Teaching of Maxim Braverman for the  

Course Information
MTH 5122 Geometry 1
Instructor: Maxim Braverman   457 LA, ext. 8769   
Class hours: T,R 7:30 - 9:00    Location:   509 LA  
Textbook:   The course will be based on several books. I will distribute notes and printouts
      
Prerequisites: Analysis 2, Linear Algebra.       
Grading: Weekly homework problems may be done collaboratively. The take-home final exam must be entirely your own work.      
Main topics to be covered:
   
 
  1. Manifolds. Smooth functions on manifolds. Tangent vectors. Maps between manifolds. Submanifolds. Wittney embedding theorem.
  2. Vector bundles. Tangent and cotangent bundles. 
  3. Vector fields. Flow of vector fields. Lie bracket. Frobenius theorem.
  4. Introduction to Lie groups.
  5. Principal bundles.
  6. Differential forms. De Rham complex. De Rham cohomology.
  7. Connection on principal and vector bundle. Parallel transport. 
  8. Covariant derivative. Twisted De Rham complex. Curvature of a connection. Geometric interpretation of curvature. 
  9. Chern-Weil theory and characteristic classes. 
  10. Hermitian metric on a vector bundle. Hermitian connection. 
  11.  Affine connections (connections on the tangent bundle). Torsion of a connection. 
  12. Riemannian metric. Existence of a Riemannian metric. Length of curves. Geodesics. 
  13. Riemannian connections. Christoffel symbols. Exponential map. 
  14. Levi-Civita connection. Normal coordinates. 
  15. Riemannian curvature. Ricci curvature. Scalar curvature. 
  16. Spaces of constant curvature. 
  17. Laplacian. Harmonic forms and cohomology (without proofs).
     
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.


Homework assignments:




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Created: August 17, 2008.  Last modified: 

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