This is the main page for **Math 392C (Riemannian Geometry)**, unique number **56950**.

I am Andy Neitzke; my office is RLM 9.134. My office hours are by appointment.

This course meets MWF from 11am-12pm, in RLM 11.176.

The main text will be **John M. Lee: "Riemannian Manifolds: An Introduction to Curvature."** Other useful
resources are **Peter Petersen: "Riemannian Geometry"**; **Jurgen Jost: "Riemannian Geometry and
Geometric Analysis"**; **Michael Spivak: "A Comprehensive Introduction to Differential Geometry", volumes 1-2**.

I will assign exercises; I strongly encourage you to do them. I will mention a few exercises during lecture, but also will post slightly more organized exercise sheets (below.)

If you need a grade for the course, you should turn in (at least) 1 problem from each exercise sheet. The due date is flexible, but you will learn the material better if you keep up with the exercises.

- 01: Bases and coordinate computations
- 02: Basic constructions with Riemannian metrics
- 03: Basic constructions with metrics and with connections
- 04: More connections, especially orthogonal ones
- 05: Properties of normal coordinates and a few explicit computations

This course will be an introduction to Riemannian geometry, the study of smooth manifolds equipped with Riemannian metrics. A Riemannian metric is a geometric structure which determines (among other things) notions of distance and curvature on a manifold. There is a fascinating interplay between the local Riemannian geometry and the global topology of a manifold. For example,

1) the Gauss-Bonnet theorem says that if we equip a surface with a Riemannian metric, then the total integral of the scalar curvature over the surface will be equal to the Euler characteristic of the surface; the Gauss-Bonnet-Chern theorem extends this to higher dimensions;

2) the Cartan-Hadamard theorem says that any complete simply connected Riemannian manifold with sectional curvature everywhere non-positive must be diffeomorphic to R^n;

3) the Hodge theorem says that we can determine the real cohomology groups of a manifold by equipping it with a Riemannian metric and studying the kernel of a p-form Laplacian operator defined by that metric;

4) Thurston's geometrization program is based on the idea that even if one is interested in purely topological questions about 3-manifolds, it is useful to equip them with Riemannian metrics.

The course will be divided roughly into two parts. The first part will consist of material which is really mandatory for anyone interested in the subject: Riemannian metrics; connections in vector and principal bundles; the Levi-Civita connection; geodesics; Hopf-Rinow Theorem; Riemann curvature and its various relatives (Ricci, Weyl, scalar, sectional curvatures); Bochner identities; geometry of homogeneous spaces; Riemannian submanifolds. The second part will consist of some applications. I would like at least to prove the Gauss-Bonnet-Chern and Cartan-Hadamard theorems mentioned above, and to give a sketch of what is involved in proving the Hodge theorem. Beyond this the topics are not fixed, and what we do will depend on the remaining time and the tastes of the class; three possibilities would be to do a bit of Lorentzian geometry (enough to discuss Einstein's equation and the Schwarzschild solution), a bit of special holonomy/calibrated geometry, or a bit of comparison geometry.

The main prerequisite is the Differential Topology prelim.

Below are my notes from the lectures so far (the mapping between files and lectures is not necessarily 1-1). They are updated to correct errors/omissions where they were helpfully pointed out or where I notice them later. I apologize for whatever errors may remain.

- 00.1: Introduction
- 01.2: Linearity principle
- 01.5: Coordinate computations
- 02.0: Riemannian metrics
- 03.0: Basic constructions with Riemannian metrics
- 04.0: Model spaces: spheres and hyperbolic spaces
- 04.5: Miscellaneous questions answered
- 05.1: Connections (covariant derivatives) in vector bundles
- 06.0: Curvature
- 07.0: Levi-Civita connection (guest lecture by Prof. Dan Freed)
- 08.0: Geodesics
- 09.0: Normal coordinates
- 10.0: Riemannian distance
- 11.0: Geodesics are (locally) minimizers
- 11.5: Proof of uniformly normal neighborhood lemma (movie)
- 11.5: Proof of uniformly normal neighborhood lemma (notes)
- 12.0: Hopf-Rinow Theorem
- 13.0: Riemann curvature
- 14.0: Curvature for Riemannian submanifolds
- 15.0: Sectional curvature
- 16.0: Jacobi fields
- 17.0: Conjugate points
- 18.0: Second variation
- 19.0: Comparison theorems
- 20.0: Laplacians and the statement of the Hodge theorem
- 20.5: Comments on the scalar Laplacian
- 21.0: Weitzenbock formula
- 22.0: Bochner method
- 23.0: Sobolev spaces
- 24.0: Garding inequality
- 25.0: Proof of the Hodge theorem (link to proof of spectral theorem)
- 26.0: Elliptic operators
- 27.0: Principal bundles
- 28.0: Connections in principal bundles
- 29.0: Gauss-Bonnet theorem
- 30.0: Gauss-Bonnet-Chern theorem
- 31.0: Killing fields
- 32.0: Lorentzian metrics and the Schwarzschild solution
- 33.0: A brief introduction to special holonomy
- 34.0: A very brief advertisement for Kahler geometry

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.