Math 865 Advanced Topics in Geometry (Fall 2009)
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Lectures
Lectures are by
Jeff Viaclovsky on Tuesdays and Thursdays
at 02:30-03:45 PM in Van Vleck B337.
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Recommended Textbooks
- Riemannian Geometry by Peter Petersen.
- Differential Geometric Structures by Poor.
- Einstein Manifolds by Besse.
- Lecture notes in Geometric Analysis.
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Mailing List
The class mailing list is math865-1-f09 "at" lists.wisc.edu.
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Lectures
- Lecture 1: Thursday, September 3
- Introduction.
- Lecture 2: Tuesday, September 8
- Riemannian metrics, vectors, and one-forms.
- The musical isomorphisms.
- Inner product on forms and tensor bundles.
- Connections on vector bundles.
- Fundamental Theorem of Riemannian Geometry.
- Lecture 3: Thursday, September 10
- Covariant derivative of tensor fields.
- Gradient and Hessian operators.
- Curvature of connections on vector bundles.
- Pull-back bundles and pull-back connection.
- Structure equation: pull-back of curvature tensor.
- Lecture 4: Tuesday, September 15
- Curvature of the Riemannian connection.
- Sectional curvature, Ricci tensor, and scalar curvature.
- Differential Bianchi Identity.
- Lecture 5: Thursday, September 17
- Discussion on wedge and symmetric products.
- Algebraic study of the curvature tensor.
- Lecture 6: Tuesday, September 22
- Algebraic study of the curvature tensor cont'd.
- Orthogonal decomposition of the curvature tensor.
- Lecture 7: Thursday, September 24
- Commuting covariant derivatives.
- Lecture 8: Tuesday, September 29
- Rough Laplacian and gradient.
- Commuting Hessian and Laplacian.
- An application to PDE.
- Lecture 9: Thursday, October 1
- Integration and adjoints.
- Lecture 10: Tuesday, October 6
- Formula for Hodge d and \delta operators.
- \delta = - div on p-forms.
- Lecture 11: Tuesday, October 13
- Bochner and Weitzenbock formulae.
- Manifolds with positive curvature operator.