Math 112: Introduction to Riemannian Geometry
Spring 2006

Tu. & Th. 10:00 - 11:50 AM
(X-Hour: Mon. 2-3PM)

Instructor:
Craig Sutton

E-mail: craig.j.sutton AT dartmouth DOT edu
Phone: 646-1059
Office Hours:
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 Sn. 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, Sn, Lie groups, RPn, CPn 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: 1st and 2nd 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
Prerequisites: Familiarity with smooth manifolds (i.e., Math 124 or the equivalent).

Disabilities: If you have a disability and require disability related accommodations please speak to me and Cathy Trueba, Director of Student Disability Services, in the Academic Skills Center as soon as possible, so we can find a remedy.

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:
100%

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

Syllabus/Homework

WEEK 1

March 28

• Topics:  Overview of Riemannian Geometry
• Reading (Due Next Class): Chp. 0 (do Carmo)

March 30
• Topics: Review of Manifolds
• Reading (Due Next Class): Chp. 0 & 1 (do Carmo), Quotients
• HW1 (Due April 6) New Due date = April 10

WEEK 2

Apr. 4
• Topics: Review of Manifolds (cont'd), Riemannian Metrics
• Reading (Due Next Time): Chp. 1 & 2

Apr. 6
• Topics: Riemannian Metrics, Isometries , Lie groups & homogeneous spaces (definitions)
• Reading (Due Next Time):  Chp. 1 & 2

WEEK 3

Apr. 10 (X-hour)
• Topics: Review of Smooth manifolds. We'll use this hour to help people new to manifolds catch up.
• HW 1 Due Today
Apr. 11
• Topics: Flat tori; Lie Groups, Lie Algebras & One-parameter subgroups
• Reading (Due Next Time): Chp. 2
Apr. 13
• Topics: Lie Algebras & One-parameter subgroups; Killing Fields;  Connections
• Reading (Due Next Time): Chp. 2 & 3
• HW2 (Due April 20)

WEEK 4

Apr. 17 (X-hour)
• Topics: Connections & Parallel Transport
• Reading (Due next Time): Chp. 2 & 3
Apr. 18
• Topics: Parallelism; Torsion and Curvature Tensor of a Connection, Flat Connections; Levi-Civita Connection
• Reading (Due Next Time): Chp. 2 & 3
Apr. 20
• Topic: Levi-Civita Connection & Geodesics
• Reading (Due Next Time): Chp. 3 & 4
• HW3 (Due April 27)

WEEK 5

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)

WEEK 6

May 2
• Topic: Riemannian submersions & group actions; Riemannian submersions, horizontal lifts and geodesics.

May 4
• Topic: Riemannian submersions, horizontal lifts and geodesics; Curvature: covariant derivative of tensors, curvature tensor, sectional curvature

WEEK 7

May 9
• Topic: Riemannian Submersions & Sectional Curvature: the A-tensor and O'Neills' formula
• Reading (Due Next Time): Chp. 5
May 11
• 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.)

WEEK 8

May 15 (X-hour)
• 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)

WEEK 9

May 22 (X-hour)
• 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.)
May 25
• Topic: No Class

WEEK 10

May 29 (X-hour)
• Topic: Isometric Immersions and the Second Fundamental Form

May 30

• Topic: Geometry of Lie Groups