MTH
5122 Geometry 1
| Instructor: |
Maxim
Braverman |
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457 LA, ext. 8769 |
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| Class hours: |
T,R 7:30 - 9:00 |
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Location: 509 LA |
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| Textbook: |
The course will be based
on several books. I will distribute notes and printouts
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| Prerequisites: |
Analysis 2, Linear
Algebra. |
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| Grading: |
Weekly homework problems
may be done collaboratively. The take-home final exam must be entirely
your own work. |
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| Main topics to be covered: |
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- Manifolds. Smooth
functions on manifolds. Tangent vectors. Maps between manifolds.
Submanifolds. Wittney embedding theorem.
- Vector
bundles. Tangent and cotangent bundles.
- Vector
fields. Flow of vector fields. Lie bracket. Frobenius theorem.
- Introduction to Lie groups.
- Principal bundles.
- Differential
forms. De Rham complex. De Rham cohomology.
- Connection on principal and vector bundle. Parallel
transport.
- Covariant
derivative. Twisted De Rham complex. Curvature of a connection.
Geometric interpretation of curvature.
- Chern-Weil
theory and characteristic classes.
- Hermitian
metric on a vector bundle. Hermitian connection.
- Affine connections (connections on the tangent
bundle). Torsion of a connection.
- Riemannian metric. Existence of a Riemannian metric.
Length of curves. Geodesics.
- Riemannian
connections. Christoffel symbols. Exponential map.
- Levi-Civita
connection. Normal coordinates.
- Riemannian
curvature. Ricci curvature. Scalar curvature.
- Spaces of
constant curvature.
- Laplacian.
Harmonic forms and cohomology (without proofs).
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Notes and printouts:
From Warner's book: Manifolds,
Tangent and
cotangent space,
suubmanifolds, Implicit
function theorem, vector fields, Frobenius theorem
Lecture notes by Professor E. Meinrenken:
Differentiable
manifolds: Sections 1-2 provides some introduction to manifolds,
section 3 deals with vector fields, section 13 introduces fiber and
vector bundles.
Riemannian
Geometry: Sections 1-4 briefly review the notions of
manifolds, tangent vectors, vector fields, differential and
push-forward map, section 6 discusses the tangent
bundles, lectures 7-9 deal with vector fields and their brackets.
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| Homework assignments: |
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