Class: MW, 1:30-3:00 PM, LH-1. On the infrequent occasions when LH-1 is occupied due to seminars or conferences, we will use LH-3
Instructor: Ved Datar
Email: vv lastname at math.iisc.ac.in, no spaces
Office: X05
Office hours: W 11:30-12:30PM, or by appointment (if you have another class during office hours).
Reference books: John Lee, Riemannian Geometry - An introduction to curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997.
Supplementary reading: Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian geometry, Third edition., Universitext. Springer-Verlag, Berlin, 2004
Pre-requisites: You should have taken a course comparable to the Calculus on manifolds or Differential Manifolds and Lie Groups courses offered at IISc. You should have also taken basic courses in linear algebra and real analysis. If you do not meet this criteria, but would still like to credit, please send me an email.
Topics to be covered: Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems. Special topics if time permits - Comparison geometry (theorems of Rauch, Toponogov, Bishop-Gromov), and Bochner technique.
Homeworks - 20%, Midterms - 30%, Final - 50%
There will be 6 homeworks. The best five will be counted towards the grade. There is no late submission of homeworks.
There will be one midterm of 60 points each and a final exam of 100 points. To pass the class, you have to take the final exam.
Starting the week of August, 12 the class has been moved to MW, 1:30-3:00PM. We will no longer meet on Tuesdays and Thursdays
| Number | Date | Topic | Homework | Notes |
| 0 | Th 08/01 | Introduction, smooth manifolds | ||
| 1 | Tu 08/06 | Review of tensors, Lie derivative | ||
| 2 | Th 08/08 | Linear connections | A1 (due 26/08/19) | |
| M 08/12 | holiday | |||
| 3 | W 08/14 | Connections (cont.), Parallel transport | Lecture-3 | |
| 4 | M 08/19 | Riemannian metrics, Isometries, examples | Lecture-4 | |
| 5 | W 08/21 | Elemntary constructions involving Riemannian metrics | Lecture-5 | |
| 6 | M 08/26 | Levi-Civita connection | A2 (due 09/09/19) | Lecture-6 |
| 7 | W 08/28 | geodesics, existence and uniqueness, examples | Lecture-7 | |
| M 09/02 | holiday | |||
| 8 | W 09/04 | exponential map and normal coordinates | Lecture-8 | |
| 9 | M 09/09 | distance function, first variation formula | Lecture-9 | |
| 10 | W 09/11 | local behaviour of geodesics | A3 (due 30/09/19) | Lecture-10 |
| 11 | M 09/16 | Local behaviour of geodesics (cont.), Hopf-Rinow theorem | Lecture-11 | |
| 12 | W 09/18 | Proof of Hopf-Rinow | Lecture-12 | |
| M 09/23 | no class | |||
| W 09/25 | midterm | |||
| 13 | M 09/30 | cut locus and conjugate locus | ||
| 14 | W 10/09 | regularity of distance function | ||
| 15 | F 10/11 | curvature tensors, symmetries, algebraic Bianchi | Lecture-15 | |
| 16 | M 10/14 | differential Bianchi, sectional, Ricci and scalar curvatures | A4 (due 23/10/19) | Lecture-16 |
| 17 | W 10/16 | second fundamental form, sectional and Gauss curvature | Lecture-17 | |
| 18 | M 10/21 | decomposition of curvature tensor | Lecture-18 | |
| 19 | W 10/23 | second variation formula, Jacobi fields | Lecture-19 | |
| 20 | M 10/28 | Characterization of Jacobi fields, local characterization of space forms | Lecture-20 | |
| 21 | W 10/30 | Conjugate points, index of the index form, Jacobi comparison | A5 (due 13/11/19) | Lecture-21 |
| 22 | M 11/04 | Ambrose-Cartan-Hadamard theorem, characterization of space forms | Lecture-22 | |
| 23 | M 11/11 | Rauch comparison theoerm and consequences | Lecture-23 | |
| 24 | W 11/13 | Ricci curvature comparison - I | Lecture-24 | |
| 25 | F 11/15 | Ricci curvature comparison - II | Lecture-25 | |
| 26 | M 11/18 | Bochner technique - I | A6 (due 28/11/19) | Lecture-26 |
| 27 | W 11/20 | Bochner technique - II | Lecture-27 |