- Required Text:
- Boothby, An Introduction to Differentiable
Manifolds and Riemannian Geometry, 2nd Ed, Academic Press
- Recommended Texts:
- Morgan, Riemannian Geometry: A
Beginner's Guide, 2nd Ed, A K Peters
Bishop and Goldberg, Tensor Analysis on Manifolds, Dover
Milnor, Topology from the Differentiable Viewpoint, U P Virginia
Spivak, Calculus on Manifolds, Benjamin/Cummings
- Outline:
-
This is a first course in manifolds and global analysis, which will
present the basic tools for those interested in, or curious about,
differential geometry or global analysis, or those who want to apply
differentiial geometric methods in other areas such as PDE, topology,
mathematical physics, and dynamical systems.
The course will cover the following topics:
- Manifolds:
- Differentiable manifolds, implicit function
theorem, rank theorem, tangent spaces, tangent bundles, vector bundles.
- Calculus on manifolds:
- Vector fields, flows, Lie bracket,
Lie derivatives, Frobenius theorem.
- Differential forms:
- Differential forms, exterior calculus,
orientability, Poincaré lemma, deRham complex.
- Integration theory:
- Stokes' theorem.
- Riemannian geometry:
- Riemannian metrics, distance, first
variation and geodesics, Riemannian connection, curvature, connections
on vector bundles.
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