Tsemo Aristide

PKFOKAM Institute OF EXCELLENCE

P.O. BOX 11646, Yaounde, Cameroon.

<EMAIL_ADDRESS>

Automorphisms of projective manifolds.

Abstract.

Let $(M,P\nabla_{M})$ be a compact projective manifold and
$Aut(M,P\nabla_{M})$ its group of automorphisms. The purpose of this paper is
to study the topological properties of $(M,P\nabla_{M})$ if
$Aut(M,P\nabla_{M}))$ is not discrete by applying the results that I have
shown in [13] and the Benzekri’s functor which associates to a projective
manifold a radiant affine manifold. This enables us to show that the orbits of
the connected component of $Aut(M,P\nabla_{M})$ are immersed projective
submanifolds. We also classify $3$-dimensional compact projective manifolds
such that $dim(Aut(M,P\nabla_{M}))\geq 2$.

1\. Introduction

The purpose of this paper is to study the group of automorphisms of projective
manifolds. Firstly we recall the definition of $(X,G)$ manifolds, their group
of automorphisms and morphisms between $(X,G)$-structures. We applied the
results described in the general framework of $(X,G)$-manifolds to the
category of affine manifolds and projective manifolds. Benzekri has
constructed a functor which associates to a projective manifold
$(M,P\nabla_{M})$ a radiant affine manifold $(B(M),\nabla_{B(M)})$ whose
underlying topological space is $M\times S^{1}$. It enables us to show that
there exists a surjective morphism between the connected component
$Aut(B(M),\nabla_{B(M)})_{0}$ of the group of affine automorphisms of
$(B(M),\nabla_{B(M)})$ and the connected component $Aut(M,P\nabla_{M})_{0}$ of
the group of projective automorphisms of $(M,P\nabla_{M})$.

Let $(M,\nabla_{M})$ be a compact affine manifold, in [13], I have studied the
relations between $Aut(M,\nabla_{M})$ and the topology of $M$. This enables us
to show that the orbits of $Aut(M,P\nabla_{M})_{0}$ are projective immersed
submanifolds. In the last section, we study the automorphisms group of $2$ and
$3$ dimensional projective manifolds. We remark that a $2$-dimensional
projective manifold whose group of automorphisms is not discrete is
homeomorphic to the sphere, the $2$-dimensional projective space or the two
dimensional torus. Finally we show that a $3$-dimensional projective manifold
$(M,P\nabla_{M})$ whose developing map is injective and such that
$dim(Aut(M,P\nabla_{M})\geq 2$ is homeomorphic to a spherical manifold,
$S^{2}\times S^{1}$, or a finite cover of $M$ is the total space of a torus
bundle.

Remark that $(X,G)$ manifolds play an important role in low dimensional
topology: seven of the eight geometry of Thurston are examples of projective
geometry (see Cooper and Goldman [8] p. 1220). In [12] p.17, Sullivan and
Thurston note that the existence of a $(X,G)$-structure on every $3$-manifold
implies the Poincare conjecture.

2\. $(X,G)$-manifolds.

A $(X,G)$ model is a finite dimensional differentiable manifold $X$, endowed
with an effective and transitive action of a Lie group $G$ which satisfies the
unique extension property. This is equivalent to saying that: two elements
$g,g^{\prime}$ of $G$ are equal if and only if their respective restriction to
a non empty open subset of $X$ are equal.

A $(X,G)$ manifold $(M,X,G)$ is a differentiable manifold $M$, endowed with an
open covering $(U_{i})_{i\in I}$ such that for every $i\in I$, there exists a
differentiable map $f_{i}:U_{i}\rightarrow X$ which is a diffeomorphism onto
its image and $f_{i}\circ f_{j}^{-1}$ coincides with the restriction of an
element $g_{ij}$ of $G$ to $f_{j}(U_{i}\cap U_{j})$. The map $f_{i}$ is called
a $(X,G)$ chart.

A $(X,G)$ structure defined on $M$ can be lifted to the universal cover
$\tilde{M}$ of $M$. This structure is defined by a local diffeomorphism
$D_{M}:\tilde{M}\rightarrow X$. This implies that a $(X,G)$ chart of this
structure is an open subset $U$ of $\tilde{M}$ such that the restriction of
$D_{M}$ to $U$ is a diffeomorphism onto its image.

Let $(X,G)$ and $(X^{\prime},G^{\prime})$ be two models, $\phi:X\rightarrow
X^{\prime}$ a differentiable map and $\Phi:G\rightarrow G^{\prime}$ a morphism
of groups such that for every $g\in G$, the following diagram is commutative:

$\matrix{X&{\buildrel
g\over{\longrightarrow}}&X\cr\phi\downarrow&&\downarrow\phi\cr
X^{\prime}&{\buildrel\Phi(g)\over{\longrightarrow}}&X^{\prime}}$

Let $(M,X,G)$ (resp. $(M^{\prime},X^{\prime},G^{\prime})$) be an $(X,G)$
manifold (resp. an $(X^{\prime},G^{\prime})$ manifold). A
$(\Phi,\phi)$-morphism
$f:(M,X,G)\rightarrow(M^{\prime},X^{\prime},G^{\prime})$ is a differentiable
map: $f:M\rightarrow M^{\prime}$ such that for every chart $(U_{i},f_{i})$ of
$M$ such that $f(U_{i})$ is contained in the chart $(V_{j},f^{\prime}_{j})$ of
$M^{\prime}$, there exists an element $g\in G$ such that the restrictions of
$f^{\prime}_{j}\circ f\circ f_{i}^{-1}$ and $\Phi(g)\circ\phi$ to
$f_{i}(U_{i})$ coincide.

We will denote by $Aut(X,M,G)$ the group of $(Id_{G},Id_{X})$-automorphisms of
$(M,X,G)$ and by $Aut(M,X,G)_{0}$ its connected component. It is a Lie group
endowed with the compact open topology. For every element $g\in
Aut(\tilde{M},X,G)$, the developing map defines a representation
$H_{M}:Aut(\tilde{M},X,G)\rightarrow G$ such that the following diagram is
commutative:

$\matrix{\tilde{M}&{\buildrel g\over{\longrightarrow}}&\tilde{M}\cr
D_{M}\downarrow&&\downarrow D_{M}\cr X&{\buildrel
H_{M}(g)\over{\longrightarrow}}&X}$

Remark that the group of Deck transformations that we identify to the
fundamental group $\pi_{1}(M)$, of $M$, is a subgroup of $Aut(\tilde{M},X,G)$.
The restriction $h_{M}$ of $H_{M}$ to the fundamental group $\pi_{1}(M)$, of
$M$ is called the holonomy representation of the $(X,G)$ manifold $(M,X,G)$.

