Class times: MWF 10:00am in AP&M 7421
OVERVIEW: In this quarter we learn about the basics of differentiable manifolds and Riemannian geometry. Topics include: tangent and cotangent bundle, vector fields, tensors and differential forms, Lie derivative, Riemannian metric, Levi-Civita connection, Riemann curvature tensor, Ricci and scalar curvatures, Jacobi fields, completeness and Hopf-Rinow theorem, and Bochner formula for 1-forms. We closely follow do Carmo's Riemaannian Geometry book, with a couple of additional topics.
Homework #1. Assigned 10-15-2012. Due 10-29-2012 in class.
Solutions to Homework #1.
10-01 Differentiable manifold
10-03 Sphere, products, Lie groups, tangent space of a surface in R^3
10-05 Tangent space
10-08 Vector fields
10-10 and 10-12 Diffeomorphisms, differential of a map, vector fields, Lie bracket, Lie derivative, Jacobi identity
10-15 Riemannian metric and manifold: constant curvature examples (simply-connected)
10-17 Left-invariant vector fields and Riemannian metrics on Lie groups, Euclidean covariant derivative
10-19 and 10-22 Levi-Civita connection
10-24 Christoffel symbols
10-26 Covariant differentiation of vector fields along a path
10-29, 10-31, 11-02 Length, distance, geodesics, exponential map
11-05 and 11-07 The Gauss lemma
11-09 First Variation of arc length formula
11-14 and 11-16 Riemann curvature tensor, tensorial property, algebraic identities it satisfies, and local coordinate formula
11-19 Sectional curvature, Ricci curvature, and scalar curvature
11-21 Hypersurfaces in Euclidean space, second fundamental form, Gauss and Codazzi equations
11-30 Gauss and Codazzi equations for hypersurfaces in Riemannian manifolds (derived differently than in class, these notes use the 12-03 lecture on moving frames)
12-03 Orthonormal frame fields, connection 1-forms, curvature 2-forms, and the Cartan structure equations.
12-05 Applications of moving frames in dimension 2