Riemannian and Complex Geometry
Possible Texts: do Carmo (F), Wells (W), MacDuff & Solomon (S)
Fall: Riemannian Geometry
Text: do Carmo, Riemannian Geometry (plus …?)
1. Riemannian Geometry
— Levi-Civita connection
— geodesics, existence
— completeness
— geodesics as flows on the tangent bundle
— Jacobi fields and curvature
— Riemann normal coordinates
— Riemann curvature tensor and its symmetries
— sectional curvature, scalar curvature
2. Connections on Vector Bundles
— Sasake Metric on the Tangent Bundle
3. Connections on Principal Bundles
— example: frame bundle
— induced connections on associated bundles
— horizontal distribution implies affine connection
4. Curvature on vector bundles and principal bundles
Winter: Geometric Topology
Texts: do Carmo, Wells, Milnor's Morse Theory
5. The Sphere Theorem or Synge's Theorem
(or local-to-global theorems of that type)
6. Morse Theory
— Morse index theorem
— Morse theory
7. de Rham complex
— Harmonic Forms
— Hodge Decomposition Theorem
8. Chern-Weil Theory
(8.5. The Dirac operator)
Spring: Symplectic and Complex Manifolds
Text: MacDuff and Solomon, Boothby, Kobayashi and Nomizu,
Griffiths and Harris
9. Symplectic Manifolds
— The Cotangent Bundle
— Hamiltonian flow
— Classical Mechanics
— Darboux Coordinates
— Moser Rigidity
— (Non-squeezing theorems)
10. Complex Manifolds
— Distributions and Frobenius's Theorem
— curvature as nonintegrability
— Almost complex structures, compatibility
— Newlander-Niremberg Theorem (analytic case)
11. Kahler Geometry