinate system $\\{x^{\mu},w\\}$ as

$\displaystyle\tilde{g}_{ab}$ $\displaystyle=$
$\displaystyle\tilde{g}_{AB}dx^{A}_{a}dx^{B}_{b}=(g_{\mu\nu}+\phi^{2}A_{\mu}A_{\nu})dx^{\mu}_{a}dx^{\nu}_{b}$
(4)
$\displaystyle+2\phi^{2}A_{\mu}dx^{\mu}_{(a}dw_{b)}+\phi^{2}dw_{a}dw_{b}\;,$

where the parentheses in the indices of a tensor denote symmetrization of the
tensor about the indices inside the parentheses. The Einstein summation
convention for tensor components is used, i.e., an index appearing in both
subscripts and superscripts is summed over all dimensions represented by the
index.

The inverse of the $5\times 5$ matrix in equation (3), which is also the
component matrix of the inverse five-dimensional metric tensor
$\tilde{g}^{ab}$, is

$\displaystyle\tilde{g}^{AB}=\left(\begin{array}[]{cc}g^{\mu\nu}&-A^{\mu}\\\\[2.84526pt]
-A^{\nu}&\phi^{-2}+A_{\rho}A^{\rho}\end{array}\right)\;,$ (7)

where the $4\times 4$ matrix $g^{\mu\nu}$ is the inverse of $g_{\mu\nu}$,
i.e.,

$\displaystyle g_{\mu\nu}g^{\nu\rho}=\delta_{\mu}^{\;\;\rho}$ (8)

where $\delta_{\mu}^{\;\;\nu}=1$ if $\mu=\nu$ and $0$ otherwise; and

$\displaystyle A^{\mu}\equiv g^{\mu\nu}A_{\nu}\;.$ (9)

Equations (8) and (9) automatically imply

$\displaystyle A_{\mu}=g_{\mu\nu}A^{\nu}\;.$ (10)

By equation (7), the inverse five-dimensional metric tensor is represented as

$\displaystyle\tilde{g}^{ab}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{a}\left(\frac{\partial}{\partial
x^{\nu}}\right)^{b}-2A^{\mu}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{(a}\left(\frac{\partial}{\partial w}\right)^{b)}$ (11)
$\displaystyle+\left(\frac{1}{\phi^{2}}+A_{\rho}A^{\rho}\right)\left(\frac{\partial}{\partial
w}\right)^{a}\left(\frac{\partial}{\partial w}\right)^{b}\;.$

It can be verified that the following reciprocal relation is satisfied

$\displaystyle\tilde{g}_{ab}\tilde{g}^{bc}=\tilde{\delta}_{a}^{\;\;c}\equiv\delta_{\mu}^{\;\;\nu}dx^{\mu}_{a}\left(\frac{\partial}{\partial
x^{\nu}}\right)^{c}+dw_{a}\left(\frac{\partial}{\partial w}\right)^{c}\;,$
(12)

where $\tilde{\delta}_{a}^{\;\;c}$ is the identity map in the five-dimensional
spacetime.

In fact, the $5\times 5$ matrices $\tilde{g}_{AB}$ and $\tilde{g}^{AB}$ in
equations (3) and (7) are inverse to each other if and only if (a) the
$4\times 4$ matrices $g^{\mu\nu}$ and $g_{\mu\nu}$ are inverse to each other
(eq. 8), and (b) the $A^{\mu}$ and $A_{\mu}$ are related by equations (9) and
(10). It should be noted that equations (8), (9), and (10) are not
independent, since any two of them can give rise to the other.

The KK 4-metric tensor $g_{ab}$ and the electromagnetic potential dual
4-vector $A_{a}$ are, respectively, a tensor and a vector in the five-
dimensional spacetime $(\tilde{\cal M},\tilde{g}_{ab})$. The question is how
they are expressed in coordinate components in the five-dimensional coordinate
system $\\{x^{\mu},w\\}$. Since $g_{\mu\nu}$ are interpreted as the components
of the four-dimensional metric in coordinates $\\{x^{\mu}\\}$, the $\mu$-$\nu$
components of $g_{ab}$ must be $g_{\mu\nu}$. Then, the general form of
$g_{ab}$ must be

$\displaystyle g_{ab}=g_{\mu\nu}dx^{\mu}_{a}dx^{\nu}_{b}+2g_{\mu
4}dx^{\mu}_{(a}dw_{b)}+g_{44}dw_{a}dw_{b}\;,$ (13)

where $g_{\mu 4}$ and $g_{44}$ are to be determined. Similarly, since
$A_{\mu}$ is interpreted as the coordinate component of $A_{a}$ in
$\\{x^{\mu}\\}$, and $A^{\mu}=g^{\mu\nu}A_{\nu}$ as the coordinate component
of $A^{a}$, we must have

$\displaystyle A_{a}=A_{\mu}dx^{\mu}_{a}+A_{4}dw_{a}$ (14)

and

$\displaystyle A^{a}=A^{\mu}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{a}+A^{4}\left(\frac{\partial}{\partial w}\right)^{a}\;,$ (15)

where $A_{4}$ and $A^{4}$ are to be determined.