The pullback $p_{M}(f)$ of an element $f$ of $Aut(M,X,G)$, by the universal
covering map, $p_{M}:\tilde{M}\rightarrow M$ is an element of
$Aut(\tilde{M},X,G)$ which belongs to the normalizer $N(\pi_{1}(M))$ of
$\pi_{1}(M)$ in $Aut(\tilde{M},X,G)$. Conversely, every element $g$ of
$N(\pi_{1}(M))$ induces an element $A_{M}(g)$ of $Aut(M,X,G)$ such that the
following diagram is commutative:

$\matrix{\tilde{M}&{\buildrel g\over{\longrightarrow}}&\tilde{M}\cr
p_{M}\downarrow&&\downarrow p_{M}\cr M&{\buildrel
A_{M}(g)\over{\longrightarrow}}&M}.$

The kernel of the morphism $A_{M}:N(\pi_{1}(M))\rightarrow Aut(M,X,G)$ is
$\pi_{1}(M)$ and $A_{M}$ is a local diffeomorphim. We will denote by
$N(\pi_{1}(M))_{0}$ the connected component of $N(\pi_{1}(M))$, it is also the
connected component of the commutator of $\pi_{1}(M)$ in $Aut(\tilde{M},X,G)$.
Since $A_{M}$ is locally invertible, it induces an isomorphism between the Lie
algebra $n(\pi_{1}(M))$ of $N(\pi_{1}(M))$ and the Lie algebra $aut(M,X,G)$ of
$Aut(M,X,G)$. If $(M,X,G)$ is a compact $(X,G)$ manifold, $aut(M,X,G)$ is
isomorphic to space of elements of ${\cal G}$, the Lie algebra of $G$, which
are invariant by $h_{M}(\pi_{1}(M))$.

3\. Affine and projective structures.

Let $\mathbb{R}^{n}$ be the $n$-dimensional real vector space. We denote by
$Gl(n,\mathbb{R})$ the group of linear automorphisms of $\mathbb{R}^{n}$ and
by $Aff(n,\mathbb{R})$ its group of affine transformations. If we fix an
origin $0$ of $\mathbb{R}^{n}$, for every element $f\in Aff(n,\mathbb{R})$, we
can write $f=(L(f),a_{f})$ where $L(f)$ is an element of $Gl(\mathbb{R}^{n})$
and $a_{f}=f(0)$. The couple $(\mathbb{R}^{n},Aff(n,\mathbb{R}))$ is a model.
A $(\mathbb{R}^{n},Aff(n,\mathbb{R}))$ manifold is also called an affine
manifold. Equivalently, an $(\mathbb{R}^{n},Aff(n,\mathbb{R}))$ manifold is a
$n$-dimensional differentiable manifold $M$ endowed with a connection
$\nabla_{M}$ whose curvature and torsion tensors vanish identically.

Remark that the linear part $L(h_{M})$ of the holonomy representation $h_{M}$
of an affine manifold $(M,\nabla_{M})$ is the holonomy of the connection
$\nabla_{M}$. We say that the $n$-dimensional affine manifold $(M,\nabla_{M})$
is radiant if its holonomy $h_{M}$ fixes an element of $\mathbb{R}^{n}$, this
is equivalent to saying that $h_{M}$ and $L(h_{M})$ are conjugated by a
translation.

The $n$-dimensional real projective space $\mathbb{R}P^{n}$ is the quotient of
$\mathbb{R}^{n+1}-\\{0\\}$ by the equivalence relation defined by $x\simeq y$
if and only there exists $\lambda\in\mathbb{R}$ such that $x=\lambda y$. If
$x$ is an element of $\mathbb{R}^{n+1}-\\{0\\}$, we will denote by
$[x]_{\mathbb{R}P^{n}}$ its equivalent class. The group $Gl(n+1,\mathbb{R})$
acts transitively on $\mathbb{R}P^{n}$ by the action defined by
$g.[x]_{\mathbb{R}P^{n}}=[g.x]_{\mathbb{R}P^{n}}$ the kernel of this action is
the group $H_{n+1}$ of homothetic maps. We denote by $PGl(n+1,\mathbb{R})$ the
quotient $Gl(n+1,\mathbb{R})$ by $H_{n}$. The couple
$(\mathbb{R}P^{n},PGl(n+1,\mathbb{R}))$ is a model. A
$(\mathbb{R}P^{n},PGl(n+1,\mathbb{R}))$ is also called a projective manifold.
Equivalently, a projective manifold can be defined by a differentiable
manifold $M$ endowed with a projectively flat connection $P\nabla_{M}$. We
will denote it by $(M,P\nabla_{M})$.

The $n$-dimensional sphere $S^{n}$ is the quotient of
$\mathbb{R}^{n+1}-\\{0\\}$ by the equivalence relation defined by $x\simeq y$
if and only if there exists $\lambda>0$ such that $x=\lambda y$. Let $x$ be an
element of $\mathbb{R}^{n+1}-\\{0\\}$, we will denote by $[x]_{S^{n}}$ its
equivalence class for this relation. Remark that if $\langle,\rangle$ is an
Euclidean metric defined on $\mathbb{R}^{n+1}$, there exists a bijection
between the unit sphere
$S^{n}_{\langle,\rangle}=\\{x:x\in\mathbb{R}^{n+1},\langle x,x\rangle=1\\}$
and $S^{n}$ defined by the restriction of the equivalence relation to
$S^{n}_{\langle,\rangle}$.

There exists a map $D_{S^{n}}:S^{n}\rightarrow\mathbb{R}P^{n}$ such that for
every element $x$ of $\mathbb{R}^{n+1}-\\{0\\}$,
$[x]_{\mathbb{R}P^{n}}=D_{S^{n}}([x]_{S^{n}}).$ The map $p_{n}$ is a covering,
thus is the developing map of a projectively flat connection $P\nabla_{S^{n}}$
defined on $S^{n}$.

A $p$-dimensional projective submanifold $(F,P\nabla_{F})$ of the projective
manifold $(M,P\nabla_{M})$ is a $p$-dimensional submanifold $F$ of $M$ endowed
with a structure of a projective manifold, such that the canonical embedding
$i_{F}:(F,P\nabla_{F})\rightarrow(M,P\nabla_{M})$ is a morphism of projective
manifolds.

Let $\hat{F}$ be the universal cover of $F$, we can lift $i_{F}$ to a
projective map $\hat{i}_{F}:\hat{F}\rightarrow\hat{M}$. The image of
$D_{M}\circ\hat{i}_{F}$ is contained in a $p$-dimensional projective subspace
$U_{F}$ of $\mathbb{R}P^{n}$. The map
$D_{M}\circ\hat{i}_{F}:\hat{F}\rightarrow U_{F}$ is a developing map of $F$.
There exists a canonical morphism $\pi_{F}:\pi_{1}(F)\rightarrow\pi_{1}(M)$
induced by $i_{F}$. Let $\gamma$ be an element of $\pi_{1}(F)$, the holonomy
$h_{F}(\gamma)$ is the restriction of $h_{M}(\pi_{F}(\gamma))$ to $U_{F}$. If
there is no confusion, we are going to denote $h_{M}(\pi_{F}(\gamma))$ by
$h_{M}(\gamma)$.

