Math 568: Differential Geometry

Instructor: Kris Tapp

Class time: Mon and Fri 12:40-2:00 in Math 5-127

Office Hours: by appointment

Main text: “Riemannian Geometry” by Do Carmo

Topics:

This course provides a basic introduction to Riemannain Geometry. We begin by reviewing smooth manifolds, Lie Brackets, and elementary facts about Lie Groups. We then introduce the basic structure of Riemannian manifolds, including geodesics, the Levi-Civita connection, curvature, the exponential map, and Jacobi fields. Extra topics include isometric immersions, Riemannian submersions, constant curvature spaces, and the Hopf-Rinow Theorem. We will also develop basic facts about vector bundles, connections in vector bundles, and the curvature of a connection in a vector bundle.

Homework:

I will hand out a homework sheet roughly every two weeks in lecture. The homework problems are carefully chosen to help you struggle with and master the basic material of the course.

Problem-solving sessions:

I will not collect homework. Rather, I will ask you to present your solutions to each other in a once-weekly problem-solving session. The tentative time for these sessions is Thursdays, 1-2, in Math 5-127 (beginning Sept 6).

Homework sheets:

(1) Homework #1 covering Do Carmo chapters 0 and 1.

(2) Homework #2 covering “connections in vector bundles” including Do Carmo chapter 2.

(3) Homework #3 covering Do Carmo chapter 3.

(4) Homework #4 covering Do Carmo chapter 4.

(5) Homework #5 covering Do Carmo chapter 5 and 7.