By equations (11) and (14), we have

$\displaystyle A^{a}$ $\displaystyle=$
$\displaystyle\tilde{g}^{ab}A_{b}=\left(g^{\mu\nu}A_{\nu}-A^{\mu}A_{4}\right)\left(\frac{\partial}{\partial
x^{\mu}}\right)^{a}$ (16)
$\displaystyle-\left[A_{\rho}A^{\rho}-\left(\frac{1}{\phi^{2}}+A_{\rho}A^{\rho}\right)A_{4}\right]\left(\frac{\partial}{\partial
w}\right)^{a}\;.\hskip 8.5359pt$

By equation (9), comparison of equation (16) to equation (15) leads to

$\displaystyle A_{4}=0\;,\hskip 28.45274ptA^{4}=-A_{\rho}A^{\rho}\;.$ (17)

Thus, we must have

$\displaystyle A_{a}=A_{\mu}dx^{\mu}_{a}\;,$ (18)

and

$\displaystyle A^{a}=A^{\mu}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{a}-A_{\rho}A^{\rho}\left(\frac{\partial}{\partial
w}\right)^{a}\;.$ (19)

By equations (13) and (15), we have

$\displaystyle g_{ab}A^{b}=\left(A_{\mu}+g_{\mu
4}A^{4}\right)dx^{\mu}_{a}+\left(g_{\mu
4}A^{\mu}+g_{44}A^{4}\right)dw_{a}\hskip 4.26773pt$ (20)

after submission of equation (10). Since $g_{ab}$ and $A^{a}$ are interpreted
as, respectively, the metric tensor and the electromagnetic potential vector
in a four-dimensional spacetime, we must have
$g_{ab}A^{b}=\tilde{g}_{ab}A^{b}=A_{a}$. Then, comparison of equation (20) to
equation (18) leads to

$\displaystyle g_{\mu 4}=g_{44}=0\;.$ (21)

Thus, in coordinates $\\{x^{\mu},w\\}$ the 4-metric tensor $g_{ab}$ is
represented as

$\displaystyle g_{ab}=g_{\mu\nu}dx^{\mu}_{a}dx^{\nu}_{b}\;.$ (22)

Then, by $g^{ab}=\tilde{g}^{ac}\tilde{g}^{bd}g_{cd}$ we get the inverse
4-metric tensor

$\displaystyle g^{ab}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{a}\left(\frac{\partial}{\partial
x^{\nu}}\right)^{b}-2A^{\mu}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{(a}\left(\frac{\partial}{\partial w}\right)^{b)}$ (23)
$\displaystyle+A^{\rho}A_{\rho}\left(\frac{\partial}{\partial
w}\right)^{a}\left(\frac{\partial}{\partial w}\right)^{b}\;.$

From equations (22) and (23) we get

$\displaystyle g_{a}^{\;\;c}$ $\displaystyle=$ $\displaystyle
g_{ab}g^{bc}=\delta_{\mu}^{\;\;\nu}dx^{\mu}_{a}\left(\frac{\partial}{\partial
x^{\nu}}\right)^{c}-A_{\mu}dx^{\mu}_{a}\left(\frac{\partial}{\partial
w}\right)^{c}$ (24) $\displaystyle=$ $\displaystyle g^{c}_{\;\;a}\;,$

just as being expected.

Therefore, given the 4+1 decomposition of the five-dimensional metric in
equation (3), the four-dimensional metric tensor $g_{ab}$ and the
electromagnetic potential vector $A^{a}$ are uniquely determined by equations
(22) and (19). They are determined by the assumed form of the five-dimensional
metric and the self-consistency requirement of the theory, without additional
assumptions.

##  III Geometric Interpretation of the Kaluza-Klein Variables

For the KK theory to be meaningful, the 4-metric tensor $g_{ab}$ and the
electromagnetic potential 4-vector $A_{a}$ assumed in the KK theory, and the
four-dimensional quantities derived from them (e.g., the four-dimensional
Ricci tensor $R_{ab}$ and the electromagnetic field antisymmetric tensor
$F_{ab}$) must be defined on some four-dimensional manifold—or a four-
dimensional submanifold embedded in the five-dimensional manifold $\tilde{\cal
M}$. Such a submanifold ${\cal M}$ should be a hypersurface in $\tilde{\cal
M}$, since $\dim{\cal M}=4=\dim{\cal\tilde{M}}-1$. Assuming that such a
hypersurface has a unit normal $n^{a}$, which must be spacelike since $({\cal
M},g_{ab})$ is supposed to be a four-dimensional spacetime. That is, all
vectors in a vector space tangent to ${\cal M}$ are orthogonal to $n^{a}$, and
$\tilde{g}_{ab}n^{a}n^{b}=n^{a}n_{a}=1$. Then, the 4-metric $g_{ab}$ on ${\cal
M}$ must be related to the 5-metric $\tilde{g}_{ab}$ on $\tilde{\cal M}$ by
$g_{ab}=\tilde{g}_{ab}-n_{a}n_{b}$, or, equivalently,

$\displaystyle g^{ab}=\tilde{g}^{ab}-n^{a}n^{b}\;.$ (25)

The questions are: does such a hypersurface exist? If yes, how is it defined?

It appears that neither of the above two questions has been seriously
considered in the literature, at least to the knowledge of the present author.
In his original paper (Kaluza, 1921) (English translation in Appelquist et al.
(1987), page 61), Kaluza only wrote that “we are certainly free to consider
our space-time to be a four-dimensional part of an $R_{5}$”. Kaluza used
$R_{5}$ to denote a five-dimensional spacetime. In (Klein, 1926a) (English
translation in Appelquist et al. (1987), page 76), Klein only stated that
“four of the coordinates, $x^{1}$, $x^{2}$, $x^{3}$, $x^{4}$, say, are to
characterize the usual space-time.” (Klein’s $x^{1}$, $x^{2}$, $x^{3}$,
$x^{4}$ are equivalent to our $x^{0}$, $x^{1}$, $x^{2}$, $x^{3}$ respectively,
and his $x^{0}$ corresponds to our $w$ coordinate.) How is “a four-dimensional
part of an $R_{5}$” defined? What is the exact meaning of “the usual space-
time” in mathematics? These questions have never been clearly answered.