Proposition 3.1. The group of automorphisms of the $n$-dimensional projective
manifold $S^{n}$ is isomorphic to $Sl(n+1,\mathbb{R})$, the group of
invertible $(n+1)\times(n+1)$ matrices such that for every element $A\in
Sl(n+1,\mathbb{R}),|det(A)|=1$.

Proof. Let $g$ be an element of $Sl(n+1,\mathbb{R})$. For every
$[x]_{S^{n}}\in S^{n}$, we write $u_{g}(x)=[g(x)]_{S^{n}}$. Let $[g]$ be the
image of $g$ by the quotient map $Sl(n+1,\mathbb{R})\rightarrow
PGl(n+1,\mathbb{R})$, we have $[g]\circ D_{S^{n}}=D_{S^{n}}\circ u_{g}$. This
implies that $u_{g}$ is an element of $Aut(S_{n},P\nabla_{S^{n}})$. Suppose
that $u_{g}=Id_{S^{n}}$, it implies that for every $[x]\in S^{n}$,
$g(x)=\lambda(x)x,\lambda(x)>0$, we deduce that $g(x)=\lambda
Id_{\mathbb{R}^{n}},\lambda>0$, and $\lambda^{n+1}=1$ since $g\in
Sl(n+1,\mathbb{R})$. This implies that $\lambda=1$. We deduce that
$u:Sl(n+1,\mathbb{R})\rightarrow Aut(S^{n},P\nabla_{S^{n}})$ defined by
$u(g)=u_{g}$ is injective. Let $f$ be an element of
$Aut(S^{n},P\nabla_{S^{n}})$, there exists an element $[g]\in
PGl(n+1,\mathbb{R})$ such that $[g]\circ D_{S^{n}}=D_{S^{n}}\circ f$. Consider
an element $g\in Sl(n+1,\mathbb{R})$ whose image by the quotient map is $[g]$,
$f=u_{g}$. This implies that $u$ is an isomorphism.

The Benzecri correspondence.

Consider the embedding $i_{n}^{G}:Gl(n,\mathbb{R})\rightarrow
Gl(n+1,\mathbb{R})$ defined by $i_{n}^{G}(A)=\pmatrix{A&0\cr 0&1}$ and the
open embedding $i_{n}:\mathbb{R}^{n}\rightarrow\mathbb{R}P^{n}$ defined by
$i_{n}(x_{1},...,x_{n})=[x_{1},...,x_{n},1]$. For every elements
$x\in\mathbb{R}^{n}$ and $g\in Gl(n,\mathbb{R})$, we have
$i_{n}(g(x))=i_{n}^{G}(g)(i_{n}(x))$. We deduce that for every affine manifold
$(M,\nabla_{M})$ whose developing map is $D_{M}$, there exists a projective
structure defined on $M$ whose developing map is $i_{n}\circ D_{M}$.

Benzecri [4] p.241-242 has defined a functor between the category of
projective manifolds of dimension $n$ and the category of radiant affine
manifolds of dimension $n+1$ which can be described as follows:

Firstly, we remark that since the universal cover $\tilde{M}$ of the
projective manifold $M$ is simply connected and $p_{n}:S^{n}\rightarrow
P\mathbb{R}^{n}$ is a covering map, the theorem 4.1 of Bredon [5] p.143
implies that the development map $D_{M}:\tilde{M}\rightarrow P\mathbb{R}^{n}$,
can be lifted to a local diffeomorphism $D^{\prime}_{M}:\tilde{M}\rightarrow
S^{n}$ which is a projective morphism. Let $N(\pi_{1}(M))$ be the normalizer
of $\pi_{1}(M)$ in $Aut(\tilde{M},P\nabla_{\tilde{M}})$, for every $g\in
N(\pi_{1}(M))$, there exists $H^{\prime}_{M}(g)\in Aut(S^{n},P\nabla_{S^{n}})$
such that the following diagram is commutative:

$\matrix{\tilde{M}&{\buildrel g\over{\longrightarrow}}&\tilde{M}\cr
D^{\prime}_{M}\downarrow&&\downarrow D^{\prime}_{M}\cr S^{n}&{\buildrel
H^{\prime}_{M}(g)\over{\longrightarrow}}&S^{n}}.$

We will denote by $h^{\prime}_{M}$ the restriction of $H^{\prime}_{M}$ to
$\pi_{1}(M)$.

There exists a local diffeomorphism $D_{\tilde{M}\times
S^{1}}:\tilde{M}\times\mathbb{R}^{*}_{+}\rightarrow\mathbb{R}^{n+1}-\\{0\\}$
defined by $D_{\tilde{M}\times S^{1}}(x,t)=tD^{\prime}_{M}(x)$, which is the
developing map of a radiant structure defined on $M\times S^{1}$ whose
holonomy representation $h_{M\times S^{1}}:\pi_{1}(M\times S^{1})\rightarrow
Gl(n+1,\mathbb{R})$ is defined by $h_{M\times
S^{1}}(\gamma,n)=2^{n}h^{\prime}_{M}(\gamma)$. This radiant affine manifold
$M\times S^{1}$ is the construction of Benzecri, we will often denote this
affine structure by $(B(M),\nabla_{B(M)})$ and by $p_{B(M)}:M\times
S^{1}\rightarrow M$ the projection on the first factor.

Let $f:(M,P\nabla_{M})\rightarrow(N,P\nabla_{N})$ be a morphism between
$n$-dimensional projective manifolds; $f$ can lifted to the projective the
morphism $\tilde{f}:\tilde{M}\rightarrow\tilde{N}$. We deduce the existence of
a morphism of affine manifolds
$f^{\prime}:\tilde{M}\times\mathbb{R}_{+}^{*}\rightarrow\tilde{N}\times\mathbb{R}_{+}^{*}$
defined by $f^{\prime}(x,t)=(\tilde{f},t)$. The morphism $f^{\prime}$ is
equivariant with respect to the action of $\pi_{1}(B(M))$ on
$\tilde{M}\times\mathbb{R}^{*}_{+}$ and $\pi_{1}(B(N))$ on
$\tilde{N}\times\mathbb{R}^{*}_{+}$, and covers a morphism
$b(f):B(M)\rightarrow B(N)$.

Let $(N,\nabla_{N})$ be a $n$-dimensional radiant affine manifold. We suppose
that the holonomy of $N$ fixes the origin of $\mathbb{R}^{n}$. The vector
field defined on $\mathbb{R}^{n}$ by $X_{R}^{\mathbb{R}^{n}}(x)=x$ is
invariant by the holonomy. Its pullback by the developing map is a vector
field $X_{R}^{\tilde{N}}$ of $\tilde{N}$ invariant by $\pi_{1}(N)$. We deduce
that $X_{R}^{\tilde{N}}$ is the pullback of a vector field $X_{R}^{N}$ of $N$
called the radiant vector field of $N$.

