Gianmarco Vega-Molino (Uni Bergen)
In the field of Riemannian geometry, the incredible Gauss-Bonnet theorem was the starting point for Index Theory, arguably one of the most important and successful endeavors in modern differential geometry. Roughly speaking, this theory shows that it is possible to recover topological information from the smooth and Riemannian structures on a manifold, which a priori are freely chosen. Index theorey sits at the intersection of differential geometry, operator theory, homology, and topology, and today continues to be a fruitful source of new ideas.
In this talk we will discuss these ideas in the setting of sub-Riemannian geometry. Whereas in Riemannian geometry one equips a smooth manifold with a symmetric, non-degenerate bilinear form that acts as an "inner product" and allows one to define the length of vector fields and curves, in sub-Riemannian geometry the coresponding form is degenerate. By insisting on a Hormander-type condition (also called bracket-generating) one can guarantee that it is possible to connect any two points by curves with well-defined lengths. The overarching question is which among well-understood notions in Riemannian geometry can be transposed to this setting.
In particular, we will discuss a Gauss-Bonnet theorem for an important subclass of sub-Riemannian manifolds called H-type foliations. While the original methods developed in Index Theory used entirely algebraic and differential methods, later developments in the 70s and 80s led to stochastic methods. Our method reflects the latter, and makes use of the heat-kernel to recover the Euler characteristic using purely sub-Riemannian information.
The result which is a joint work with F. Baudoin (U. Connecticut) and E. Grong (U. Bergen) is published as "A horizontal Chern-Gauss-Bonnet formula on totally geodesic foliations".
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