| January 13: |
Riemannian metrics
Riemannian volume form
Divergence of a vector field
Riemannian gradient of a function
Laplace-Beltrami operator
Harmonic functions
Conformal diffeomorphisms
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| January 15: |
Existence of isothermal coordinates on Riemannian surfaces
Vector bundles
Connections
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| January 20: |
Linear connections
Existence and uniqueness of the Levi-Civita connections
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| January 22: |
Exercises:
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| January 27: |
Exercises:
Solutions:
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| January 29: |
Exercises:
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| February 3: |
Geodesics
Geodesic vector field and geodesic flow
Exponential map
Injectivity radius
Positivity of the injectivity radius of compact sets
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| February 5: |
Exercises:
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| February 10: |
Variational characterization of geodesics
Gauss Lemma
Short geodesics are the unique minimizers of the length function
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| February 12: |
Exercises:
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| February 17: |
Hopf-Rinow Theorem
Riemann tensor
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| February 19: |
Exercises:
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| March 9: |
Parallel transport
Flat Riemannian manifolds
Symmetries of the Riemann tensor
Ricci tensor
Scalar curvature
Einstein manifolds
Sectional curvature
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| March 19: |
Exercises:
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| March 16: |
Notes:
Riemannian submanifolds
Second fundamental form
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| March 18: |
Exercises:
Solutions:
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March 20:
(15h30, Amphi A) |
Notes:
Embedded surfaces in the Euclidean 3-space
Gaussian curvature
Gauss' Theorema Egregium
Einstein manifolds of higher dimension have constant scalar curvature
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| March 23: |
Notes:
Hopf Umlaufsatz
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| March 25: |
Exercises:
Solutions:
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| March 30: |
Notes:
Gauss-Bonnet's Theorem
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| April 1: |
Exercises:
Solutions:
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| April 6: |
Notes:
Hessian of the length functional
Conjugate points
Jacobi fields
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| April 8: |
Exercises:
Solutions:
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| April 15: |
Exercises:
Solutions:
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