Course Syllabus

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This is an introductory course in differential geometry. The first part of the course covers basic ideas from differential topology and calculus on manifolds, including the Frobenius theorem and basics about Lie groups. The second part will introduce concepts from Riemannian geometry, such as metrics, connections, curvature, and geodesics. In the last part of the course we will prove some of the big theorems of Riemannian geometry relating curvature to topology.  These include the Cartan-Hadamard theorem, Bonnet-Myers' theorem, and if time permits various comparison theorems.

MATH 405 (linear algebra), 411 (advanced calculus), and 730 (basic topology) or equivalents are prerequisites for this course.

Here is some more basic information about MATH 740:

  • Time & Place: MWF 12:00-12:50 pm, MTH 1311.

  • Instructor: Professor Richard A. Wentworth, 3109 Mathematics Building, (301) 405-5130.

  • Office Hours: W 10-11:30 am, and by appointment.

  • Texts: Riemannian Geometry, Manfredo do Carmo (recommended). Foundations of Differentiable Manifolds and Lie Groups, Frank Warner (recommended).

  • Additional References: Introduction to Smooth Manifolds, J. Lee, Lecture notes on Riemannian geometry by Bishop, Lecture notes on comparison theorems by Eschenburg, Comparison theorems in Riemannian geometry, by J. Cheeger and D. Ebin.

  • Homework: There will be periodic homework assignments. These will not be graded.

  • Quizzes: There will be three in class quizzes. These will be on: Feb 27, Apr 1,  and May 8.

  • Make-up. If you cannot be present for one of the quizzes you must let me know in advance. We will schedule an oral exam as a make-up.

  • Grading: The final grade will depend on your participation and on your performance on the quizzes.

The material above is dated January 27.

Below is a more detailed syllabus of the topics I want to cover. I will update this periodically.

1, Smooth manifolds lecture notes

  • immersions and embeddings
  • vector fields, tangent and vector bundles
  • Frobenius theorem
  • tensors and forms
  • Lie derivative and integration

2. Riemannian structures

  • Connections on vector bundles
  • Riemannian metrics, volume, Hodge star
  • Fundamental theorem, normal coordinates
  • Curvature and properties
  • Parallel transport, holonomy
  • Submanifolds

3. Geodesics

  • geodesic flow, convex neighborhoods
  • Jacobi fields, conjugate points
  • Hopf-Rinow theorem

4. Cartan-Ambrose-Hicks theorem

  • Spaces of constant curvature
  • Cartan's theorem

5. Rauch comparison theorem

6. Bishop comparison theorems

Course Summary:

Course Summary
Date Details Due