Chapter 5
Invariant Koszul Form of Homogeneous
Bounded Domains and Information
Geometry Structures
Frédéric Barbaresco
Abstract In 1955, Jean-Louis Koszul has written a seminal paper entitled “Sur la
forme hermitienne canonique des espaces homogènes complexes“. Let G be a Lie
group and let B be a closed subgroup of G, Koszul introduced a hermitian form that
he called canonical hermitian form of the complex homogeneous space G/B with an
invariant volume form. In this seminal paper, Koszul has investigated homogeneous
spaces and necessary conditions to carry a nondegenerate 2-form derived from the
invariant volume element. The advantage of this form for the determination of homogeneous bounded domains was underlined by Elie Cartan in last sentence of his 1932
paper “Sur les domaines bornés de l’espace de n variables complexes”, observing
that a necessary condition for G/B to be a bounded domain is that this form is positive definite. Hirohiko Shima has first observed that this geometry of Koszul Hessian
structures is linked with Information Geometry. Jean-Louis Koszul also developed
these structures in a Lecture “Exposés sur les Espaces Homogènes Symétriques”
given at Sao-Paulo in 1958. We will synthetize these Jean-Louis Koszul works of 1955
and 1958 on this invariant form. We will put these elements in the context of more
recent researches 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. In last part, we develop the role of homogeneous bounded domains
and invariant Koszul form in the framework of Information Geometry with use-case
of Gaussian laws.
Keywords Invariant koszul form · Homogeneous bounded domains · Information
geometry
F. Barbaresco (B)
Thales Land & Air Systems, Voie Pierre-Gilles de Gennes, F91470 Limours, France
e-mail: frederic.barbaresco@thalesgroup.com
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_5
89
Invariant Koszul Form of Homogeneous
Bounded Domains and Information
Geometry Structures
Frédéric Barbaresco
Abstract In 1955, Jean-Louis Koszul has written a seminal paper entitled “Sur la
forme hermitienne canonique des espaces homogènes complexes“. Let G be a Lie
group and let B be a closed subgroup of G, Koszul introduced a hermitian form that
he called canonical hermitian form of the complex homogeneous space G/B with an
invariant volume form. In this seminal paper, Koszul has investigated homogeneous
spaces and necessary conditions to carry a nondegenerate 2-form derived from the
invariant volume element. The advantage of this form for the determination of homogeneous bounded domains was underlined by Elie Cartan in last sentence of his 1932
paper “Sur les domaines bornés de l’espace de n variables complexes”, observing
that a necessary condition for G/B to be a bounded domain is that this form is positive definite. Hirohiko Shima has first observed that this geometry of Koszul Hessian
structures is linked with Information Geometry. Jean-Louis Koszul also developed
these structures in a Lecture “Exposés sur les Espaces Homogènes Symétriques”
given at Sao-Paulo in 1958. We will synthetize these Jean-Louis Koszul works of 1955
and 1958 on this invariant form. We will put these elements in the context of more
recent researches 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. In last part, we develop the role of homogeneous bounded domains
and invariant Koszul form in the framework of Information Geometry with use-case
of Gaussian laws.
Keywords Invariant koszul form · Homogeneous bounded domains · Information
geometry
F. Barbaresco (B)
Thales Land & Air Systems, Voie Pierre-Gilles de Gennes, F91470 Limours, France
e-mail: frederic.barbaresco@thalesgroup.com
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_5
89