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