MATH 537 Riemannian Geometry, Time MW 12301345 SMLC124.
Instructors: Lectures: MW 12:3013:45 SMLC124, Dimiter Vassilev, Associate Professor; Office: SMLC, Office 326; Email: vassilev@unm.edu
Office Hours: Monday, Wednesday 2pm3pm. Feel free to stopby anytime if you have a quick question.
Catalog Description: Theory of connections, curvature, Riemannian metrics, HopfRinow theorem, geodesics. Riemannian submanifolds. Prerequisite: 536.
HOMEWORK: Homework and course related material will be posted on UNMLearn. The general rule is that you should do the homework problems in order to maximize what you learn in the class.
ATTENDANCE: Attendance at UNM is mandatory, see policy.
Textbooks:








Schedule:
Class Week  Topics 
1. Aug 21  Connection in a vector bundle 
2. Aug 28  Christoffel symbols; connection matrix (of 1forms) 
3. Sep 4 
Induced connections on tensor products and dual spaces. Vector valued differential forms. 
4. Sep 11  Covariant derivative; Differential Bianchi identity. 
5. Sep 18  Parallel transport and holonomy group. 
6. Sep 25  AmbroseSinger' theorem; holonomy principle. Linear connections  torsion tensor. Ricci identities. 
7. Oct 2  Bianchi identities. Bianchi and Ricci identities in the language of formsCartan's structure equations. 
8. Oct 9  Local calculations. Geodesics of a linear connection. 
9. Oct 16  Examples of computations of geodesics. The geodesic vector field (spray) on TM. 
10. Oct 23  Examples. The exponential map. 
11. Oct 30  Normal coordinates and their properties. Convex normal neighborhoodsWhitehead's theorem. 
12. Nov 6

Finish the proof of Whitehead's theorem. Variations of a (geodesic) curve. Jacobi's equation and fields. Conjugate points. Riemannian metric and the LeviCivita connection. 
13. Nov 13

Riemannian vector bundles and their holonomy. Equivalence of the metric and manifold topologies. The Heisenberg group and Lie groups  left invariant vector fields and the Lie algebra, leftinvariant metrics and their LeviCivita connection. 
14. Nov 20

Computations of the connection and curvature using forms and Cartan's structure equations hyperbolic space example. Properties of the curvature tensor. Spaces of constant curvature. The musical isomorphism. Sectional curvature. Proof that the sectional curvatures determine the curvature. 
15. Nov 27

Ricci and scalar curvatures. KulkarniNomizu product and the decomposition of the curvature tensor. Einstein manifolds. Schur's teorem. Contructed differential Bianchi identity. Riemannian submanifolds  Gauss' equation and the second fundamental form, the induced connection on the normal bundle. 
16. Dec 4  Weingarten's equation. Gauss' equation for curvature. The Gauss and Weingarten maps for a hypersurface in Euclidean space; principle curavtures. Gauss' lemma and HopfRinow's theorem. 
11Dec 