Riemannian geometry, Fall 2013
Text:
Jurgen Jost, Riemannian Geometry and Geometric Analysis
Auxiliary Text:
M. P. do Carmo, Riemannian geometry
Homework set 1, due Oct 1
Homework set 2, due Nov 6
Homework set 3, due Dec 12
Table of Contents
1 Preliminaries
1.1 Manifolds and smooth maps
1.1.1 Examples of manifold
1.1.2 The tangent space and the differential of a map
1.1.3 Submanifolds
1.2 Riemannian metrics pp 13-17
1.3 Vector bundles pp 33-38
1.3.1 Tensors
1.4 Differential forms pp 40-43
1.4.1 Exterior differential
1.4.2 Hodge star operator pp 83-85
1.5 Integral curves of vector fields
1.5.1 Lie derivative of functions and tensors
1.6 Lie groups
2 Geodesics I (pp 18-32)
2.1 Geodesics as the critical points of the energy functional
2.2 Normal coordinates
2.3 Hopf-Rinow theorem
3 Connections and curvature
3.1 Connections in vector bundles (as covariant derivative)
3.2 Parallel transport and holonomy
3.3 Covariant exterior derivative
3.3.1 Second Bianchi identity
3.3.2 Chern classes
3.4 Connections in fiber bundles (brief)
3.5 Properties of Riemannian curvatures
3.5.1 Symmetries of R
3.5.2 Sectional curvature
3.5.3 Ricci and sectional curvature
3.5.4 Spaces of constant curvature
3.6 Geometric meaning of curvatures
3.7 The Yang-Mills functional
3.7.1 Self-duality in dimension 4
4 Geodesics II
4.1 1st and 2nd variations of arc length
4.2 Jacobi fields
4.3 Conjugate points
4.4 Space forms