The texts we will be referring to are as follows :
1. John Lee, Riemannian Geometry - An introduction to curvature (primary text).
2. Peter Petersen, Riemannian geometry.
The course description (along with pre-requisites) can be found on this webpage.
Instructors :
Vamsi Pritham Pingali, vamsipingali@iisc.ac.in
Classroom and timings : Tue and Thu 3:30-5:00 in LH-5
The Grading policy : 20% for the HW (to be submitted either in the class or by email), 30% for the Midterm, and 50% for the Final/Project (depending on the size of the class).
Exams:
The Midterm shall be held on 17 September in the usual classroom from 3:00-5:00. The syllabus is everything done until that point (including Riemann, Ricci and scalar curvatures).
The Final shall be held on Nov 28 from 2-5 PM in LH-5. The syllabus is everything in this course (except the Bochner technique).
Ethics: Read the information on the
IISc student ethics page. In short, cheating is a silly thing. Don't do
it. As for the quizzes based on HW, write them up on your own. You are NOT allowed to
discuss them amongst yourselves.
Here
is the tentative schedule. (It is subject to changes and hence
visiting this webpage regularly is one of the best ideas in the
history of best ideas.)
|
Week |
Dates |
Syllabus covered |
|
1 |
29 July - 4 Aug |
Introduction and Hopf rotation theorem for planar curves (Tuesday notes) |
|
2 |
5 Aug - 11 Aug |
Geometry of surfaces (Tuesday notes), Review of manifolds-Definitions, Tangent bundle (Thursday notes) |
|
3 |
12 Aug - 18 Aug |
Review of forms, vector fields, and integration (Tuesday notes), Thursday is a holiday |
|
4 |
19 Aug - 25 Aug |
Riemannian metrics and examples (Tuesday notes), Quotient manifold theorem and Riemannian submersions (Thursday notes) |
|
5 |
26 Aug - 1 Sept |
Induced metrics on tensors, volume form, and the distance function (Tuesday notes), Distance function is a metric, geodesic equation (Thursday notes) |
|
6 |
2 Sep - 8 Sept |
Connections - Definition (Tuesday notes), Connections - local behaviour, induced connections (Thursday notes) |
|
7 |
9 Sep - 15 Sept |
Connections - parallel transport, and the Levi-Civita connection (Tuesday notes), Riemann tensor and its symmetries, Ricci and scalar curvatures (Thursday notes) |
|
9 |
23 Sep - 29 Sept |
Riemann's theorem, sectional curvature(Tuesday notes), Model spaces and geometry of submanifolds(Thursday notes) |
|
10 |
30 Sept - 6 Oct |
Gauss-Codazzi equation, First variation formula, Definition of exponential map (Tuesday notes), Gauss lemma (Thursday notes) |
|
11 |
7 Oct - 13 Oct |
Hopf-Rinow theorem (Tuesday notes), Consequences of Hopf-Rinow (Thursday notes) |
|
12 |
14 Oct - 20 Oct |
Second variation and Jacobi fields (Tuesday notes), Geodesics stop minimising beyond the first conjugate point (Thursday notes) |
|
13 |
21 Oct - 27 Oct |
Proof of the geodesics minimising property, cut time, and injectivity radius (Tuesday notes), Bonnet-Myers theorem (Thursday notes) |
|
14 |
28 Oct - 3 Nov |
Sectional curvature comparison (Tuesday notes), Thursday was a holiday |
|
15 |
4 Nov - 10 Nov |
Local Bishop-Gromov (Tuesday notes), Bishop-Gromov and Cheng's rigidity (Thursday notes) |
|
16 |
11 Nov - 17 Nov |
Synge's theorem and orientation double covers (Tuesday notes), Preissmann's theorem (Thursday notes) |
|
HW number |
HW to be submitted by |
Homework (subject to changes; please check regularly) |
|
1 |
13 Aug |
|
|
2 |
29 Aug |
|
|
3 |
5 Sept |
|
|
4 |
Sept 24 |
|
|
5 |
3 Oct |
|
|
6 |
11 Oct |
|
|
7 |
18 Oct |
|
|
8 |
25 Oct |
|
|
9 |
8 Nov |