MATH 567- FALL 2011-COURSE LOG

8/18 Th  definition of C^k manifold/ S^n via stereographic proj/ Hausdorff property for quotient spaces/ real projective space

8/23 Tu  projective space, Grasssmanians (outline)/ diffeomorphisms, discontinuous group actions/
conditions for Hausdorff quotient, example

8/23  Th  submanifolds of R^n/ preimage of regular value/injective immersions vs. embddings/proper maps

8/25 Tu Orthogonal group, Tangent vector, tangent bundle,  differentials

9/1  Th local flows of vector fields, lie brackets, diffeomorphisms defined by commuting vector fields

9/6  Tu  orientability, commuting vector fields (end)

9/8  Th general topology: paracompactness, precompact exhaustions, metrizability

9/13 Tu partitions of unity, Riemannian metrics, metric space structure

9/15  Th Connections, parallel transport

9/20 Tu orientation: discussion of HW problems

9/22 Th connections on submanifolds, Levi-Civita connection

9/27 Tu Problems on connections and parallel transport; definition of geodesic

HANDOUTS

Some results from General Topology

Completely integrable geodesic flows (two examples)

asymptotic isoperimetric quotient for a cone

some multilinear algebra

curvature tensor

HOMEWORK PROBLEMS

Problem set 1

From do Carmo's Riemannian Geometry:

Chapter 0: 3, 5, 7, 8, 9

Chapter 2: 1, 2, 3, 4, 5, 7, 8

Chapter 3: 5, 6, 7, 8, 9, 11, 12

Chapter 4: 4, 7, 8, 10

Chapter 7: 3, 5, 6, 7, 9, 10, 11