In some references the authors have explicitly identified the four-dimensional
spacetime with the hypersurface defined by $w=\mbox{const}$ without any proof
or argument. For example, in (Einstein & Bergmann, 1938) Einstein and Bergmann
wrote that “We consider a four dimensional surface cutting each of the
$A$-lines once and only once. We introduce on this surface 4 coördinates
$x^{a}(a=1...4)$ and assume $x^{0}$ equal zero on this surface.” (Section I of
(Einstein & Bergmann, 1938), subsection “The Special Coördinate System”). Note
that their coordinate $x^{0}$ corresponds to our $w$, and their $x^{a}$
correspond to our ${x^{\mu}}(\mu=0,1,2,3)$. Their “$A$-lines” correspond to
our $w$-lines. To distinguish it from the electromagnetic potential vector
$A^{a}$ used in this paper, let us denote the 5-vector “$A$” used in (Einstein
& Bergmann, 1938) by $\tilde{A}^{a}$. In our notations,
$\tilde{A}^{a}=\phi^{-2}(\partial/\partial w)^{a}=\phi^{-1}n^{a}$, which is
related to the 4-vector potential $A^{a}$ by $\tilde{A}_{a}=A_{a}+dw_{a}$ (see
eq. 37 below, Einstein and Bergmann chose $\phi=1$ so that
$\tilde{g}_{ab}\tilde{A}^{a}\tilde{A}^{b}=1$). Similarly, in (Thiry, 1948)
(English translation in (Appelquist et al., 1987), page 108), Thiry wrote that
“Kaluza’s attempt at a unified theory consists of considering space-time as
the $x^{0}=\mbox{const.}$ subspace of a five-dimensional Riemann space, and of
assuming this subspace cylindrical with respect to the fifth coordinate
$x^{0}$.” (Thiry’s $x^{0}$ is equal to our $w$ coordinate.)

There are also people taking different views. For example, Coquereaux &
Esposito-Farese (1990) interpreted the four-dimensional spacetime as a
hypersurface orthogonal to the $w$-lines by stating that “Locally, the
4-dimensional space orthogonal to this vector will be interpreted as the usual
space-time”. Their “this vector” corresponds to our
$n^{a}=\phi^{-1}(\partial/\partial w)^{a}$, i.e., the vector $n^{a}$ in
equation (37) below. However, Coquereaux & Esposito-Farese did not provide
evidence supporting their views. They did not even consider whether “the
4-dimensional space orthogonal to this vector” exists or not. As will be shown
later in this paper, such a hypersurface does not exist in general.

The view that the submanifold supporting the four-dimensional variables in the
KK theory coincides with the hypersurface defined by $w=\mbox{const}$ was
disproved in (Li, 2016). Let us use ${\cal S}$ to denote the hypersurface
defined by $w=\mbox{const}$, and write its unit normal as $s^{a}$. The $s^{a}$
is defined by $\tilde{g}_{ab}s^{a}s^{b}=1$ and
$\tilde{g}_{ab}s^{a}(\partial/\partial x^{\mu})^{a}=0$ for $\mu=0...3$. Thus
we must have

$\displaystyle
s_{a}=\left(\tilde{g}^{44}\right)^{-1/2}dw_{a}=\frac{1}{\sqrt{\phi^{-2}+A_{\rho}A^{\rho}}}dw_{a}\;.$
(26)

By equations (19) and (23), we have

$\displaystyle
A^{a}s_{a}=-\left(\tilde{g}^{44}\right)^{-1/2}A_{\rho}A^{\rho}=-\left(\tilde{g}^{44}\right)^{-1/2}A_{a}A^{a}\;,\hskip
8.5359pt$ (27)

and

$\displaystyle g^{ab}s_{b}$ $\displaystyle=$
$\displaystyle\left(\tilde{g}^{44}\right)^{-1/2}\left[-A^{\mu}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{a}+A^{\rho}A_{\rho}\left(\frac{\partial}{\partial
w}\right)^{a}\right]$ (28) $\displaystyle=$
$\displaystyle-\left(\tilde{g}^{44}\right)^{-1/2}A^{a}\;.$

Hence, the four-dimensional variables $g_{ab}$ and $A^{a}$ are not orthogonal
to $s^{a}$ unless $A_{a}=0$.

By equation (4) we have the metric tensor on ${\cal S}(w=0)$

$\displaystyle\hat{g}_{ab}$ $\displaystyle=$
$\displaystyle\tilde{g}_{ab}-s_{a}s_{b}=\left(g_{\mu\nu}+\phi^{2}A_{\mu}A_{\nu}\right)dx^{\mu}_{a}dx^{\nu}_{b}$
(29)
$\displaystyle+2\phi^{2}A_{\mu}dx^{\mu}_{(a}dw_{b)}+\frac{\phi^{4}A_{\rho}A^{\rho}}{1+\phi^{2}A_{\rho}A^{\rho}}dw_{a}w_{b}\;.$

After restriction of the action of $\hat{g}_{ab}$ on vectors tangent to ${\cal
S}$, we get the metric tensor on ${\cal S}$

$\displaystyle\hat{g}_{ab}=\left(g_{\mu\nu}+\phi^{2}A_{\mu}A_{\nu}\right)dx^{\mu}_{a}dx^{\nu}_{b}\;.$
(30)

The four-dimensional components of $\hat{g}_{ab}$ are not $g_{\mu\nu}$, but
$g_{\mu\nu}+\phi^{2}A_{\mu}A_{\nu}$. Hence, the KK theory can be treated as
approximately being defined on ${\cal S}$ only if the electromagnetic field is
sufficiently weak so that $|\phi^{2}A_{\mu}A_{\nu}|\ll|g_{\mu\nu}|\sim 1$,
i.e., only if

$\displaystyle|\phi A_{\mu}|\ll A_{\rm
crit}\equiv\frac{c^{2}}{G^{1/2}}=3.48\times 10^{24}\,{\rm statvolt}\;.$ (31)

If $|\phi A_{\mu}|\gtrsim A_{\rm crit}$, the $g_{ab}$ and $A_{a}$ are not
orthogonal to $s^{a}$ and the KK theory cannot be treated as being defined on
the hypersurface defined by $w=0$.

