5 Invariant Koszul Form of Homogeneous Bounded Domains …
99
Koszul studied symmetric homogeneous spaces and defines the relation between
invariant flat affine connections and the affine representations of Lie algebras and
invariant Hessian metrics characterized by affine representations of Lie algebras.
Koszul provides a correspondence between symmetric homogeneous spaces with
invariant Hessian structures using affine representations of Lie algebras, and proves
that a symmetric homogeneous space simply connected with an invariant Hessian
structure is a direct product of a Euclidean space and of a homogeneous dual-cone.
Let G be a connected Lie group and G/K a homogeneous space over which G acts
effectively. Koszul gives a bijective correspondence between all planar G -invariantes
connections on G/K and all of a certain class of affine representations of the Lie
algebra of G. The main theorem of Koszul is:
Koszul’s Theorem: Let G/K be a homogeneous space of a connected Lie group
G and be gand kthe Lie algebras of G and K, assuming that G/K has G-invariant
connection, then admits an affine representation (f, q) on the vector space E.
Conversely, assume that G is simply connected and has an affine representation,
then G/K admits a flat G-invariant connection.
5.3 Contextualization with Last Advanced Works
Every Kähler manifold has a symplectic structure, but the converse that every closed
symplectic manifold has also a Kähler structure is not true. Based on an observation
of Paulette Liberman, W. P. Thurston [84] has produced some counter-examples
of symplectic manifolds which are not Kähler (Kodaira-Thurston manifold). But,
recently Alberto Della Vedova has observed that co-adjoint orbits, endowed with their
canonical Kirillov-Kostant-Souriau symplectic structure [85–87], serve as covering
spaces of symplectic manifolds admitting homogeneous non Chern-Ricci flat special
compatible almost complex structures. This result has been developed in PhD of
Alice Gatti, supervised by Alberto Della Vedova that adjoint orbits of non-compact
semisimple Lie groups turn out to be naturally almost-Kähler manifolds endowed
with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almostcomplex structure, providing explicit formulae for the Chern-Ricci form.
Let us denote by ρ the Ricci curvature form, then we have ([X, Y ]) = ρ(X, Y ).
This relation imply that a symplectic manifold admitting a homogeneous special
compatible almost complex structure with non-zero Hermitian scalar curvature is,
up to coverings, a coadjoint orbit equipped with the KKS (Kirillov-Kostant-Souriau)
symplectic form [88–90].
Theorem [84, 88] Let (M, ω) be a symplectic manifold admitting a homogeneous
compatible almost complex structure satisfying ρ = λω for some λ = 0. Then M is
a covering space of a coadjoint orbit and ω is the pull-back via the covering map of
the canonical symplectic form:
([X, Y ]) = λσ (X, Y ) = λB(v, [X, Y ]) with B(X, Y ) = T r(ad X ad Y ) killing
form where K is the isotropy subgroup of v with respect to the adjoint representation
99
Koszul studied symmetric homogeneous spaces and defines the relation between
invariant flat affine connections and the affine representations of Lie algebras and
invariant Hessian metrics characterized by affine representations of Lie algebras.
Koszul provides a correspondence between symmetric homogeneous spaces with
invariant Hessian structures using affine representations of Lie algebras, and proves
that a symmetric homogeneous space simply connected with an invariant Hessian
structure is a direct product of a Euclidean space and of a homogeneous dual-cone.
Let G be a connected Lie group and G/K a homogeneous space over which G acts
effectively. Koszul gives a bijective correspondence between all planar G -invariantes
connections on G/K and all of a certain class of affine representations of the Lie
algebra of G. The main theorem of Koszul is:
Koszul’s Theorem: Let G/K be a homogeneous space of a connected Lie group
G and be gand kthe Lie algebras of G and K, assuming that G/K has G-invariant
connection, then admits an affine representation (f, q) on the vector space E.
Conversely, assume that G is simply connected and has an affine representation,
then G/K admits a flat G-invariant connection.
5.3 Contextualization with Last Advanced Works
Every Kähler manifold has a symplectic structure, but the converse that every closed
symplectic manifold has also a Kähler structure is not true. Based on an observation
of Paulette Liberman, W. P. Thurston [84] has produced some counter-examples
of symplectic manifolds which are not Kähler (Kodaira-Thurston manifold). But,
recently Alberto Della Vedova has observed that co-adjoint orbits, endowed with their
canonical Kirillov-Kostant-Souriau symplectic structure [85–87], serve as covering
spaces of symplectic manifolds admitting homogeneous non Chern-Ricci flat special
compatible almost complex structures. This result has been developed in PhD of
Alice Gatti, supervised by Alberto Della Vedova that adjoint orbits of non-compact
semisimple Lie groups turn out to be naturally almost-Kähler manifolds endowed
with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almostcomplex structure, providing explicit formulae for the Chern-Ricci form.
Let us denote by ρ the Ricci curvature form, then we have ([X, Y ]) = ρ(X, Y ).
This relation imply that a symplectic manifold admitting a homogeneous special
compatible almost complex structure with non-zero Hermitian scalar curvature is,
up to coverings, a coadjoint orbit equipped with the KKS (Kirillov-Kostant-Souriau)
symplectic form [88–90].
Theorem [84, 88] Let (M, ω) be a symplectic manifold admitting a homogeneous
compatible almost complex structure satisfying ρ = λω for some λ = 0. Then M is
a covering space of a coadjoint orbit and ω is the pull-back via the covering map of
the canonical symplectic form:
([X, Y ]) = λσ (X, Y ) = λB(v, [X, Y ]) with B(X, Y ) = T r(ad X ad Y ) killing
form where K is the isotropy subgroup of v with respect to the adjoint representation