94
F. Barbaresco
affine transformations leaving stable a non-degenerate convex open domain and a
homogeneous domain.
In the paper «Variétés localement plates et convexité» [75] of 1965, Koszul established the following theorem: let M be a locally related differentiable manifold. If
the universal covering of M is isomorphic as a flat manifold with a convex open
domain containing no straight line in a real affine space, then there exists on M a
closed differential form α such that Dα (D linear covariant derivative of zero torsion)
is defined positive in all respects and which is invariant under every automorphism
of M. If G is a group of automorphisms of M such that G\M is quasi-compact and
if there exists on M a closed 1-differential form α invariant by G and such that Dα
is definite positive at any point, then the universal covering of M is isomorphic as
a flat manifold with a convex open domain that does not contain a straight line in a
real affine space.
In the paper «Déformations des variétés localement plates» [81] of 1968, Koszul
provided other proofs of theorems introduced in [75]. Koszul considered related
differentiable manifolds of dimension n and TM the fibered space of M. The linear
connections on M constitute a subspace of the space of the differentiable applications
of the TMxTM fiber product in the space T(TM) of the TM vectors. Any locally flat
connection D (the curvature and the torsion are zero) defines a locally flat connection
on the covering of M, and is hyperbolic when universal covering of M, with this
connection, is isomorphic to a sharp convex open domain (without straight lines) in
R
n . Koszul showed that, if M is a compact manifold, for a locally flat connection on
M to be hyperbolic, it is necessary and sufficient that there exists a closed differential
form of degree 1 on M whose covariant differential is positive definite.
In the paper «Trajectoires convexes de groupes affines unimodulaires» [82] in
1970, Koszul demonstrated that a convex sharp open domain in R
n that admits a
unimodular transitive group of affine automorphisms is an auto-dual cone. This is a
more geometric demonstration of the results shown by Ernest Vinberg [64] on the
automorphisms of convex cones.
We will also introduce in the following more recent results as a recent work of Della
Vedova on covering spaces of symplectic manifolds (co-adjoint orbits, endowed with
their canonical Kirillov-Kostant-Souriau symplectic structure) admitting homogeneous non Chern-Ricci flat special compatible almost complex structures, or Biquard
extension of Kostant-Sekiguchi-Vergne correspondance.
5.2 Invariant Koszul Form for Homogeneous Bounded
Domains
Koszul considered on G/B an invariant complex structure tensor I. All the invariant
volumes on G/B, equal up to a constant factor, define with the complex structure the
same invariant Hermitian form on G/B, called Hermitian canonical form, denoted
h. Let E be a differentiable fiber space of base M and let p be the projection of E
F. Barbaresco
affine transformations leaving stable a non-degenerate convex open domain and a
homogeneous domain.
In the paper «Variétés localement plates et convexité» [75] of 1965, Koszul established the following theorem: let M be a locally related differentiable manifold. If
the universal covering of M is isomorphic as a flat manifold with a convex open
domain containing no straight line in a real affine space, then there exists on M a
closed differential form α such that Dα (D linear covariant derivative of zero torsion)
is defined positive in all respects and which is invariant under every automorphism
of M. If G is a group of automorphisms of M such that G\M is quasi-compact and
if there exists on M a closed 1-differential form α invariant by G and such that Dα
is definite positive at any point, then the universal covering of M is isomorphic as
a flat manifold with a convex open domain that does not contain a straight line in a
real affine space.
In the paper «Déformations des variétés localement plates» [81] of 1968, Koszul
provided other proofs of theorems introduced in [75]. Koszul considered related
differentiable manifolds of dimension n and TM the fibered space of M. The linear
connections on M constitute a subspace of the space of the differentiable applications
of the TMxTM fiber product in the space T(TM) of the TM vectors. Any locally flat
connection D (the curvature and the torsion are zero) defines a locally flat connection
on the covering of M, and is hyperbolic when universal covering of M, with this
connection, is isomorphic to a sharp convex open domain (without straight lines) in
R
n . Koszul showed that, if M is a compact manifold, for a locally flat connection on
M to be hyperbolic, it is necessary and sufficient that there exists a closed differential
form of degree 1 on M whose covariant differential is positive definite.
In the paper «Trajectoires convexes de groupes affines unimodulaires» [82] in
1970, Koszul demonstrated that a convex sharp open domain in R
n that admits a
unimodular transitive group of affine automorphisms is an auto-dual cone. This is a
more geometric demonstration of the results shown by Ernest Vinberg [64] on the
automorphisms of convex cones.
We will also introduce in the following more recent results as a recent work of Della
Vedova on covering spaces of symplectic manifolds (co-adjoint orbits, endowed with
their canonical Kirillov-Kostant-Souriau symplectic structure) admitting homogeneous non Chern-Ricci flat special compatible almost complex structures, or Biquard
extension of Kostant-Sekiguchi-Vergne correspondance.
5.2 Invariant Koszul Form for Homogeneous Bounded
Domains
Koszul considered on G/B an invariant complex structure tensor I. All the invariant
volumes on G/B, equal up to a constant factor, define with the complex structure the
same invariant Hermitian form on G/B, called Hermitian canonical form, denoted
h. Let E be a differentiable fiber space of base M and let p be the projection of E