To find a submanifold ${\cal M}$ supporting the KK four-dimensional variables,
we submit equations (11) and (23) into equation (25). We get

$\displaystyle n^{a}n^{b}=\frac{1}{\phi^{2}}\left(\frac{\partial}{\partial
w}\right)^{a}\left(\frac{\partial}{\partial w}\right)^{b}\;,$ (32)

which immediately leads to a unique solution (up to a sign)

$\displaystyle n^{a}=\phi^{-1}w^{a}\;,\hskip
22.76228ptw^{a}\equiv\left(\frac{\partial}{\partial w}\right)^{a}\;.$ (33)

Thus, if the submanifold ${\cal M}$ exists, its unit normal $n^{a}$ must be a
unit vector tangent to the coordinate lines of the extra dimension. It is easy
to verify that both $g_{ab}$ and $A^{a}$ are orthogonal to $n^{a}$.

Let us denote the tangent space of the five-dimensional manifold $\tilde{M}$
at a point $p\in\tilde{M}$ by $\tilde{\cal T}_{p}$, $\dim\tilde{\cal
T}_{p}=\dim\tilde{\cal M}=5$. The disjoint union of $\tilde{\cal T}_{p}$ at
all points of $\tilde{M}$ is called the tangent bundle of $\tilde{\cal M}$ and
denoted as $\tilde{\cal T}=\tilde{\cal T}(\tilde{\cal M})$. Let ${\cal T}$ be
a rank-4 subbundle of $\tilde{\cal T}$ (called a rank-4 distribution or
tangent distribution, or tangent subbundle (Lee, 2012)), described by a
disjoint union of subspaces containing all vectors and tensors orthogonal to
$w^{a}\propto n^{a}$ at all points of $\tilde{\cal M}$, i.e., ${\cal
T}=\coprod_{p\in\tilde{\cal M}}{\cal T}_{p}$ with $\dim{\cal T}_{p}=4$. The
${\cal T}$ is a smooth distribution in the sense that for each
$p\in\tilde{\cal M}$ we can find an open neighborhood $\tilde{\cal O}$ of $p$
such that in $\tilde{\cal O}$, ${\cal T}$ is spanned by smooth vector and
tensor fields orthogonal to $w^{a}$. We have $g_{ab}$, $g_{a}^{\;\;b}$,
$g^{ab}\in{\cal T}$. The $g_{ab}$ is the metric tensor field in ${\cal T}$.
The $g^{ab}$ is the inverse metric tensor field, and $g_{a}^{\;\;b}$ the
identity map in ${\cal T}$.333The
$g_{a}^{\;\;b}=\tilde{g}_{a}^{\;\;b}-n_{a}n^{b}$ is also the projection
operator mapping a vector (and a tensor) in $\tilde{\cal T}$ to a vector (and
a tensor) in ${\cal T}$. Since $A_{a}w^{a}=0=A^{a}w_{a}$, we have $A_{a}$,
$A^{a}\in{\cal T}$, too.

By equations (4) and (18), we have

$\displaystyle\phi^{2}A_{a}=\phi^{2}A_{\mu}dx^{\mu}_{a}=\tilde{g}_{bc}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{b}\left(\frac{\partial}{\partial
w}\right)^{c}dx^{\mu}_{a}\;.$ (34)

By equation (12), we have

$\displaystyle dx^{\mu}_{a}\left(\frac{\partial}{\partial
x^{\mu}}\right)^{b}=\tilde{\delta}_{a}^{\;\;b}-dw_{a}\left(\frac{\partial}{\partial
w}\right)^{b}\;.$ (35)

Hence, we get

$\displaystyle\phi^{2}A_{a}$ $\displaystyle=$
$\displaystyle\tilde{g}_{bc}\left(\frac{\partial}{\partial
w}\right)^{c}\left[\tilde{\delta}_{a}^{\;\;b}-dw_{a}\left(\frac{\partial}{\partial
w}\right)^{b}\right]$ (36) $\displaystyle=$
$\displaystyle\tilde{g}_{ac}\left(\frac{\partial}{\partial
w}\right)^{c}-\phi^{2}dw_{a}\;,$

where we have used $\tilde{g}_{bc}(\partial/\partial w)^{b}(\partial/\partial
w)^{c}=\tilde{g}_{44}=\phi^{2}$. Submitting the definition of $n^{a}$ in
equation (33) into equation (36), we get the relation

$\displaystyle A_{a}=\phi^{-1}n_{a}-dw_{a}\;.$ (37)

Equation (37) states that $A^{a}$ is obtained from orthogonal decomposition of
the vector field $\mu^{a}\equiv\tilde{g}^{ab}dw_{b}=(g^{44})^{1/2}s^{a}$: one
component is along the direction of $w^{a}$ (i.e., the direction of $n^{a}$ by
eq. 33), the other (which is $\propto-A^{a}$) is in the direction orthogonal
to $w^{a}$.

