College of Liberal Arts and Sciences

# Department of Mathematics

## PDE and Differential Geometry Seminar

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Title: Sub-Riemannian model spaces
Speaker: Erlend Grong (Universite Paris Sud, Orsay, France)
Time: Monday, September 25, 2017 at 2:30 pm
Place: MONT 214Abstract: Model spaces in Riemannian geometry are complete, simply connected Riemannian manifolds with constant sectional curvature. The only such spaces are Euclidean space, spheres and the hyperbolic space, and each of these are uniquely determined by their dimension and sectional curvature. Such spaces are important in results such as the Laplacian and volume comparison theorem. We want to search for similarly ideal spaces in sub-Riemannian geometry, where the concept of sectional curvature is less understood. Sub-Riemannian manifolds can be though of as Riemannian manifolds where we have certain permissible directions for curves, which changes the distance and the geometry. Using the idea of model spaces as spaces with maximal isometry groups, we find that these spaces have a canonical partial connection, but that the curvature of this connection does not determine the sub-Riemannian manifold uniquely. We also give some classification results.

Organizer: Lihan Wang