Instructor: Prof. Ben Chow

Office: 7438 AP&M Phone: 858 534-7690

Office hours: MWF 11:00am or by appointment.

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.

Lecture notes:

09-28 Introduction

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