Proposition 3.2. Let $(M,P\nabla_{M})$ be a compact projective manifold. There
exists a surjective morphism of groups between the connected component of
$Aut(B(M),\nabla_{B(M)})$ and the connected component of $Aut(M,P\nabla_{M})$.

Proof. Let $f$ be an element of $Aut(B(M),\nabla_{B(M)})_{0}$, the connected
component of $Aut(B(M),\nabla_{B(M)})$. Consider an element $\tilde{f}$ of
$Aut(\tilde{M}\times\mathbb{R}^{*}_{+})_{0}$ over $f$. For every
$\tilde{x}\in\tilde{M}$ and $t\in\mathbb{R}^{*}_{+}$, we can write
$\tilde{f}(\tilde{x},t)=(\tilde{g}(\tilde{x},t),h(\tilde{x},t))$. The flow of
$X_{R}^{\mathbb{R}^{n+1}}$ is in the center of $Gl(n+1,\mathbb{R})$, we deduce
that $\tilde{f}$ commutes with the flow $X_{R}^{\tilde{B}(M)}$,
$g(\tilde{x},t)$ does not depend of $t$ and $h(\tilde{x},t)=th(\tilde{x},1)$.

Let $\gamma$ be an element of $\pi_{1}(M)$, since
$(\gamma,2).(\tilde{x},t)=(\gamma(\tilde{x}),2t)$ is an element of
$\pi_{1}(B(M))$ and $\tilde{f}$ is an element of $N(\pi_{1}(B(M)))_{0}$, we
deduce that $(\gamma,2)$ commutes with $\tilde{f}$ and $\tilde{g}$ commute
with $\gamma$. This implies that there exists an element $g$ of
$Aut(M,P\nabla_{M})$ whose lifts is $\tilde{g}$. Remark that since $\tilde{f}$
is an affine transformation, $h(x,1)$ is a constant. The correspondence
$P:Aut(B(M),\nabla_{B(M)})_{0}\rightarrow Aut(M,P\nabla_{M})_{0}$ defined by
$P(f)=g$ is well defined and is surjective morphism of groups since for every
element $f\in Aut(M,P\nabla_{M})_{0}$, $P(b(f))=f$.

Let $(M,P\nabla_{M})$ be a projective manifold $M$, the orbits of the radiant
flow $\phi_{t}^{B(M)}$ of $X_{R}^{B(M)}$ are compact. The images of the
elements of $\phi_{t}^{B(M)}$ by $P$ are the identity on $(M,P\nabla_{M})$.
This implies that $dim(Aut(M,P\nabla_{M}))+1\leq
dim(Aut(B(M),\nabla_{B(M)})))$. We deduce that if $(M,P\nabla_{M})$ is a
projective manifold, such that $Aut(M,P\nabla_{M})$ is not discrete, the
dimension of $Aut(B(M),\nabla_{B(M)})$ is superior or equal to $2$.

4\. Automorphisms of projective manifolds and automorphisms of radiant affine
manifolds.

Let $(N,\nabla_{N})$ be an affine manifold. In [13], I have shown that
$aut(N,\nabla_{N})$, the Lie algebra of $Aut(N,\nabla_{N})$ is endowed with an
associative product defined by $X.Y={\nabla_{M}}_{X}Y$. We deduce that
$\nabla_{B(M)}$ defines on $aut(B(M),\nabla_{B(M)})$ an associative structure
which can be pulled back to $n(\pi_{1}(B(M))$. It results that the Lie algebra
$H_{B(M)}(n(B(M),\nabla_{B(M)}))$ of the image of $N(\pi_{1}(B(M)))$ by
$H_{B(M)}$ is stable by the canonical product of matrices which is the image
of the associative product of $n(\pi_{1}(B(M)))$ by $H_{B(M)}$. Remark that
$H_{B(M)}(n(B(M),\nabla_{B(M)}))$ is isomorphic to $n(B(M),\nabla_{B(M)})$.
The theorem 23 of chap. III of [1] implies that we can write:
$H_{B(M)}(n(B(M),\nabla_{B(M)}))=S_{M}\oplus N_{M}$ where $S_{M}$ is a semi-
simple associative algebra and $N_{M}$ a nilpotent associative algebra.

In [14], by using this associative product, I have shown that the orbits of
the canonical action of $Aff(N,\nabla_{N})_{0}$ on $N$ are immersed affine
submanifolds of $(N,\nabla_{N})$ and are the leaves of a (singular) foliation.
This leads to the following result:

Proposition 4.1. Let $(M,P\nabla_{M})$ be a projective manifold. The orbits of
the action of $Aut(M,P\nabla_{M})_{0}$ on $M$ are immersed projective
submanifolds and are the leaves of a singular foliation.

Proof. The orbits of $Aut(B(M),\nabla_{B(M)})_{0}$ are immersed affine
submanifolds of $B(M)$. The proposition 3.2 shows that there exists a
surjective map $P:Aut(B(M),\nabla_{B(M)})_{0}\rightarrow
Aut(M,P\nabla_{M})_{0}$ such that, for every $g\in
Aut(B(M),\nabla_{B(M)})_{0}$ and $x\in B(M)$, $p_{B(M)}(g(x))=P(g)(p_{B}(x))$.
This implies that the orbits of $Aut(M,P\nabla_{M})_{0}$ are the images of the
orbits of $Aut(B(M),\nabla_{B(M)})_{0}$ by the quotient map $B(M)\rightarrow
M$.

Theorem 4.1 Let $(M,P\nabla_{M})$ be a compact oriented projective manifold of
dimension superior or equal to $2$. Suppose that $H_{M}(N(\pi_{1}(M)))$ acts
transitively on $\mathbb{R}P^{n}$, then $(M,P\nabla_{M})$ is isomorphic to a
finite quotient of $\mathbb{K}P^{m}$ by a subgroup of $\mathbb{K}$ where
$\mathbb{K}$ is the field of real numbers, complex numbers, quaternions or
octonions. The action of $\pi_{1}(M)$ on $\mathbb{K}P^{n}$ is induced by its
action on $\mathbb{K}^{m+1}$ by homothetic maps.

