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)
Prerequisites:
MAT 201A; MAT 239; MAT 250A & MAT 250B highly recommended; intended primarily for second-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