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Modern Geometry I
I Differential Manifolds
- Smooth
manifolds
- Tangent spaces,
tangent bundles
- Smooth maps
(submersions, immersions, embeddings)
- Vector fields (flows,
Frobenius's theorem)
- Lie groups and
homogeneous spaces
- Tensors,
differential forms, Stokes's theorem
II Riemannian Geometry
- Riemannian
metrics and connections
- Geodesics
- Curvature
- Jacobi fields
- Isometric
immersions (second fundamental form)
- Hopf-Rinow and
Cartan-Hadamard theorems
- Manifolds of
constant curvature
- Bonnet-Myers
theorem
Modern Geometry II
I Differential Topology
- Transversality
- Tubular
neighborhoods
- Intersection
theory (mod 2 and oriented)
- Degrees
- Poincare-Hopf
index theorem
- Lefschetz
fixed-point theorem
- de Rham
cohomology
- Poincare
duality
II Vector Bundles and Principal Bundles
- Real and
complex vector bundles
- Metrics,
connections, and curvature on vector bundles
- Chern,
Pontryagin, and Euler classes
- Principal
bundles
- Connections and
curvature on principal bundles
- Parallel
transport and holonomy
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