Proof. The fact that $H_{M}(N(\pi_{1}(M)))$ acts transitively on
$\mathbb{R}P^{n}$ implies that $H^{\prime}_{M}(N(\pi_{1}(M)))$ acts
transitively on $S^{n}$. The theorem of Montgomery Zipplin [11] p.226 implies
that a connected compact subgroup $K^{\prime}$ of
$H^{\prime}_{M}(N(\pi_{1}(M)))$ acts transitively on $S^{n}$. The theorem I p.
456 of Montgomery and Samelson [10] implies that a connected compact simple
subgroup $C^{\prime}$ of $K^{\prime}$ acts transitively on $S^{n}$. The Lie
algebra of the connected component $C$ of ${H^{\prime}}_{M}^{-1}(C^{\prime})$
is isomorphic to the Lie algebra of $C^{\prime}$ since the kernel of
$H^{\prime}_{M}$ is discrete. This implies that $C$ is compact. Remark that
the orbits of the action of $C$ on $\tilde{M}$ are open. We deduce that $C$
acts transitively on $\tilde{M}$ and $\tilde{M}$ is compact. This implies that
$D^{\prime}_{M}:\tilde{M}\rightarrow S^{n}$ is a covering since it is a local
diffeomorphism defined between compact manifolds. This implies that
$D^{\prime}_{M}$ is a diffeomorphim since $S^{n}$ and $\tilde{M}$ are simply
connected.

We can write $\mathbb{R}^{n}=\oplus_{i\in I}U_{i}$ where $U_{i}$ is an
irreducible component of the action of $h^{\prime}_{M}(\pi_{1}(M))$ since
$\pi_{1}(M)$ is finite.

Let $i,j\in I$, consider two non zero elements $x_{i}\in U_{i},x_{j}\in
U_{j}$, since $H^{\prime}_{M}(N(\pi_{1}(M))_{0})$ acts transitively on $S^{n}$
there exists $B\in H^{\prime}_{M}(N(\pi_{1}(M))_{0})$ such that
$B(x_{i})=cx_{j}$, $B(U_{i})\cap U_{j}$ is invariant by
$H^{\prime}_{M}(\pi_{1}(M))$, we deduce that $B(U_{i})=U_{j}$ since $U_{j}$ is
irreducible and $B$ is an isomorphism.

The group of automorphisms of the irreducible representation $U_{i}$ is
$\mathbb{K}$ where $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H}$ or
$\mathbb{O}$. We deduce that $\mathbb{R}^{n+1}$ is a $\mathbb{K}$ vector space
and the action of $H^{\prime}_{M}(\pi_{1}(M))_{0})$ on $\mathbb{K}^{n}$ is
induced by its action on $\mathbb{K}$ by right multiplication of elements of
$\mathbb{K}$.

Remark.

Suppose that the dimension of $M$ is even, and $H_{M}(N(\pi_{1}(M))_{0}))$
acts transitively on $\mathbb{R}P^{n}$. The proof of the previous theorem can
be simplified as follows: Every element of $H^{\prime}_{M}(\pi_{1}(M))$ has a
fixed point since every element of $Gl(2n+1,\mathbb{R})$ has a real
eigenvalue. We deduce that $H^{\prime}_{M}(\pi_{1}(M))$ is the identity and
there exists a map $f:M\rightarrow\mathbb{R}P^{n}$ such that $D_{M}=f\circ
p_{M}$. This implies that $f$ is a covering map and $M$ is homeomorphic to
$S^{n}$ or $\mathbb{R}P^{n}$.

Let $(M,P\nabla_{M})$ be a projective manifold, suppose that
$Aut(M,P\nabla_{M})_{0}$ is not solvable. This implies that
$Aut(B(M),\nabla_{B(M)})_{0}$ and the connected component of the normalizer
$N(\pi_{1}(B(M)))_{0}$ of $\pi_{1}(B(M))$ in
$Aut(\widetilde{B(M)},\nabla_{\widetilde{B(M)}})$ are not solvable. We deduce
that the image of $N(\pi_{1}(B(M))$ by $H_{B(M)}$ contains a subgroup
$H_{S^{1}}$ isomorphic to $S^{1}$. We denote by $X^{\prime}_{B(M)}$, a vector
field which generates the Lie algebra of $H_{S^{1}}$, its pullback by the
developing map $D_{B(M)}$ of $B(M)$ is a vector field $\tilde{X}_{B(M)}$
invariant by the fundamental group of $B(M)$. We deduce that there exists a
vector field $X_{B(M)}$ of $B(M)$ whose pullback by the universal covering map
is $\tilde{X}_{B(M)}$. Suppose that the developing map is injective, the flow
of $X_{B(M)}$ defines an action of $S^{1}$ on $B(M)$ which is transverse and
commutes with the radial flow. This implies there exists a vector $X_{M}$ on
$M$ which is the image of $X_{B(M)}$ by the map induced by
$p_{B(M)}:B(M)\rightarrow M$. The vector field $X_{M}$ induces an action of
$S^{1}$ on $M$: We have:

Proposition 4.2. Let $(M,P\nabla_{M})$ be a compact projective manifold whose
developing map is injective, suppose that $Aut(M,P\nabla_{M})_{0}$ is not
solvable, then $M$ is endowed with a non trivial action of $S^{1}$.

5\. Automorphisms of projective manifolds of dimension $2$ and $3$.

In dimension $2$, we have the following result:

Proposition 5.1. Let $(M,P\nabla_{M})$ be a $2$-dimensional compact connected
oriented projective manifold, suppose that $Aut(M,P\nabla_{P})$ is not
discrete, then $M$ is homeomorphic to the $2$-dimensional torus or to the
sphere.

Proof. Suppose $N_{M}\neq 0$, there exists a non zero element $A_{M}\in N_{M}$
such that $A_{M}^{2}=0$, we deduce that $dim(ker(A_{M}))=2,dim(Im(A_{M}))=1$.
Remark that $Im(A_{M})$ is fixed by the holonomy.

Suppose that $N_{M}=0$, we deduce that $dim(S_{M})\geq 2$, there exists a non
zero element distinct of the identity $e_{M}\in S_{M}$ such that
$e^{2}_{M}=e_{M}$. To see this remark that $S_{M}$ contains either an
associative algebra isomorphic to the associative algebra of $2\times 2$ real
matrices or two idempotents which are linearly independent. The linear map
$e_{M}$ is diagonalizable and its eigenvalues are equal to $0$ and $1$. Since
the flow of $e_{M}$ is distinct of the radial flow, we deduce that $0$ is an
eigenvalue of $e_{M}$. This implies that either the dimension of the
eigenspace associated to $0$ is $1$, or the the dimension of the eigenspace
associated to $1$ is $1$. We deduce that the holonomy preserves a vector
subspace of dimension $1$.

We conclude that if $Aut(M,P\nabla_{M}))$ is not discrete, its holonomy fixed
a point of $P\mathbb{R}^{2}$. The lemma 2.5 p. 808 in Goldman [9] implies that
the Euler number of $M$ is positive.

Dimension $3$.

In this section, we study the group of automorphisms of a connected
$3$-dimensional compact projective manifold $(M,P\nabla_{M})$ whose group of
automorphisms is not discrete.

$Aut(M,P\nabla_{M})_{0}$ is not solvable.

