250130 VO Metric Geometry (2023W)

This is a first course on metric geometry. The central idea of this field is to describe geometric properties (such as length, angles and curvature) in terms of metric distances alone. As it turns out, many notions familiar from differential geometry can indeed be captured in such "synthetic" terms alone.

The foundational notion is that of a length space, i.e., a metric space where the metric distance between two points is given by the infimum of the length of all connecting curves. Key examples are Riemannian manifolds and polyhedra.

Curvature bounds in such spaces are based on comparison with triangles in certain model spaces. E.g., the sphere has positive curvature because triangles are fatter than Euclidean triangles of the same sidelengths. Spaces with a curvature bound in this sense from below/above are called Alexandrov/CAT(k) spaces.

Metric geometry, and in particular the theory of length spaces, is a vast and very active field of research that has found applications in diverse mathematical disciplines, such as differential geometry, group theory, dynamical systems and partial differential equations. It has led to identifying the ‘metric core’ of many results in differential geometry, to clarifying the interdependence of various concepts, and to generalizations of central notions in the field to low regularity situations.

The prerequisites for following this course are mild and I will soley assume konwledge of (metric) topology. To fully cherish the final chapter familarity with Riemannian geometry or elementary differential geometry is, however, useful.