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