By $\tilde{\nabla}_{[a}dw_{b]}=0$, where the brackets in the indices of a
tensor denote antisymmetrization of the tensor about the indices inside the
brackets, equation (37) implies that
$\tilde{\nabla}_{[a}A_{b]}=\tilde{\nabla}_{[a}\left(\phi^{-1}n_{b]}\right)$,
i.e.,

$\displaystyle\tilde{\nabla}_{[a}A_{b]}=n_{[b}\tilde{\nabla}_{a]}\phi^{-1}+\phi^{-1}\tilde{\nabla}_{[a}n_{b]}\;.$
(38)

Although $\tilde{\nabla}_{a}$ is the derivative operator associated with the
metric $\tilde{g}_{ab}$, equation (38) remains valid if $\tilde{\nabla}_{a}$
is replaced by any derivative operator. By equation (38), we have the
antisymmetric tensor of the electromagnetic field

$\displaystyle F_{ab}\equiv
2g_{a}^{\;\;c}g_{b}^{\;\;d}\tilde{\nabla}_{[c}A_{d]}=2\phi^{-1}g_{a}^{\;\;c}g_{b}^{\;\;d}\tilde{\nabla}_{[c}n_{d]}\;,$
(39)

where we have used the identity $g_{a}^{\;\;b}n_{b}=0$. By definition,
$F_{ab}\in{\cal T}$ since $F_{ab}w^{b}=0=F_{ab}w^{a}$.

If $w^{a}$ were timelike and $n_{a}n^{a}=-1$, the quantity

$\displaystyle\omega_{ab}\equiv
g_{a}^{\;\;c}g_{b}^{\;\;d}\tilde{\nabla}_{[d}n_{c]}$ (40)

would be called the vorticity tensor of the congruence of the integral curves
of the vector field $w^{a}$ (Hawking & Ellis, 1973). Similarly, the expansion
tensor would be defined by

$\displaystyle\theta_{ab}\equiv
g_{a}^{\;\;c}g_{b}^{\;\;d}\tilde{\nabla}_{(c}n_{d)}=\frac{1}{2}\tilde{\pounds}_{n}g_{ab}\;,$
(41)

where $\tilde{\pounds}_{n}$ is the Lie derivative generated by $n^{a}$ (as
usual, the tilde above $\pounds_{n}$ indicates that the operation is in the
five-dimensional spacetime). Now $w^{a}$ is spacelike and $n_{a}n^{a}=1$. We
can still define the $\omega_{ab}$ and $\theta_{ab}$ by equations (40) and
(41) and call them the vorticity tensor and the expansion tensor,
respectively, of the integral curves of $w^{a}$. Both $\omega_{ab}$ and
$\theta_{ab}$ are $\in{\cal T}$, since they both are orthogonal to $w^{a}$.
Then, equation (39) is equivalent to

$\displaystyle F_{ab}=-2\phi^{-1}\omega_{ab}\;.$ (42)

Therefore, the electromagnetic field antisymmetric tensor in the KK theory is
proportional to the vorticity tensor of the congruence of the extra dimension
coordinate curves.

##  IV Submanifold Supporting the Effective Four-Dimensional Theory does not
Exist

As shown in section III, the KK four-dimensional field variables $g_{ab}$ and
$A_{a}$ are contained in a rank-4 distribution ${\cal T}$ with members
orthogonal to the extra dimension coordinate vector field $w^{a}$. In order
that there exists an integral submanifold of ${\cal T}$, i.e., a submanifold
${\cal M}\subseteq\tilde{\cal M}$ whose tangent space at any $p\in{\cal M}$
coincides with ${\cal T}_{p}$, the distribution ${\cal T}$ must be involutive,
i.e., for any $X^{a},Y^{a}\in{\cal T}$ the commutator (Lie bracket)
$[X,Y]^{a}\in{\cal T}$ (Frobenius’s theorem, (Lee, 2012)). The condition that
the tangent subbundle ${\cal T}$ is integrable is equivalent to that the
vector field $w^{a}$ is hypersurface orthogonal, i.e., there exists a
hypersurface orthogonal to $w^{a}$.

By Frobenius’s theorem, $w^{a}$ is hypersurface orthogonal if and only if
$[X,Y]^{a}\in{\cal T}$ for all $X^{a},Y^{a}\in{\cal T}$, which is
mathematically equivalent to the condition that there is a dual vector $v_{a}$
such that (Wald, 1984; Straumann, 2013)

$\displaystyle\tilde{\nabla}_{[a}w_{b]}=w_{[a}v_{b]}\;.$ (43)

Equation (43) is equivalent to the condition that the congruence of the
spacelike curves tangent to $w^{a}$ is vorticity-free, which is proved as
follows. Since $w^{a}=\phi n^{a}$, equation (43) is equivalent to

$\displaystyle\tilde{\nabla}_{[a}n_{b]}=n_{[a}v^{\prime}_{b]}\;,$ (44)

where $v^{\prime}_{a}=v_{a}+\tilde{\nabla}_{a}\ln\phi$. Then we have

$\displaystyle
g_{a}^{\;\;c}g_{b}^{\;\;d}\tilde{\nabla}_{[c}n_{d]}=g_{a}^{\;\;c}g_{b}^{\;\;d}n_{[c}v^{\prime}_{d]}=0\;,$
(45)

since $g_{a}^{\;\;c}n_{c}=0$. By equation (40) we get then

$\displaystyle\omega_{ab}=0\;.$ (46)

Equation (44) can also be derived from equation (46), since by
$\tilde{\nabla}_{a}n_{b}=\theta_{ab}+\omega_{ba}+n_{a}a_{b}$ we have

$\displaystyle\tilde{\nabla}_{[a}n_{b]}=\omega_{ba}+n_{[a}a_{b]}\;,$ (47)

where $a_{a}\equiv n^{b}\tilde{\nabla}_{b}n_{a}$. Thus, $\omega_{ab}=0$
implies $\tilde{\nabla}_{[a}n_{b]}=n_{[a}a_{b]}$, i.e., equation (44) if we
take $v^{\prime}_{b}=a_{b}$. Therefore, a necessary and sufficient condition
for a congruence of timelike or spacelike curves to be hypersurface orthogonal
is that the congruence is vorticity-free.

