**Math 112: Introduction to
Riemannian Geometry
Spring 2006**

(X-Hour: Mon. 2-3PM)

Instructor: Craig Sutton

Mon. 10AM - Noon

Wed. 4-5PM

(Also by Appointment)

Syllabus/Homework

General Information

Textbooks:

*Riemannian Geometry*, Manfredo do Carmo. (required)- An
Introduction to Lie Groups and the Geometry of Homogeneous Spaces, A.
Arvanitoyeorgos (required)

**Course Description:** Riemannian geometry is the study of
smooth manifolds equipped with a Riemannian metric. With this
additional structure one is able to define geometric concepts such as
geodesics, connections and curvature.
The Riemannian geometer frequently explores the relationship between
these concepts and the topological and differentiable structures of the
underlying manifold. For example, the celebrated sphere theorem tells
us that if M is a compact, simply-connected, smooth manifold of
dimension n that admits a Riemannian metric g with sectional curvature
satisfying 1/4 < Sec_{(M, g)} <= 1, then M is
homeomorphic to S^{n}.
As a corollary we can conclude that the
universal cover of any compact manifold which admits a Riemannian
metric g with 1/4 < Sec_{(M, g)} <= 1 must be a sphere.

This course will serve as an introduction to this classical and
vibrant area of research with an aim towards developing both
theoretical and computational proficiency. It should be of relevance to
students with interests in geometry, topology and (mathematical)
physics. The topics covered will include some of the following.

- Riemannian
Basics: Riemannian metrics; Affine & Levi-Civita Connections
(Christoffel Symbols); geodesics, the exponential map, and the geodesic
flow; curvature tensor, sectional curvature, Ricci & Scalar
curvature; Curvature on compact Lie groups; Killing Fields.

- Computations:
Computation of Riemannian basics on surfaces of revolution, S
^{n}, Lie groups, RP^{n}, CP^{n}and other homogeneous spaces, and warped products. - Geometry of Isometric Immersions: Immersions; The second fundamental form; Principal directions & curvatures; totally geodesic submanifolds; Sectional Curvature & the Gaussian curvature of surfaces; Minimal surfaces; The equations of Gauss, Ricci and Codazzi (i.e., the fundamental eqs.).
- Jacobi Fields \& Interpretations of Curvature: Jacobi equation and Jacobi Fields; Curvature and the spreading of geodesics; Conjugate points and singularities of the exponential map.
- Variations of Energy &
Geodesics: 1
^{st}and 2^{nd}Variation Formulas; Geodesics are locally minimizing; Theorems of Bonnet-Meyers and Synge. - Complete Manifolds & the theorems of Hadamard and Hopf-Rinow
- Spaces of Constant Curvature: Theorem of Cartan on Curvature and Metrics; Space forms \& totally discontinuous group actions.
- Manifolds of Negative Curvature: Existence of closed geodesics; Preissman's theorem; Non-existence of Killing fields; Ergodicity of the geodesic flow (optional).
- Sphere Theorem
- Isometry Groups & Isometric Actions: Isometry groups as Lie groups; Killing fields; group actions; action/fundamental vector fields.
- The Geometry of Riemannian Submersions: submersions; the A- and T-tensors; O'Neill's formula; Principal Bundles \& Connections.
- The Geometry of Lie Groups
and other Homogeneous Spaces

Office Hours: Office Hours are a good time to flesh out material you're having trouble with or to go beyond the syllabus. Please don't hesitate to stop by in either case.

Homework: Homework will be assigned and collected every fortnight. You are encouraged to work together, but the solutions should be your own. No late homework will be accepted.

Honor Principle:

A. Homework: I encourage you to form
study groups to discuss course material and homework problems. However,
the assignments you turn in should be in your own words and
handwriting. Also, the names of others you consulted with should appear
at the top of your assignment.

B. Exams:
You should not give or receive help during exams. All exams are closed
book unless otherwise stated.

Homework:

I will announce the definitive grading guide in the coming weeks.

March 28

- Topics: Overview of Riemannian Geometry

- Reading (Due Next Class):
Chp. 0 (do Carmo)

- Topics: Review of Manifolds

- Reading (Due Next Class):
Chp. 0 & 1 (do Carmo), Quotients

- HW1 (Due April 6) New Due date = April 10

- Topics: Review of Manifolds (cont'd), Riemannian Metrics
- Reading (Due Next Time):
Chp. 1 & 2

- Topics: Riemannian Metrics, Isometries , Lie groups &
homogeneous spaces (definitions)

- Reading (Due Next Time): Chp. 1 & 2

- Topics: Review of Smooth manifolds. We'll use this hour to help people new to manifolds catch up.
- HW 1 Due Today

- Topics: Flat tori; Lie Groups, Lie Algebras & One-parameter
subgroups

- Reading (Due Next Time):
Chp. 2

- Topics: Lie Algebras & One-parameter subgroups; Killing
Fields; Connections

- Reading (Due Next Time):
Chp. 2 & 3

- HW2 (Due April 20)

- Topics: Connections & Parallel Transport

- Reading (Due next Time): Chp. 2 & 3

- Topics: Parallelism; Torsion and Curvature Tensor of a
Connection, Flat Connections; Levi-Civita Connection

- Reading (Due Next Time): Chp. 2 & 3

- Topic: Levi-Civita Connection & Geodesics

- Reading (Due Next Time):
Chp. 3 & 4

- HW3 (Due April 27)

Apr. 25

- Topic: More on Geodesics: normal neighborhoods & minimizing
properties

- Reading (Due Next Time): Chp. 4

Apr. 27

- Topic: Minimizing properties of geodesics; Riemannian submersions
& group actions

- Reading (Due Next Time): Chp. 4
- HW4 (Due May 9)

- Topic: Riemannian submersions
& group actions; Riemannian submersions, horizontal lifts and
geodesics.

- Reading (Due Next Time):

- Topic: Riemannian submersions, horizontal lifts and geodesics; Curvature: covariant derivative of tensors, curvature tensor, sectional curvature
- Reading (Due Next Time):

- Topic: Riemannian Submersions & Sectional Curvature: the
A-tensor and O'Neills' formula

- Reading (Due Next Time):
Chp. 5

- Topic: Sectional
Curvature of CP^n; Ricci and Scalar Curvature, the curvature operator;
Jacobi Fields

- Reading (Due Next Time): Chp. 5 & 9 (Note: You should start reading Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

- Topic: Jacobi Fields
- Reading (Due Next Time): Chp. 9 (Note: You should start reading Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

May 16

- Topic: Jacobi Fields and Conjugate Points, Geodesics w/o
conjugate points

- Reading (Due Next Time):
Chp. 9 (Note:
Read Chps. 7 & 8. We will most likely not cover them in class, but
you should be able to understand them on your own.)

May 18

- Class: Variations of Energy & the Index Form
- Reading (Due Next Time): Chp. 6 (Note: Read Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)
- HW5 (due May 30)

- Topic: The Theorems of Bonnet-Meyers, Weinstein and Synge
- (Due Next Time): Chp. 6 (Note: Read Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

May 23

- Topic: Isometric Immersions and the Second Fundamental Form

- Reading (Due Next Time): Chp. 6 (Note: Read Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

- Topic: No Class
- Reading (Due Next Time):

- Topic: Isometric Immersions and the Second Fundamental Form

May 30

- Topic: Geometry of Lie Groups

- Reading (Due Next Time):