1.Jacobi fields, 2nd variation, Jacobi equation, conjugate points,minimizing property of geodesics, Index Lemma, Jacobi’s theorem, two proofs
2. Myers-Bonnet theorem, Cartan-Hadamard theorem, Rauch comparison theorem with applications to injectivity radius
3. Space of constant curvature, group theory viewpoints, geodesics, Jacobi fields
4. Cartan-Ambrose-Hicks Theorem
5, 6, 7. 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
8. vector bundles, bundle maps, pull-back bundles, complex vector bundles
9. connection, curvature form, Bianchi identity
10. Chern classes, invariant polynomials, Chern character, unitary connection
11. Examples and application of Chern classes, immersions and embeddings in complex projective spaces
12. Pontrjagin classes, Euler class, relation with Chern classes,Todd class, A-hat genus
13. star operator, Hodge decomposition theorem, Poincare duality,
14. Kunneth formula, Bochner-Weitzenbock formula, proof
15. divergence, application of B-W formula to topology of manifolds, index of de Rham complex, remark on Index theorem
16. Gauge theory, Erlanger Program, historical remarks, principle bundles
17. Examples of Lie groups, SU(2)-bundles, Yang-Mills equation, self-duality equation