| Speakers: 1st Week |
| 1 |
AM |
Anandateertha Mangasuli |
| 2 |
CSA |
C S Aravinda |
| 3 |
GS |
G Santhanam |
| 4 |
HS |
Harish Seshadri |
| Speakers: 2nd Week |
| 1 |
KS |
K Shankar |
| 2 |
KV |
Kaushal Verma |
| 3 |
SG |
Siddhartha Gadgil |
| 4 |
TNV |
T N Venkataramana |
| 5 |
GB |
Gautam Bharali |
Schedule of Lectures
| Date |
0930-1100 |
1130-1300 |
1400-1500 |
1530-1700 |
| 19/7 (Monday) |
L1(CSA) |
L2(AM) |
Discussion |
L3(HS) |
| 20/7 (Tuesday) |
L4(AM) |
L5(HS) |
Discussion |
L6(AM) |
| 21/7 (Wednesday) |
L7(AM) |
L8(GS) |
Discussion |
L9(HS) |
| 22/7 (Thrusday) |
L10(HS) |
L11(CSA) |
Discussion |
L12(CSA) |
| 23/7 (Friday) |
L13(CSA) |
L15(GS) |
Discussion |
L14(CSA) |
| 24/7 (Saturday) |
L16(GS) |
L17(AM) |
Discussion |
L18(HS) |
| Date |
0930-1100 |
1130-1300 |
1400-1530 |
1600-1700 |
| 26/7 (Monday) |
KV1 |
KS1 |
SG1 |
GB(GL1) |
| 27/7 (Tuesday) |
KV2 |
KS2 |
SG2 |
HS1(BT)* |
| 28/7 (Wednesday) |
HS2(BT) |
KS3 |
SG3 |
TNV1(GL2) |
| 29/7 (Thrusday) |
CSA1(RF) |
KS4 |
HS1(RF) |
TNV2(GL3) |
| 30/7 (Friday) |
SG4 |
HS2(RF) |
CSA2(RF) |
KS(GL4) |
* This lecture will be from 1600 to 1730 hrs.
First week
- Lecture 1: Smooth manifolds - definition and examples; smooth functions, bump functions (smooth urysohn lemma); tangents vectors, vector fields, tensor fields - definition and properties (of tensors).
- Lecture 2: Metric tensor, Riemannian manifolds, covariant differentiation, curvature tensors and curvatures; lengths of curves, distance function, geodesics, parallel transport and exponential map.
- Lecture 3: Hopf-Rinow theorem.
- Lecture 4: First and second variations of length and energy functionals; Jacobi fields, Gauss lemma.
- Lecture 5: Cartan-Hadamard and Bonnet-Myers theorems.
- Lecture 6: Models of constant curvature; Cartan’s theorem on the determination of the metric by (constant) curvatures.
- Lecture 7: Rauch and Toponogov comparison theorems (include proof of Rauch but only state Toponogov).
- Lectures 8, 9 and 10: Klingenberg’s injectivity radius estimate, Synge’s theorem, Bishop and Bishop-Gromov volume comparison theorems.
- Lectures 11, 12, 13 and 14: Preissmann (and Flat-torus) theorems, Eberlein-O’Neill compactification, Busemann functions, classification of isometries (into elliptic, parabolic and axial or hyperbolic).
- Lectures 15 and 16: Riemannian immersions, submersions; immersion and submersion equations; second fundamental form.
- Lectures 17 and 18: Symmetric spaces of compact and noncompact type; their curvatures.
Second week
- Shankar (KS) will give 4 lectures on the Soul theorem, the splitting the- orem, structure of fundamental groups in non-negative and positive curva- ture. About 4 lectures on Bochner technique (BT) (2 by KV and 2 by HS), 4 lectures on Ricci flow (2 by CSA and 2 by HS) and about 4 lectures on Gromov-Hausdorff convergence, by SG.
Besides the above main series lectures, we will have two guest lectures by T N Venkataramana on the construction of compact and finite-volume quotients of real hyperbolic space forms, one guest lecture by Gautam Bharali on uniformization of surfaces and one guest lecture by K Shankar.
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