Suppose that $Aut(M,P\nabla_{M})_{0}$ is not solvable, then
$Aut(B(M),\nabla_{B(M)})_{0}$ and $N(\pi_{1}(B(M)))_{0}$ are not solvable. We
deduce that the connected subgroup of $Gl(n+1,\mathbb{R})$,
$H_{B(M)}(N(\pi_{1}(M))_{0})$ contains a subgroup $H"$ isomorphic to $S^{1}$.
We denote by $X"_{B(M)}$ a vector field which generates the Lie algebra of
$H"$. The pullback $X^{\prime}_{B(M)}$ of $X"_{B(M)}$ by $D_{B(M)}$ is the
pullback of a vector field $X_{B(M)}$ of $B(M)$ by $p_{B(M)}$.

Suppose that the set of fixed points of $H"$ is not empty, we can write
$\mathbb{R}^{4}=U\oplus V$ where $U$ is a $2$-dimensional vector subspace
corresponding to the non trivial irreducible submodule of $H"$ and $V$ the set
of fixed points. Remark that $h_{B(M)}(\pi_{1}(B(M)))$ preserves $U$ and $V$
since it commutes with $H"$. This implies that there exists a foliation ${\cal
F}_{U}$ (resp. ${\cal F}_{V}$) on $B(M)$ whose pullback by the universal
covering map is the pullback by $D_{B(M)}$ of the foliation of
$\mathbb{R}^{4}$ whose leaves are $2$-dimensional affine spaces parallel to
$U$ (resp. parallel to $V$).

Proposition 5.2. Suppose that $V\cap D_{B(M)}(\widetilde{B(M)})$ is empty.
Then a finite cover of $M$ is a total space of a fibre bundle over $S^{1}$
whose fibre is $T^{2}$.

Proof. The vector field defined by $Y"(u,v)=u;u\in U,v\in V$ is invariant by
the holonomy of $B(M)$. To show this, remark that the restriction of $H"$ to
$U$ defines on it a complex structure and since $h_{B(M)}(\pi_{1}(B(M)))$
commutes with $H"$, its restriction to $U$ are morphisms of that complex
structure. The pullback of $Y"$ by $D_{B(M)}$ is the pullback of a vector
field $Y_{B(M)}$ of $B(M)$ by the universal covering map. The image $Y_{M}$ of
$Y_{B(M)}$ by $p_{B(M)}$ and $X_{M}$ commute and generate a locally free
action of $\mathbb{R}^{2}$ on $M$ since $V\cap D_{B(M)}(\widetilde{B(M)})$ is
empty. Chatelet, Rosenberg and Weil [6] implies that $M$ is the total space of
a fibre bundle over $S^{1}$ whose fibre is $T^{2}$.

If ${\cal F}_{V}$ has compact leaf, we have the following result:

Proposition 5.3. Let $(M,P\nabla_{M})$ be a $3$-dimensional compact projective
manifold whose developing map is injective. Suppose that
$Aut(M,P\nabla_{M})_{0}$ is not solvable and $V\cap
D_{B(M)}(\widetilde{B(M)})$ is not empty. Then the holonomy of
$(M,P\nabla_{M})$ is solvable.

Proof. Let $\hat{F}_{0}$ be a connected component of $V\cap
D_{B(M)}(\widetilde{B(M)})$ its image by the universal covering map is a
compact leaf $F_{0}$ compact leaf of ${\cal F}_{V}$ which is a $2$-dimension
compact affine manifold, we deduce that its fundamental group is solvable. Let
$r$ be the restriction of $h(\pi_{1}(B(M))$ to $V$, since
$h_{B(M)}(\pi_{1}(B(M))$ preserves $V$, we have an exact sequence:

$1\rightarrow Ker(r)\rightarrow h_{B(M)}(\pi_{1}(B(M))\rightarrow
Im(r)\rightarrow 1$

The groups $Ker(r)$ is solvable since it restriction to $U$ commutes with a
non trivial linear action of $S^{1}$. The group $Im(r)$ is also solvable since
it is contained in $h_{F_{0}}(\pi_{1}(F_{0}))$, we deduce that
$h_{B(M)}(\pi_{1}(B(M))$ is solvable.

$Aut(M,P\nabla_{M})_{0}$ is solvable.

In this section we study $3$-dimensional projective manifolds whose group of
automorphisms is solvable. We can decompose the associative algebra
$n(\pi_{1}(B(M))$ by writing: $n(\pi_{1}(B(M))=S_{M}\oplus N_{M}$, where
$S_{M}$ is a semi-simple associative algebra and $N_{M}$ a nilpotent
associative algebra. We deduce that $S_{M}$ is the direct product of
associative algebras isomorphic to either $\mathbb{R}$ or $\mathbb{C}$ and is
commutative. It results that the fact that $Aut(M,P\nabla_{M})_{0}$ is not
commutative implies that $N_{M}$ is not commutative.

$Aut(M,P\nabla_{M})_{0}$ is solvable and is not commutative.

Theorem 5.1. Suppose that $N_{M}$ is not commutative, then
$h_{B(M)}(\pi_{1}(B(M)))$, the image of the holonomy of $B(M)$ is solvable.

Proof.

First step:

Suppose that the square of every element of $N_{M}$ is zero.

Let $A,B\in N_{M}$ such that $AB\neq BA$. Suppose that $dim(ker(A))=3$. It
implies that $dim(Im(A))=1$. Since $(A+B)^{2}=0$, we deduce that $AB+BA=0$ and
$B(Ker(A))\subset Ker(A),B(Im(A))\subset Im(A))$, we deduce that the
restriction of $B$ to $Im(A)$ is zero since $B$ is nilpotent and
$dim(Im(A))=1$. This implies that $BA=0$, we deduce that $AB=0$ and $AB=BA$.
Contradiction.

Suppose that that $dim(Ker(A))=dim(Im(A))=dim(Ker(B))=dim(Im(B))=2$. We deduce
that $Im(A)=Ker(A),Im(B)=Ker(B)$ since $A^{2}=B^{2}=0$. If $Ker(A)\cap
Ker(B)=0$, $\mathbb{R}^{4}=Ker(A)\oplus Ker(B)$ and $AB=BA=0$ since $AB+BA=0$.
If $Ker(A)=Ker(B)$, $AB=BA=0$ since $Im(A)=Im(B)=Ker(A)=Ker(B)$.
Contradiction.

We deduce that $dim(Ker(A)\cap Ker(B))=1$. We can write
$\mathbb{R}^{4}=Vect(e_{1},e_{2},e_{3},e_{4})$ where $Vect(e_{1})=Ker(A)\cap
Ker(B)$, $Vect(e_{1},e_{2})=Ker(A)$ and $Vect(e_{1},e_{3})=Ker(B)$. Every
element in $h_{B(M)}(\pi_{1}(B(M))$ preserves $Ker(A)\cap Ker(B),Ker(A)$ and
$Ker(A)+Ker(B)$ since it commutes with $A$ and $B$. We deduce that
$\pi_{1}(B(M))$ is solvable since it preserves a flag.