Only if $w^{a}$ is hypersurface orthogonal the distribution ${\cal T}$ in a
neighborhood of any point is integrable and can be spanned by coordinate base
vector fields (Wald, 1984). By equations (42) and (46), $w^{a}$ is
hypersurface orthogonal if and only if

$\displaystyle F_{ab}=0\;.$ (48)

Thus, the submanifold ${\cal M}$ supporting the KK variables exists if and
only if the electromagnetic field antisymmetric tensor vanishes. This
restriction on the KK theory is too strong, as there can be no unification of
the gravitational and electromagnetic interactions if the electromagnetic
field vanishes.

##  V Action and Compactification of the Extra Dimension

When the cylinder condition is satisfied, i.e., all components of the five-
dimensional metric tensor are independent of the extra dimension coordinate
(Kaluza, 1921), the Maxwell equations can easily be derived from the action
principle.444If the cylinder condition is dropped the derived field equations
are much more complicated, see, e.g., (Overduin & Wesson, 1997). The cylinder
condition is equivalent to the requirement that $w^{a}$ is a Killing vector of
the five-dimensional spacetime, i.e.,

$\displaystyle\tilde{\pounds}_{w}\tilde{g}_{ab}=0\;.$ (49)

When the above condition is satisfied, the Ricci scalar $\tilde{R}$ associated
with the five-dimensional bulk metric $\tilde{g}_{ab}$ is related to the Ricci
scalar $R$ associated with the KK four-dimensional metric $g_{ab}$ by (Bailin
& Love, 1987; Coquereaux & Esposito-Farese, 1990; Williams, 2015)

$\displaystyle\tilde{R}=R-\frac{\phi^{2}}{4}F_{ab}F^{ab}+\tilde{\nabla}_{a}v^{a}\;,$
(50)

where $v^{a}$ is a vector.

The determinant of the five-dimensional metric,
$\tilde{g}=\det\tilde{g}_{AB}$, is related to the determinant of the four-
dimensional metric, $g=\det g_{\mu\nu}$, by $\tilde{g}=\phi^{2}g$. Thus, we
have $\sqrt{-\tilde{g}}=\phi\sqrt{-g}$ and the five-dimensional action of
gravity

$\displaystyle I_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{G}}\int\tilde{R}\sqrt{-\tilde{g}}\,d^{4}xdw$
(51) $\displaystyle=$ $\displaystyle\frac{1}{\tilde{G}}\int
dw\int\phi\left(R-\frac{\phi^{2}}{4}F_{ab}F^{ab}\right)\sqrt{-g}\,d^{4}x\;,\hskip
11.38092pt$

where $\tilde{G}$ is the five-dimensional gravitational constant. The
divergence term in equation (50), $\tilde{\nabla}_{a}v^{a}$, has been dropped
off since it has no contribution to the action integral under appropriate
boundary conditions.

Since $w^{a}$ is a Killing vector field, the five-dimensional spacetime can be
compactified along the direction of extra dimension, i.e., the direction of
$w$-lines (Fig. 1). That is, a spacetime point $\\{x^{\mu},w\\}$ in
$(\tilde{M},\tilde{g}_{ab})$ is identified with the spacetime point
$\\{x^{\mu},w+L\\}$ in $(\tilde{M},\tilde{g}_{ab})$, where $L>0$ is a
constant. Then, the value of $w$ is restricted to the region of $[0,L)$, which
leads to $\int dw=L$. To have the circumference of the extra dimension—which
is $C_{w}\equiv\phi\int dw=\phi L$—to be constant, $\phi$ must be constant
(Bailin & Love, 1987).555It should be noted that ${\phi}$ cannot be constant
if the vacuum Einstein field equation $\tilde{R}_{ab}=0$ is imposed. This
follows because the equation $\tilde{R}_{ww}=0$ entails that
$\tilde{\nabla}^{a}\tilde{\nabla}_{a}\phi=\frac{\phi^{3}}{4}F^{ab}F_{ab}\;,$
as first noted by Jordan Jordan (1947) and Thiry Thiry (1948). For the extra
dimension to be unaccessible to current experiments the length $C_{w}=\phi L$
must be sufficiently small (Klein, 1926a). Then, equation (51) becomes

$\displaystyle
I_{g}=\frac{C_{w}}{\tilde{G}}\int\left(R-\frac{\phi^{2}}{4}F_{ab}F^{ab}\right)\sqrt{-g}\,d^{4}x\;.$
(52)

Figure 1: Under the cylinder condition the five-dimensional spacetime can be
compactified along the direction of extra dimension [i.e., the direction of
$w^{a}=(\partial/\partial w)^{a}$]. This way, the hypersurface ${\cal S}(w=0)$
in $(\tilde{M},\tilde{g}_{ab})$ is identified with the ${\cal
S}^{\prime}(w=L)$ under the map generated by $w^{a}$. The black dot on ${\cal
S}$ is identified with the black dot on ${\cal S}^{\prime}$, the circle on
${\cal S}$ identified with the circle on ${\cal S}^{\prime}$, and so on (as
indicated by dashed lines). The KK variables $A^{a}$ and $g_{ab}$ are
orthogonal to $w^{a}$ hence not tangent to ${\cal S}$, since in general
$w^{a}$ is not orthogonal to ${\cal S}$. In fact, $w^{a}$ is not orthogonal to
any hypersurface unless the electromagnetic field $F_{ab}$ vanishes. Note that
to make the extra dimension unaccessible to current experiments its
circumference $C_{w}=\phi L$ has to be small.

