Code: 01URG Introduction to Riemannian geometry
Lecturer: prof. Mgr. David Krejčiřík Ph.D., DSc. Weekly load: 2+0 Completion: EX
Department: 14101 Credits: 2 Semester: W
Description:
This lecture is intended for an advanced undergraduate having possibly (but not necessarily) already taken a basic course on topological and differential manifolds. In addition to understanding the geometric meaning of curvature and its intimate relationship to topology, the student will learn the basic apparatus of Riemannian geometry suitable for further study of modern parts of mathematics and mathematical physics. Possible extension of this lecture is the geometric analysis of partial differential equations on Riemannian manifolds.
Contents:
1. Motivation. The notion of curvature in classical theories of curves and surfaces.
2. Review of basic tools. Tensors, manifolds and vector bundles.
3. Riemannian metrics. The volume element and integration. Constant-curvature model spaces.
4. Connections. Covariant derivatives of tensor fields. Parallel transport along curves. Geodesics.
5. Riemannian geodesics. The Levi-Civita connection. The exponential map. Normal coordinates. Geodesics of the model spaces.
6. Geodesics and distance. Geodesics as length-minimising curves. First variation formula. The Gauss lemma. Completeness and the Hopf-Rinow theorem.
7. Curvature. Local invariants of Riemannian metrics. The curvature tensor. Flat manifolds. The Ricci and scalar curvatures.
8. Riemannian submanifolds. The second fundamental form. Hypersurfaces in the Euclidean space, the Gauss curvature and Theorema Egregium. Sectional curvatures.
9. The Gauss-Bonnet theorem. The Umlaufsatz and the Gauss-Bonnet formula. The Euler characteristic of a topological manifold.
10. Jacobi fields. The Jacobi equation. Conjugate points. The second variation formula.
11. Curvature and topology. Comparison theorems. The Cartan-Hadamard and Bonnet's theorems.
Recommended literature:
Key references:
1. J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.

Recommended references:
2. M. P. do Carmo, Riemannian geometry, Birkhauser 1992.
3. O. Kovalski, Úvod do Riemannovy geometrie, Univerzita Karlova, 1995.
4. M. Spivak, A Comprehensive Introduction to Differential Geometry, Volumes I-V, Publish or Perish, 1999.
5. P. Petersen: Riemannian Geometry. Springer, 2016.
Keywords:
Riemannian geometry, metric, connection, geodesic, curvature.

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