Step 2.

Suppose that there exists an element $A\in N_{M}$ such that $A^{2}\neq 0$.

If $dim(KerA)=1$, we have $(A^{2})^{2}=0$ implies that $Im(A^{2})\subset
Ker(A^{2})$. Remark that $x\in Ker(A^{2})$ if and only if $A(x)\in Ker(A)$ and
$x\in A^{-1}(Ker(A))$; $dim(A^{-1}(Ker(A))=2$ since $dim(Ker(A))=1$. We deduce
that $dim(Ker(A^{2}))=dim(Im(A^{2}))=2$, and $Ker(A)\subset
Ker(A^{2})=Im(A^{2})\subset Im(A)$ and $\pi_{1}(B(M))$ preserves a flag since
it commutes with $A$ and $dim(Im(A))=3$.

Suppose that $dim(ker(A))=2$, we deduce that $dim(Im(A))=2$. Suppose that
$Ker(A)\cap Im(A)=0$, we deduce that $\mathbb{R}^{4}=Ker(A)\oplus Im(A)$. This
is impossible since $A$ is nilpotent. We also remark that $Ker(A)$ is distinct
of $Im(A)$ since $A^{2}\neq 0$. This implies that $dim(Ker(A)\cap Im(A))=1$.
Every element of $\pi_{1}(B(M))$ preserves, $Ker(A)\cap Im(A),Ker(A)$ and
$Ker(A)+Im(A)$ and thus preserves a flag. We deduce that $h_{B(M)}(\pi(B(M)))$
is solvable.

$Aut(M,P\nabla_{M})_{0}$ is commutative.

Suppose that the developing map is injective and $Aut(M,P\nabla_{M})_{0}$ is
commutative and its dimension is superior or equal to $2$. Let $X_{M}$ and
$Y_{M}$ two projective vector fields linearly independent. We denote by
$X_{B(M)}$ and $Y_{B(M)}$ two affine vector fields of $B(M)$ whose respective
images by $p_{B(M)}$ are $X_{M}$ and $Y_{M}$. There exist affine vector fields
$X^{\prime}_{B(M)}$ and $Y^{\prime}_{B(M)}$ of $\mathbb{R}^{4}$ whose
respective images by the covering map are $X_{B(M)}$ and $Y_{B(M)}$. Remark
that if the group generated by $X_{M}$ and $Y_{M}$ acts freely on $M$, then
Chatelet, Rosenberg and Weil [6] implies that $M$ is the total space of a
torus bundle.

In the rest of this section we assume that the set of zero of $X_{M}$ is not
empty. This implies that we can assume that the set of zero $U$ of
$X^{\prime}_{B(M)}$ is not empty by eventually replacing $X^{\prime}_{B(M)}$
with $X^{\prime}_{B(M)}+cX_{R}$ where $c\in\mathbb{R}$ and $X_{R}$ is the
radiant flow. We denote by $B(N)$ the image of $U\cap D_{B(M)}(\tilde{M})$ by
the covering map. Remark that $B(N)$ is not empty.

Proposition 5.4. Suppose that $dim(U)=3$ then $\pi_{1}(M)$ is solvable.

Proof. Suppose that the restriction of $Y^{\prime}_{B(M)}$ to $U$ is not zero.
This implies that the restriction of $Y_{B(M)}$ to $B(N)$ is not zero. This
implies that the group of projective automorphisms of $N$, the quotient of
$B(N)$ by the radiant flow is not discrete. The proposition 5.1 implies that
$N$ has a finite cover homeomorphic to $S^{2}$ or $T^{2}$. This implies that
$\pi_{1}(B(N))$ is solvable. The restriction of $\pi_{1}({B(M)})$ to $U$
induces an exact sequence whose image is contained in the image of the
holonomy representation of $B(N)$ and whose kernel is solvable. We deduce that
$\pi_{1}(B(M))$ and $\pi_{1}(M)$ are solvable. If the restriction of
$Y^{\prime}_{B(M)}$ to $U$ vanishes, let $V$ be the image of
$X^{\prime}_{B(M)}$, if $V$ is not contained in $U$, then
$\mathbb{R}^{4}=U\oplus V$, and since $Y^{\prime}_{B(M)}$ commutes with
$X^{\prime}_{B(M)}$, it preserves $V$. This implies that $X^{\prime}_{B(M)}$
and $Y^{\prime}_{B(M)}$ are linearly dependent contradiction.

Suppose that $V$ is a subset of $U$, let $W$ be the image of
$Y^{\prime}_{B(M)}$, if $W$ is not contained in $U$, then we can apply the
previous argument to obtain a contradiction by replacing $X^{\prime}_{B(M)}$
by $Y^{\prime}_{B(M)}$. Suppose that $W$ is contained in $U$, $V\cap
W=\\{0\\}$ since $X^{\prime}_{B(M)}$ and $Y^{\prime}_{B(M)}$ are linearly
independent. We deduce that, the holonomy of $B(M)$ preserves, $V,V\oplus W$
and $U$. This implies that the holonomy of $B(M)$ and $\pi_{1}(B(M))$ are
solvable.

Proposition 5.5. Suppose that $dim(U)=2$, then $\pi_{1}(B(M))$ is solvable.

Proof. Suppose that $U\oplus Im(X^{\prime}_{B(M)})=\mathbb{R}^{4}$.

Step $1$.

If the restriction of $X^{\prime}_{B(M)}$ or $Y^{\prime}_{B(M)}$to
$Im(X^{\prime}_{B(M)})$ are not a multiple of the identity, we deduce that
$r(\pi_{1}(B(M))$, the image of the restriction of $\pi_{1}(B(M))$ to
$Im(X^{\prime}_{B(M)})$ is solvable since it commutes with $X^{\prime}_{B(M)}$
and $Y^{\prime}_{B(M)}$. The restriction of $\pi_{1}(B(M))$ to $Ker(X_{B(M)})$
is contained in the holonomy group the $2$-dimensional closed affine manifold
$B(N)$, we deduce that it is solvable. This implies $\pi_{1}(B(M))$ is
solvable.

Step $2$.

