Differential Geometry II

Course Material: The two textbook references are [Frankel] and [Lee]. The first part of the course will introduce connections on vector bundles and follow a mix of both references. The second part of the course will be on Riemannian geometry and follow [Lee].

References:
[Frankel] T. Frankel, "The Geometry of Physics"
[Lee] J. Lee, "Introduction to Riemannian Manifolds"

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Lecture Notes:
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(0) Review of manifolds

(1) Vector bundles I
[Frankel] 16.1

(1) Vector bundles II
[Frankel] 16.1

(2) Metrics
[Lee] Chapter 2

(3) Connections
[Frankel] 16.3, [Lee] Chapter 4

(4) Levi-Civita connection
[Frankel] 9.2

(5) Curvature of a connection
[Frankel] 9.3, 9.4

(6) Riemann curvature tensor
[Lee] Chapter 7

(7) de Rham cohomology
[Lee: Introduction to Smooth Manifolds] Chapter 17

(8) Chern classes
[Frankel] 22.1

(9) Yang-Mills functional
[Frankel] 20.5

(10) Geodesics
[Lee] Chapter 5

(10) Geodesics II
[Lee] Chapter 6

(11) Completeness
[Lee] Chapter 6

(12) Submanifold geometry
[Lee] Chapter 8

(13) Jacobi fields I
[Lee] Chapter 10

(14) Jacobi fields II
[Lee] Chapter 10

(15) Laplace comparison theorem
[Lee] Chapter 11