Math 250B - Differential Geometry

Welcome to Math 250b!

Course description:

This is the second in a three-sequence graduate course in differential geometry.

Topics include: connections, curvature, Riemannian geometry, Chern classes.

Instructor: Dragos Oprea, doprea "at" you-know-where dot edu

Lectures: TuTh 12:30-1:50, APM 5-402

Office hours: Tu 2:30-4:30, APM 6-101, or by appointment. The office hours on February 18 will be from 2:00-3:15 and 5:10-5:55.

Textbook:

No textbook is formally required, but you can consult do Carmo's Riemannian Geometry for parts of the course. You can also consult Lee's Riemannian Geometry.

Material for general connections in arbitrary vector bundles can be found in Madsen and Tornehave's book From calculus to cohomology. (You can find these books online.)

Prerequisites:

Math 250A. This is a graduate level course, so at times, we may use notions from related fields, including linear algebra, topology, real analysis and a bit of complex analysis. I am happy to discuss prerequisites on an individual basis. If you are unsure, please don't hesitate to contact me.

The syllabus can be found here.

Grading:

The final grade is based on 4 homework sets, regular attendance of lectures, and a final project. Each homework set is worth 18.75%, the final project is worth 25%. The final project should be an expository paper, roughly 10 pages.

The problem sets are mandatory and are a very important part of the course. The problem sets are to be uploaded on Gradescope. There will be a 48hr grace period, but no other extensions will be given.

There are several ways to upload your homework on Gradescope. A possible method is described here.

Working with your peers is acceptable, but solutions must be written independently.

Academic Integrity

Any violation of the Academic Integrity Policies is considered a serious offence at UCSD. Students caught cheating will face administrative sanctions. No credit will be given on the assignment/exam which resulted in the academic integrity violation.

Important dates:

First class: Tuesday, January 7
University Holiday: Monday, January 20
University Holiday: Monday, February 17
Last class: Thursday, March 13

Lecture Summaries

Lecture 1: Introduction, course goals, short review. Riemannian metrics. Examples of Riemannian metrics. Isometries. Length of curves. Riemannian distance. Riemannian volume form.

Lecture 2: Introduction to connections in arbitrary vector bundles. Covariant derivative. Local and point operators, and C^-infty linearity. Covariant derivatives in direct sums, tensor products, duals, endomorphism bundles.

Lecture 3: Connection forms in local trivializing charts. Local data defining connections. Pullback bundles and pullback connections. Parallel transport.

Lecture 4: Curvature of a connection. Definition and tensorial properties. Local representation of curvature in terms of the connection matrix. Bianchi identity.

Lecture 5: Geometric interpretation of curvature. Holonomy. Flat bundles and representations of the fundamental group.

Lecture 6: Chern classes and Chern character of vector bundles via curvature forms. Invariant polynomials.
Lecture 7: Chern classes and independence of the choice of the connection. Aside on the topological definition of Chern classes. Properties of Chern classes and Chern character.
Lecture 8: How to compute Chern classes of holomorphic line bundles. Example: the tangent space of CP^1. Pfaffian of a skew-symmetric matrix. Euler form of an orientable Riemannian bundle with a metric connection. Statement of Chern-Gauss-Bonnet.

Lecture 9: More on Euler classes. Affine connections. The Levi-Civita connection. Christoffel symbols. Parallel transport. Geodesics.

Lecture 10: More on geodesics and some examples. The exponential map. The derivative of the exponential map at the origin. Normal coordiantes and geodesic balls. Curvature. Symmetries of the curvature tensor.

Lecture 11: Sectional curvature. Sectional curvature determines Riemannian curvature. Riemannian curvature when sectional curvature is constant. Ricci tensor. Scalar curvature. Discussion of surfaces and Gauss-Bonnet.

Lecture 12: Second order expansion of the metric and volume form in geodesic coordinates. Further interpretation of Ricci tensor and scalar curvature. Isometric embeddings and the second fundamental form. Theorema Egregium.

Lecture 13: More on Theorema Egregium. Hypersurfaces. The shape operator. Principal curvatures. The Gauss map. Totally geodesic embeddings. Interpretation of sectional curvature as Gaussian curvature of a suitable surface. Mean curvature.

Lecture 14: Constant speed reparametrization of regular curves. Variation of length formula. Symmetry lemma. Curves which minimize length are geodesics. Totally normal neighborhoods. Points in totally normal neighborhoods are joined by geodesics.

Lecture 15: Gauss lemma. Geodesics minimize length locally. Conversely, curves which minimize length are geodesics. Statement of Hopf-Rinow.

Lecture 16: Geodesic completeness and proof of Hopf-Rinow. Variations through geodesics and Jacobi fields. Conjugate points.

Lecture 17: Existence of Jacobi fields. Normal Jacobi fields. Examples of Jacobi fields on spaces of constant sectional curvature. Second variation of length. Bonnet-Myers theorem.

Lecture 18: More on the connection between Jacobi fields and variations through geodesics. Conjugate points and critical points of the exponential map. Hadamard's theorem and complete manifolds with nonpositive sectional curvature.

Homework:

Homework 1 due Wednesday, January 22 - PDF
Homework 2 due Wednesday, January 29 - PDF
Homework 3 due Friday, February 7 - PDF
Homework 4 due Friday, February 21 - PDF

Final Projects - possible topics PDF