This class will go through the following topics in a year.

1. Motivations for manifolds; concept of manifolds; vector fields, tangent vectors, 1-forms
2. Pull-back; push-forward; submanifolds
3. Whitney Embedding theorem; immersions, submersions
4. Tangent bundles; tensors; antisymmetric tensors
5. Flow; one parameter subgroup
6. Brackets, Lie derivative, computation via coordinates
8-10. Differential forms, exterior derivative, tensor operations, contractions;
Interior product, Cartan’s formula, classical vector analysis, Stokes theorem
for differential forms
11-12. Frobenius theorem, Poincare Lemma, De Rham cohomology, differential form version of Frobenius, applications to PDE and geometry
13. Riemannian manifolds, Euclidean case, covariant derivative
14. Riemannian metric, Riemann-Christoffel symbols, parallelism, geodesic
15. Exponential mapping, normal coordinates
16. Convex neighborhoods; two proofs
17. Riemannian curvature, sectional curvature, Ricci curvature
18. Bianchi identity, hypersurface
19. 1st fundamental forms, 2nd fundamental forms determine hypersurfaces up to isometry
20. Variation formula, Gauss lemma
21. Hopf-Rinow theorem
22-24. Jacobi fields, 2nd variation, Jacobi equation, conjugate points, minimizing property of geodesics, Index Lemma, Jacobi’s theorem, two proofs
25-27. Myers-Bonnet theorem, Cartan-Hadamard theorem, Rauch comparison theorem with applications to injectivity radius
28-29. Space of constant curvature, group theory viewpoints, geodesics, Jacobi fields
30. Cartan-Ambrose-Hicks Theorem
31. Miscellaneous
a) flows and transformations
b) Killing vector fields
c) volume element and divergence
d) Ricci curvature and volume growth
e) 2nd Bianchi identity applied to Einstein manifolds
f) Cut locus, injectivity radius, Klingenberg’s lemma

References for 1st semester:
1. Do Carmo: Riemannian geometry (together with his book on “curves and surfaces”)
2. Gallot-Hulin-Lafontaines: Riemannian geometry
3. Helgason: Differential geometry, Lie groups and symmetric spaces
4.Hicks: Notes on Differential geometry
5. Cheeger-Ebin: Comparison theorems in Riemannian geometry