The appearance of the term $-(\phi^{2}/4)F_{ab}F^{ab}$ in the Lagrangian
density in equation (52) guarantees that the Maxwell equations can be derived
from the five-dimensional Einstein field equation by variation of the action
$I_{g}$ with respect to the potential vector $A^{a}$. In fact, if we identify
$\tilde{G}/C_{w}$ as the four-dimensional gravitational constant $G=1$ (i.e.,
$C_{w}=\tilde{G}$) and $\phi=2$, the action in equation (52) reduces to the
total action of gravity and electromagnetic fields in a four-dimensional
spacetime (see (Wald, 1984), Appendix E). Note that, however, all the
quantities ($F_{ab}$, $R$, and $g$) appearing in the integral of equation (52)
are defined in the tangent subbundle ${\cal T}$ orthogonal to $w^{a}$, since
they all are derived from $A_{a}$, $g_{ab}$, and the derivative operator
$\nabla_{a}$ associated with $g_{ab}$.

According to the results in section IV, the rank-4 distribution ${\cal T}$ is
not integrable and hence $g_{ab}$, $A_{a}$, and quantities derived from them
are not tangent to any four-dimensional submanifold embedded in $\tilde{\cal
M}$ unless the electromagnetic field $F_{ab}$ vanishes. Although under the
cylinder condition the four-dimensional Einstein field equation and the
Maxwell equations are successfully derived from the five-dimensional Einstein
field equation through the action principle, these equations are not supported
by a four-dimensional submanifold hence do not define a four-dimensional
spacetime.

The manifold structure of the KK theory, after the extra dimension is
compactified, is depicted in Fig. 2. The five-dimensional spacetime “tube” is
made of twisted four-dimensional “wires”, with each wire representing a
hypersurface $w=\mbox{const}$. The transverse cross-section of the spacetime
tube corresponds to the $w$-coordinate lines, i.e., curves whose tangent
vectors are $w^{a}=(\partial/\partial w)^{a}$, as indicated by the blue circle
in the figure. The KK variables $A^{a}$, $g_{ab}$, and the associated
distribution ${\cal T}$, are in the longitudinal direction along the tube
(i.e., the direction perpendicular to $w^{a}$). They are not tangent to any
four-dimensional submanifold. Thus, the action in equation (52) is defined in
the rank-4 tangent subbundle or distribution ${\cal T}$, but not defined on a
four-dimensional submanifold.

Figure 2: The Kaluza-Klein tube obtained by compactification of the extra
dimension in the direction of $w^{a}$. The five-dimensional spacetime tube is
made of twisted wires, each wire representing a hypersurface defined by
$w=\mbox{const}$. The blue circle on the right represents the $w$-coordinate
line with a length of $C_{w}=\phi L$. The KK variables $A^{a}$ and $g_{ab}$
are contained in a four-dimensional tangent subbundle ${\cal T}$ orthogonal to
$w^{a}$.

##  VI Summary and Discussion

All existing physical theories are defined on a smooth manifold with or
without a well-defined spacetime metric. In the KK theory, the five-
dimensional theory is defined on a five-dimensional manifold with a Lorentz
metric determined by the five-dimensional Einstein field equation. The four-
dimensional metric tensor and the electromagnetic potential vector assumed in
the KK theory must be defined on a four-dimensional submanifold (i.e., a
hypersurface) embedded in the five-dimensional manifold, in order for the
derived four-dimensional theory (including the four-dimensional Einstein field
equation and the Maxwell equations) to be able to describe the four-
dimensional world where we live and do physical experiments. But this is not
the case, as has been shown in this paper.

In general, the four-dimensional KK variables $g_{ab}$, $A_{a}$, and other
geometric quantities derived from them (e.g., the four-dimensional Ricci
tensor $R_{ab}$ and the electromagnetic field antisymmetric tensor $F_{ab}$)
are in a four-dimensional subbundle that is not tangent to any four-
dimensional submanifold, since by the KK construction $g_{ab}$ and $A_{a}$ are
orthogonal to the vector field $w^{a}$ generating the extra dimension but
$w^{a}$ is not hypersurface orthogonal unless the electromagnetic field
vanishes. Thus, the results presented in the paper lead us to such a paradox:
the KK theory is valid mathematically only if the electromagnetic field
derived from the KK theory vanishes. This is a general conclusion, independent
of the cylinder condition adopted for derivation of the four-dimensional field
equations.

When the electromagnetic field is weak and has a negligible effect on the
spacetime metric, i.e., when condition (31) is satisfied, the four-dimensional
metric tensor and the electromagnetic potential vector can be regarded as
approximately being defined on the hypersurface of $w=\mbox{const}$. But then
the KK theory becomes an approximate and weak-field limit theory, conflicting
the original spirit of unification of gravitational and electromagnetic
interactions. In addition, without a precisely defined four-dimensional
submanifold supporting the four-dimensional variables, it is hard to accept
the approximate theory since it is not well defined in mathematics. An
ultimate solution to the problem raised in this paper may be given by a
different 4+1 decomposition of a five-dimensional spacetime metric as having
been proposed in (Li, 2016), where the four-dimensional spacetime is defined
on a hypersurface that is not orthogonal to the extra dimension, but then the
theory is different from the KK theory since an electromagnetic field equation
with a curvature-coupled term is derived.

###### Acknowledgements.

The author thanks the anonymous adjudicator for a very enlightening report,
which has stimulated the author to think more deeply and more widely about the
problem discussed in the paper. The report has also helped to improve the
presentation of the paper. This work was supported by the NSFC grants program
(no. 11973014).

