Global Riemannian Geometry: Curvature and Topology
The small booklet contains lecture notes from two advanced courses. The first one was written by S. Markvorsen (Distance geometric analysis on manifolds). It contains a comparison theory for distance functions on immersed submanifolds in Riemannian manifolds and its relations to seemingly unrelated topics. These include isoperimetric inequalities, diffusion processes (mean exit time comparison), transience, warped products, and all that related to the Laplacian. The second lecture notes are written by M. Min-Oo (The Dirac operator in geometry and physics). The first part describes traditional results on the Dirac operator, its index formula and the Lichnerowicz formula. The second part treats the Gromov notion of K-area and the corresponding fundamental K-area inequality, while the third part discusses various aspects of the famous positive mass theorem. Both reviews are very useful for the orientation of the reader in the field, details should be found in the corresponding literature.