92
F. Barbaresco
Fig. 5.2 Three courses given by Jean-Louis Koszul at Sao-Paulo during 50’s on «Faisceaux et
Cohomologie»;«Variétés Kählériennes» and «Exposés sur les espaces homogènes symétriques»
was pushed much further by the Russian School [58–71]. It is the work of Piatetskii
Shapiro on the Siegel domains, then those of E.B. Vinberg on the homogeneous cones
that led me to the study of the affine transformation groups of the locally flat manifolds
and in particular to the convexity criteria related to invariant forms”. In particular,
J.L. Koszul main source of inspiration is given in this last sentence of Elie Cartan’s
1935 paper [72]: “It is clear that if one could demonstrate that all homogeneous
domains whose form Φ =
i, j
∂
2 log K (z,z
∗ )
∂z i ∂z
∗
j
dz i dz
∗
j is positive definite are symmetric,
the whole theory of homogeneous bounded domains would be elucidated. This is
a problem of Hermitian geometry certainly very interesting”. The work of Koszul
has also been extended and deepened by one of his student Jacques Vey in [73]
and [74]. Jacques Vey has transposed the notion of hyperbolicity, developed by W.
Kaup for Riemann surfaces, into the category of differentiable manifolds with flat
linear connection (locally flat manifolds), which makes it possible to completely
characterize the locally flat manifolds admitting as universal covering a convex open
sharp cone of R
n , which had been studied by Koszul in [75]. The links between
Koszul’s work and those of Ernest B. Vinberg [58–65] were recently developed
at the conference “Transformation groups 2017” in Moscow dedicated to the 80th
anniversary of Professor EB Vinberg, in Dmitri Alekseevsky’s talk on “Vinberg’s
theory of homogeneous convex cones: developments and applications” [76].
In the paper «Sur la forme hermitienne canonique des espaces homogènes
complexes» [77] of 1955: Koszul considers the Hermitian structure of a homogeneous G/B manifold (G related Lie group and B a closed subgroup of G, associated,
up to a constant factor, to the single invariant G, and to the invariant complex structure
by the operations of G). Koszul says “The interest of this form for the determination
of homogeneous bounded domains has been emphasized by Elie Cartan: a necessary condition for G/B to be a bounded domain is indeed that this form is positive
definite”. Koszul calculated this canonical form from infinitesimal data Lie algebra
F. Barbaresco
Fig. 5.2 Three courses given by Jean-Louis Koszul at Sao-Paulo during 50’s on «Faisceaux et
Cohomologie»;«Variétés Kählériennes» and «Exposés sur les espaces homogènes symétriques»
was pushed much further by the Russian School [58–71]. It is the work of Piatetskii
Shapiro on the Siegel domains, then those of E.B. Vinberg on the homogeneous cones
that led me to the study of the affine transformation groups of the locally flat manifolds
and in particular to the convexity criteria related to invariant forms”. In particular,
J.L. Koszul main source of inspiration is given in this last sentence of Elie Cartan’s
1935 paper [72]: “It is clear that if one could demonstrate that all homogeneous
domains whose form Φ =
i, j
∂
2 log K (z,z
∗ )
∂z i ∂z
∗
j
dz i dz
∗
j is positive definite are symmetric,
the whole theory of homogeneous bounded domains would be elucidated. This is
a problem of Hermitian geometry certainly very interesting”. The work of Koszul
has also been extended and deepened by one of his student Jacques Vey in [73]
and [74]. Jacques Vey has transposed the notion of hyperbolicity, developed by W.
Kaup for Riemann surfaces, into the category of differentiable manifolds with flat
linear connection (locally flat manifolds), which makes it possible to completely
characterize the locally flat manifolds admitting as universal covering a convex open
sharp cone of R
n , which had been studied by Koszul in [75]. The links between
Koszul’s work and those of Ernest B. Vinberg [58–65] were recently developed
at the conference “Transformation groups 2017” in Moscow dedicated to the 80th
anniversary of Professor EB Vinberg, in Dmitri Alekseevsky’s talk on “Vinberg’s
theory of homogeneous convex cones: developments and applications” [76].
In the paper «Sur la forme hermitienne canonique des espaces homogènes
complexes» [77] of 1955: Koszul considers the Hermitian structure of a homogeneous G/B manifold (G related Lie group and B a closed subgroup of G, associated,
up to a constant factor, to the single invariant G, and to the invariant complex structure
by the operations of G). Koszul says “The interest of this form for the determination
of homogeneous bounded domains has been emphasized by Elie Cartan: a necessary condition for G/B to be a bounded domain is indeed that this form is positive
definite”. Koszul calculated this canonical form from infinitesimal data Lie algebra