## References

  * Kaluza (1921) T. Kaluza, Zum Unitätsproblem der Physik, Sitzungsber. Preuss. Akad. Wiss., 966 (1921)
  * Klein (1926a) O. Klein, Quantentheorie und Fünfdimensionale Relativitätstheorie, Z. Phys. 37, 895 (1926)
  * Klein (1926b) O. Klein, The Atomicity of Electricity as a Quantum Theory Law, Nature 118, 516 (1926)
  * Nordström (1914) G. Nordström, Über die Moglichkeit, das Electromagnetische Feld und das Gravitationsfeld zu Vereinigen, Phys. Z. 15, 504 (1914)
  * Cremmer & Scherk (1976) E. Cremmer & J. Scherk, Spontaneous Compactification of Space in an Einstein-Yang-Mills-Higgs Model, Nucl. Phys. B 108, 409 (1976)
  * Cremmer & Scherk (1977) E. Cremmer & J. Scherk, Spontaneous Compactification of Extra Space Dimensions, Nucl. Phys. B 118, 61 (1977)
  * Luciani (1978) J. F. Luciani, Space-time Geometry and Symmetry Breaking, Nucl. Phys. B 135, 111 (1978)
  * Englert (1982) F. Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity, Phys. Lett. B 119, 339 (1982)
  * Li & Gott (1998) L.-X. Li & J. R. Gott, Inflation in Kaluza-Klein Theory: Relation between the Fine-Structure Constant and the Cosmological Constant, Phys. Rev. D 58, 103513 (1998)
  * van Nieuwenhuizen (1981) P. van Nieuwenhuizen, Supergravity, Phys. Rep. 68, 189 (1981)
  * Duff, Nilsson, & Pope (1986) M. J. Duff, B. E. W. Nilsson, & C. N. Pope, Kaluza-Klein Supergravity, Phys. Rep. 130, 1 (1986)
  * Green, Schwarz, & Witten (2012) M. B. Green, J. H. Schwarz, & E. Witten, Superstring Theory, 25th Anniversary Edition, Vols. I and II (Cambridge University Press, Cambridge, 2012)
  * Polchinski (2003) J. Polchinski, String Theory, Vols. I and II (Cambridge University Press, Cambridge, 2003)
  * Becker, Becker, & Schwarz (2007) K. Becker, M. Becker, & J. H. Schwarz, String Theory and M-Theory: A Modern Introduction (Cambridge University Press, Cambridge, 2007)
  * Kiritsis (2019) E. Kiritsis, String Theory in a Nutshell, 2nd edition (Princeton University Press, Princeton, 2019)
  * Randall & Sundrum (1999a) L. Randall & R. Sundrum, Large Mass Hierarchy from a Small Extra Dimension, Phys. Rev. Lett. 83, 3370 (1999)
  * Randall & Sundrum (1999) L. Randall & R. Sundrum, An Alternative to Compactification, Phys. Rev. Lett. 83, 4690 (1999)
  * Maartens & Koyama (2010) R. Maartens & K. Koyama, Brane-World Gravity, Living Rev. Relativity 13, 5 (2010)
  * Nath & Yamaguchi (1999) P. Nath & M. Yamaguchi, Probing the Nature of Compactification with Kaluza-Klein Excitations at the Large Hadron Collider, Phys. Lett. B 466, 100 (1999)
  * Bhattacharyya et al. (2009) G. Bhattacharyya, A. Datta, S. K. Majee, & A. Raychaudhuri, Exploring the Universal Extra Dimension at the LHC, Nucl. Phys. B 821, 48 (2009)
  * Deutschmann et al. (2017) N. Deutschmann, T. Flacke, & J. S. Kim, Current LHC Constraints on Minimal Universal Extra Dimensions, Phys. Lett. B 771, 515 (2017)
  * Wald (1984) R. M. Wald, General Relativity (The University of Chicago Press, Chicago, 1984)
  * de Witt (1964) B. S. de Witt, Dynamical Theory of Groups and Fields, in Relativity Groups and Topology, Les Houches 1963 (Gordon and Breach Science Publishers, 1964), p. 585
  * Kerner (1987) R. Kerner, Generalization of the Kaluza-Klein Theory for an Arbitrary non-Abelian Gauge Group, Ann. Inst. Henri Poincaré 9, 143 (1968)
  * Cho & Freund (1975) Y. M. Cho & P. G. O. Freund, Non-Abelian gauge fields as Nambu-Goldstone fields, Phys. Rev. D 12, 1711 (1975)
  * Appelquist et al. (1987) T. Appelquist, A. Chodos, & P. G. O. Freund (editors), Modern Kaluza-Klein Theories (Addison-Wesley, Amsterdam, 1987)
  * Einstein & Bergmann (1938) A. Einstein & P. Bergmann, On a Generalization of Kaluza’s Theory of Electricity, Ann. Math. 39, 683 (1938)
  * Thiry (1948) M. Y. Thiry, Les Équations de la Théorie Unitaire de Kaluza, Compt. Rend. Acad. Sci. Paris. 226, 216 (1948)
  * Coquereaux & Esposito-Farese (1990) R. Coquereaux & G. Esposito-Farese, The Theory of Kaluza-Klein-Jordan-Thiry Revisited, Ann. Inst. Henri Poincaré 52, 113 (1990)
  * Li (2016) L.-X. Li, A New Unified Theory of Electromagnetic and Gravitational Interactions, Front. Phys. 11, 110402 (2016)
  * Lee (2012) J. M. Lee, Introduction to Smooth Manifolds, 2nd edition (Springer, Berlin, 2012)
  * Hawking & Ellis (1973) S. W. Hawking & G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)
  * Bailin & Love (1987) D. Bailin & A. Love, Kaluza-Klein Theories, Rep. Prog. Phys. 50, 1087 (1987)
  * Williams (2015) L. L. Williams, Field Equations and Lagrangian for the Kaluza Metric Evaluated with Tensor Algebra Software, Journal of Gravity 2015, 901870 (2015)
  * Straumann (2013) N. Straumann, General Relativity, 2nd edition (Springer, Berlin, 2013)
  * Overduin & Wesson (1997) J. M. Overduin & P. S. Wesson, Kaluza-Klein Gravity, Phys. Rep. 283, 303 (1997)
  * Jordan (1947) P. Jordan, Erweiterung der projektiven Relativitätstheo-rie, Ann. Physik 436, 219 (1947)