Séminaire GTE - Jeudi 24 septembre, 11h15

Judit Abardia :  Integral geometry in complex space forms

Classicaly, integral geometry in Euclidean space $\R^n$ deals with two problems: the study of the
measure of planes meeting a convex domain in terms of the geometry of the convex domain, and the study
of the kinematic formula of Blashcke-Chern-Santaló.

In order to study the first problem, it is used an invariant (with respect to the isometry group) density on the space of planes. 

If we interpret the measure of planes meeting a convex domain as a functional of the space of convex domains
in $\R^n$ to $\R$, then this functional satisfies an additive property (the same as the volume), and it is called valuation.
Moreover, it is invariant under the isometry group of $\R^n$. The space of functionals satisfying these properties has structure of finite dimensional vector space.

If now we change $\R^n$ by $\C^n$, we can study the same questions (we can also consider other manifolds such
as $\CP^n$ or $\CH^n$). In this talk, using a basis of the space of invariant valuations under the isometry group of $\C^n$,
we will give analogous results in $\C^n$, and also an expression for the Gauss-Bonnet formula in $\CP^n$ and $\CH^n$.