Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 240A: Differential Geometry
Approved: 2008-09-01, Michael Kapovich

Units/Lecture:

Fall, every year; 4 units; lecture/term paper or discussion

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
M. do Carmo, Riemannian Geometry ($55)
Search by ISBN on Amazon: 0817634908

Prerequisites:

Course 201A and 239; 250AB highly recommended; intended primarily for 2nd-year graduate students.

Course Description:

Riemannian metrics, connections, geodesics, Gauss lemma, convex neighborhoods, curvature tensor, Ricci and scalar curvature, connections and curvature on vector bundles.

Suggested Schedule:

Lectures Sections Topics/Comments

First 4 chapters of do Carmo's book Riemannian metrics, connections, geodesics, Gauss lemma, convex neighborhoods, curvature tensor, Ricci and scalar curvature.


Examples of Riemannian metrics and computation of connection and curvature: sphere, compact Lie groups, hyperbolic space.


Also cover: connections and curvature on vector bundles using, for instance, Kobayashi and Nomizu.


Supplementary topics: G-structures, pseudo-Riemannian metrics, Einstein metrics, holonomy.

Additional Notes:

Supplementary Reading
  • P. Petersen, Riemannian Geometry
  • J. Jost, Riemannian Geometry and Geometric Analysis
  • S. Kobayashi, Transformation Groups in Differential Geometry. Classics in Mathematics
  • S. Kobayashi and K. Nomizu, Foundations of Differential Geometry