# Math 213B: Introduction to Riemannian Geometry Spring 2011 News
Description
Lecture time and location
Prerequisite
Office hours
Textbook
Syllabus
Class schedule
Handouts
Homework
Paper
Anonymous feedback

## News

(5/16) The final version of the notes on the intrinsic proof of the Gauß-Bonnet theorem is here.

(5/10) I just uploaded a corrected version of the notes on the intrinsic proof of the Gauß-Bonnet theorem.

(5/6) I just posted my notes on the intrinsic proof of the Gauß-Bonnet theorem.

(5/6) Here are some hints for homework 5.

(4/22) Homework 5 has been posted. It's due on the last day of class, May 16.

(4/21) I just posted a handout on the following question: suppose X1,..., Xn are smooth linearly independent vector fields on a smooth n-manifold M. When are they locally coordianate vector fields? Answer: when they mutually commute. This will be used on Monday to finish the proof that R = 0 iff M is flat.

(3/22) Homework 4 has been posted. It's due approximately on April 13.

(3/7) The class is moving from CL 127 to Sweeney Hall 240. We'll first meet there this Wednesday.

(3/5) I posted some hints for homework 3.

(3/1) Homework 3 has been posted and is due somehwhere around March 16.

(2/28) As promised, I posted a link to a short and elementary introduction to differential forms.

(2/15, part 2) I just posted a list of some potential topics for the survey paper. More to come. Also, feel free to suggest others.

(2/15) I have posted a short write-up called Three faces of the Lie bracket. It describes three equivalent characterizations of the Lie bracket of vector fields.

(2/11, part 2) I just posted a link to a handout on the coordinate-free view of the derivative of a map between Euclidean spaces.

(2/11) I just posted Homework 2 roughly due on Febraury 21.

(2/2) I left copies of relevant chapters from Boothby in an envelope hanging near my office door - help yourselves.

(2/1) My office hours have changed. They are now MW 11:45-12:45 + 3-4, Tu 2:30-3:30 (online), and by appointment.

## Description

Riemannian geometry is a generalization of the classical differential geometry of curves and surfaces you studied in Math 113 (or an equivalent course) to abstract smooth manifolds equipped with a family of smoothly varying inner products on tangent spaces. The main goal of Riemannian geometry is to understand how a manifold curves in different directions and how curvature of a manifold relates to its topology.

## Lecture time and location

MW 10:30-11:45 in SH 240

## Prerequisite

Math 213A or instructor consent.

Note: To make the course more accessible, I will not assume familiarity with abstract smooth manifolds. Instead, I will give a brief introduction to basics topics in smooth manifolds, which are necessary for this course. I'm willing to accept anyone with a basic background in geometry, multivariable calculus, and linear algebra, and a certain level of mathematical maturity.

## Office hours

MW 11:45-12:45 + 3:00-4:00
Tu 2:30-3:30 (online), and by appointment

## Textbook John M. Lee: Riemannian Manifolds, An Introduction to Curvature, Springer, GTM vol. 176, 1997

## Syllabus

Green sheet
Introduction, review and motivation. Basics of smooth manifolds, vector bundles and tensors. Definitions and examples of Riemannian metrics. Connections. Riemannian geodesics. Geodesics and distance. Curvature. Riemannian submanifolds. The Gauss-Bonnet theorem. Jacobi fields. Curvature and topology.

## Class schedule

This schedule is only approximate.

# Date Sections to be covered
1 1/26 Ch. 1, Intro. to curvature
2 1/31 Smooth manifolds (Boothby)
3 2/2 Smooth maps (Boothby)
4 2/7 Immersions and embeddings (Boothby)
5 2/9 Tangent bundle and vector fields (Boothby)
6 2/14 Vector bundles. Tensors (Boothby)
7 2/16 Differential forms (Boothby)
8 2/21 Ch. 3: Riemannian metrics and elementary constructions
9 2/23 Ch. 3: Generalizations of Riemannian metrics and model spaces
10 2/28 Ch. 4: Motivation, connections
11 3/2 Ch. 4: Vector fields along curves
12 3/7 Ch. 4: Geodesics
13 3/9 Ch. 5: The Riemannian connection. Exponential map.
14 3/14 Ch. 5: Normal coordinates. Geodesics of model spaces
15 3/16 Ch. 6: Lenghts and distances on Riem. manifolds. Geodesics as minimizers
16 3/21 Ch. 6: Completeness and Hopf-Rinow
17 3/23 Ch. 7: Local invariants and flat manifolds
* 3/28-4/1 Spring break
18 4/4 Ch. 7: Symmetries of the curvature tensor
19 4/6 Ch. 7: Ricci and scalar curvatures
20 4/11 Ch. 8: Riem. submanifolds and the second fundamental form
21 4/13 Ch. 8: Hypersurfaces in Euclidean space
22 4/18 Ch. 8: Curvature in higher dimensions
23 4/20 Ch. 9: Gauss-Bonnet formula
24 4/25 Ch. 9: Gauss-Bonnet theorem
25 4/27 Ch. 10: Jacobi equation and computation of Jacobi fields
26 5/2 Ch. 10: Conjugate points and second variation formula
27 5/4 Ch. 10: Geodesics do not minimize past conjugate points
28 5/9 Ch. 11: Some comparison theorems. Manifolds of negative curvature
29 5/11 Ch. 11: Manifolds of positive curvature
30 5/16 Ch. 11: Manifolds of constant curvature, the end!

Homework 40%, expository paper 30%, take-home final 30%.

## Handouts

Basic topology

A coordinate-free view of the derivative

Three faces of the Lie bracket

What is a differential form?

When are vector fields coordinate vector fields?

An intrinsic proof of the Gauß-Bonnet theorem

## Homework

There will be regular homework assignments, taken mostly from the textbook.

# Due date Assignment
1 2/7 HW 1
2 2/21 HW 2
3 3/16 HW 3
4 4/13 HW 4
5 5/16 HW 5

## Paper

Each student is expected to write a short expository paper in LaTeX on a topic related to Riemannian geometry. Here are some potential topics:
• Subriemannian geometry
• Semi- (or pseudo-) riemannian geometry and relativity
• Geodesic flow of a surface of negative curvature
• The Frobenius theorem on integrability of distributions
• Ricci flow: definition and basic properties
• Curvature and fundamental group
• Foliations
• Principal bundles
• Integration of differential forms on manifolds
• Etc.