Riemannian distance, geodesically convex sets

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Introduces hyperbolic geometry, covering complete metric spaces, isometries, and Gaussian curvature in dimension 2.

Introduces geodesic convexity on Riemannian manifolds and explores its properties.

Covers the concept of distance induced by the Riemannian metric on manifolds.

Explores distance, geodesics, and complete manifolds, emphasizing the existence of minimizing geodesics and the concept of metric completeness.

Explores Riemannian connections on manifolds, emphasizing smoothness and compatibility with the metric.

Explores curves with Poritsky property, Birkhoff integrability, and Liouville nets in billiards.

Explores geodesic convexity in metric spaces and its applications, discussing properties and the stability of inequalities.

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Covers geodesically convex optimization on Riemannian manifolds, exploring convexity properties and minimization relationships.

Introduces metric spaces, topology, and continuity, emphasizing the importance of open sets and the Hausdorff property.