Suppose that the restriction of $X^{\prime}_{B(M)}$ and $Y^{\prime}_{B(M)}$ to
$Im(X^{\prime}_{B(M)})$ are multiple of the identity. If the restriction of
$Y^{\prime}_{B(M)}$ to $U$ is equal to $aI_{U}$, we deduce that
$Y^{\prime}_{B(M)}$ is contained in the vector space generated by
$Id_{\mathbb{R}^{4}}$ and $X^{\prime}_{B(M)}$. This implies that the dimension
of the vector space generated by $X_{M}$ and $Y_{M}$ is $1$. Contradiction. We
deduce that the restriction of $Y^{\prime}_{B(M)}$ to $U$ is not a multiple of
the $Id_{U}$. There exists a real $a$ such that the restriction of
$Z=Y^{\prime}_{B(M)}+aId_{\mathbb{R}^{4}}$ to $Im(X^{\prime}_{B(M)})$ is zero.
The restriction of $Z$ to $U$ is distinct of a multiple of the identity, we
conclude that $\pi_{1}(B(M))$ is solvable by replacing $Y^{\prime}_{B(M)}$ by
$Z$ in the first step of the proof.

Suppose that $dim(U\cap Im(X^{\prime}_{B(M)})=1$. The vector subspaces, $U\cap
Im(X^{\prime}_{B(M)}),U,U\oplus Im(X^{\prime}_{B(M)})$ are stable by the
holonomy. We deduce that $\pi_{1}(B(M))$ preserves a flag and is solvable.

Suppose that $U=Im(X^{\prime}_{B(M)})$.

We can write $\mathbb{R}^{4}=Vect(e_{1},e_{2},e_{3},e_{4})$ where
$Vect(e_{1},e_{2})=U$ and
$X^{\prime}_{B(M)}(e_{3})=e_{1},X^{\prime}_{B(M)}(e_{4})=e_{2}$. Let $\gamma$
be an element of $\pi_{1}(B(M))$, if we write the fact that the matrix
$M(\gamma)$ of $\gamma$ commutes with the matrix of $X^{\prime}_{B(M)}$ in the
basis $(e_{1},e_{2},e_{3},e_{4})$, we obtain that:

$M(\gamma)=\pmatrix{a_{1}&b_{1}&c_{1}&d_{1}\cr a_{2}&b_{2}&c_{2}&d_{2}\cr
0&0&a_{1}&b_{1}\cr 0&0&a_{2}&b_{2}}$

Since the restriction of $\pi_{1}(B(M))$ to $U$ is contained in the holonomy
of $\pi_{1}(B(N))$ which is solvable, we deduce that $\pi_{1}(B(M))$ is
solvable.

Proposition 5.6. Suppose that $dim(U)=1$, then $\pi_{1}(B(M))$ is nilpotent.

Proof. We can write $n(\pi_{1}(B(M))=S_{M}\oplus N_{M}$ where $S_{M}$ is semi-
simple and $N_{M}$ nilpotent. Suppose that $N_{M}$ is not zero, and consider
$n\in N_{M}$, if $n^{2}\neq 0$, $(n^{2})^{2}=0$, $dim(Ker(n^{2}))\geq 2$, if
$n^{2}=0$, $dim(Ker(n))\geq 2$. We can apply the proposition 5.4 and the
proposition 5.5.

Suppose that $N_{M}=0$, $n(B(M))$ contains a non zero idempotent $u_{1}$ which
is not a multiple of the identity, since it does not generates the radiant
flow. If the eigenvalues of $u_{1}$ are $0$ or $1$, this implies that
$dim(Ker(u_{1}))\geq 2)$ or $dim(Ker(u_{1})-Id_{\mathbb{R}^{4}})\geq 2$, we
can apply the proposition 5.4 and the proposition 5.5 to deduce that
$\pi_{1}(B(M))$ is nilpotent.

Theorem 5.2. Let $(M,P\nabla_{M})$ be a $3$-dimensional projective manifold
whose developing map is injective, suppose that $dim(Aut(M,P\nabla_{M}))\geq
2$, then $M$ is homeomorphic to a spherical manifold, $S^{2}\times S^{1}$ or a
finite cover of $M$ is a torus bundle.

Proof. Suppose that $Aut(M,P\nabla_{M})_{0}$ is not solvable, the proposition
5.2 and the proposition 5.3 imply that either $M$ is the total space of a
bundle over $S^{1}$ whose fibre is homeomorphic to $T^{2}$ or $\pi_{1}(M)$ is
solvable. If $Aut(M,P\nabla_{M})_{0}$ is solvable, the theorem 5.1, the
propositions 5.4, 5.5 and 5.6 show that $\pi_{1}(M)$ is solvable. In [2], it
is shown that a $3$-dimensional closed manifold whose fundamental group is
solvable is homeomorphic to a spherical manifold, $S^{1}\times S^{2}$, a
finite cover of $M$ is a torus bundle, or $\mathbb{R}P^{3}\\#\mathbb{R}P^{3}$.
Benoist [3] and Goldman and Cooper [8] have shown that there does not exist a
projective structure on $\mathbb{R}P^{3}\\#\mathbb{R}P^{3}$.

References.

1\. Albert A. Structure of algebras. Vol. 24. American Math. Soc. 1939.

2\. Aschenbrenner, M. Friedl, S. Wilton, H. 3 manifold groups.
https://arxiv.org/pdf/1205.0202.pdf

3\. Benoist, Y. Nilvariétés projectives. Commentarii Mathematici Helvetici,
1994, vol. 69, no 1, p. 447-473.

4\. Benzecri J.P. Sur les variétés localement affines et localement
projectives. Bulletin de la S.M.F. 88 (1960) 229-332.

5\. Bredon, G. Bredon, Glen E. Topology and geometry. Vol. 139. Graduate texts
in Math.

6\. Chatelet, Gilles, Rosenberg, Harold, et Weil, Daniel. A classification of
the topological types of $\mathbf{R}^{2}$-actions on closed orientable
3-manifolds. Publications Mathématiques de l’IHÉS, 1974, vol. 43, p. 261-272

7\. Conlon, Lawrence. Transversally parallelizable foliations of codimension
two. Transactions of the American Mathematical Society, 1974, vol. 194, p.
79-102.

8\. Cooper, D. Goldman, W. A 3–Manifold with no Real Projective Structure. In
Annales de la Faculté des sciences de Toulouse: Mathématiques, vol. 24, no. 5,
pp. 1219-1238. 2015

9\. Goldman, W. Convex real projective structures on compact surfaces. J.
Differential Geometry 31 (1990) 791-845.

10\. Montgomery D. Samelson H. Transformation groups of spheres. Annals of
Mathematics 44 (1943) 454-470.

11\. Montgomery, D. Zippin, L. Topological transformation groups. Interscience
Tracts in pure and applied mathematics 1955.

12\. Sullivan, D; Thurston,, W. Manifolds with canonical coordinate charts:
some examples.” Enseign. Math 29 (1983): 15-25.

13\. Tsemo, A. Dynamique des variétés affines. Journal of the London
Mathematical Society 63 (20010 469-486.

14\. Tsemo, A. Décomposition des variétés affines. Bulletin des sciences
mathématiques 125 (2001) 71-83.

15\. A Tsemo, A. ”Linear foliations on affine manifolds.” arXiv preprint
arXiv:2008.05357 (2020).