5 Invariant Koszul Form of Homogeneous Bounded Domains …
101
induces a left- invariant foliation which is isotropic with respect to the Kirillov
symplectic form. In [95], author has given a new proof that semi-simple co-adjoint
orbits through real hyperbolic elements are symplectomorphic to cotangent bundles,
establishing a new connection between the Iwasawa horospherical projection and the
symplectic geometry of real hyperbolic co-adjoint orbits. In [96] authors have studied
orbits of coadjoint representations of classical compact Lie groups by an explicit
parameterization of the orbit by means of a generalized stereographic projection,
obtaining a Kählerian structure on the orbit, introducing basis two-forms for the
cohomology group of the orbit.
5.4 Koszul Hessian Geometric Structure of Information
Geometry
The elementary geometric structures discovered by Jean-Louis Koszul are the foundations of Information Geometry. These links were first established by Professor
Hirohiko Shima [97–102] in his 2007 book entitled “The Geometry of Hessian
Structures” [103], which is dedicated to Professor Koszul. The origin of this work
followed the visit of Koszul in Japan in 1964, for a mission coordinated with the
French government. Koszul taught lectures on the theory of flat manifolds at Osaka
University. Hirohiko Shima was then a student and attended these lectures with the
teachers Matsushima and Murakami. This lecture was at the origin of the notion of
Hessian structures and the beginning of the works of Hirohiko Shima. Henri Cartan
noted concerning Koszul’s ties with Japan, “Koszul has attracted eminent mathematicians from abroad to Strasbourg and Grenoble. I would like to mention in particular
the links he has established with representatives of the Japanese School of Differential Geometry”. Shima’s book [103] is a systematic introduction to the theory of
Hessian structures (provided by a pair of a flat connection D and an Hessian metric
g). Koszul studied flat manifolds with a closed 1-form α, such that Dα be positive
definite, where Dα is a hessian metric . However, not all Hessian metrics are globally of the form g = Dα . Shima introduces the notion of Codazzi structure for a
pair (D,g), with D a torsion-free connection, which verifies the Codazzi equation
(D X g)(Y, Z ) = (D Y g)(X, Z ). A Hessian structure is a Codazzi structure for which
connection D is flat. This is an extension of Riemannian geometry. It is then possible
to define a connection D’ and a dual Codazzi structure (D’,g) with D
= ∇− D where
∇ is the Levi-Civita connection. For a hessian structure (D, g) with g = Ddϕ, the
dual Codazzi structure
D
, g
is also a Hessian structure and g = D
dϕ
, where ϕ
is the Legendre transform of ϕ: ϕ
=
i
x
i ∂ϕ
∂ x i − ϕ. Shima observed that Information
101
induces a left- invariant foliation which is isotropic with respect to the Kirillov
symplectic form. In [95], author has given a new proof that semi-simple co-adjoint
orbits through real hyperbolic elements are symplectomorphic to cotangent bundles,
establishing a new connection between the Iwasawa horospherical projection and the
symplectic geometry of real hyperbolic co-adjoint orbits. In [96] authors have studied
orbits of coadjoint representations of classical compact Lie groups by an explicit
parameterization of the orbit by means of a generalized stereographic projection,
obtaining a Kählerian structure on the orbit, introducing basis two-forms for the
cohomology group of the orbit.
5.4 Koszul Hessian Geometric Structure of Information
Geometry
The elementary geometric structures discovered by Jean-Louis Koszul are the foundations of Information Geometry. These links were first established by Professor
Hirohiko Shima [97–102] in his 2007 book entitled “The Geometry of Hessian
Structures” [103], which is dedicated to Professor Koszul. The origin of this work
followed the visit of Koszul in Japan in 1964, for a mission coordinated with the
French government. Koszul taught lectures on the theory of flat manifolds at Osaka
University. Hirohiko Shima was then a student and attended these lectures with the
teachers Matsushima and Murakami. This lecture was at the origin of the notion of
Hessian structures and the beginning of the works of Hirohiko Shima. Henri Cartan
noted concerning Koszul’s ties with Japan, “Koszul has attracted eminent mathematicians from abroad to Strasbourg and Grenoble. I would like to mention in particular
the links he has established with representatives of the Japanese School of Differential Geometry”. Shima’s book [103] is a systematic introduction to the theory of
Hessian structures (provided by a pair of a flat connection D and an Hessian metric
g). Koszul studied flat manifolds with a closed 1-form α, such that Dα be positive
definite, where Dα is a hessian metric . However, not all Hessian metrics are globally of the form g = Dα . Shima introduces the notion of Codazzi structure for a
pair (D,g), with D a torsion-free connection, which verifies the Codazzi equation
(D X g)(Y, Z ) = (D Y g)(X, Z ). A Hessian structure is a Codazzi structure for which
connection D is flat. This is an extension of Riemannian geometry. It is then possible
to define a connection D’ and a dual Codazzi structure (D’,g) with D
= ∇− D where
∇ is the Levi-Civita connection. For a hessian structure (D, g) with g = Ddϕ, the
dual Codazzi structure
D
, g
is also a Hessian structure and g = D
dϕ
, where ϕ
is the Legendre transform of ϕ: ϕ
=
i
x
i ∂ϕ
∂ x i − ϕ. Shima observed that Information