Geometry of Anosov Flows and Rigidity - 1/4

Explore the fundamental geometry of Anosov flows through this mathematical lecture that introduces these dynamical systems where every vector experiences uniform expansion or contraction. Begin with the archetypal example of geodesic flow on manifolds with negative sectional curvature, then delve into the geometric properties including invariant manifolds, ergodicity, and regularity questions. Learn about flows that are compact group extensions of Anosov flows and discover the associated "Brin group," which functions as a Galois group of the extension. Apply these advanced techniques to understand a significant result developed with David Fisher and Ben Lowe, demonstrating that compact negatively curved real-analytic Riemannian manifolds with infinitely many totally geodesic hypersurfaces must have constant sectional curvature. Gain insights into the rigidity properties of these geometric structures and their implications for differential geometry and dynamical systems theory.