| Week of | Content | ||
| Aug 31 |
Lecture
1 and 2: Manifolds, differentiable maps, tangent vectors and tangent spaces, transport of vectors/ differentials of maps, immersions and embeddings, submanifolds, vector fields and their trajectories, commutator, (vector) distributions, Frobenius theorem, tensors, tensor fields |
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| Sept 7 |
Lecture
3 and 4: Local frames, differential forms, Cartan algebra and its derivations, Maurer-Cartan Theorem, more on Frobenius theorem, theorems of Pfaff and Darboux |
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| Sept 14 |
Lecture
5 and 6: Connections: Koszul axioms, parallelism; tensor-valued forms; connection 1-form, covariant exterior differential; curvature 2-form, torsion 2-form; Ricci formula; Ist and IInd Bianchi identities in terms of curvature 2-forms and torsion 2-forms; definition of curvature and torsion tensors in terms of Koszul notation |
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| Sept 21 |
Lecture
7 and 8: torsion/curvature 2-forms vs torsion and curvature tensors; Cartan structure equations and Bianchi identities as a closed differential system; Bianchi identities in the Koszul notation; Riemannian manifolds, pseudo-Riemannian manifolds; isometry |
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| Sept 28 | Lecture
9: Examples of metrics; left, right and biinvariant metrics on Lie groups; Lobachevski metric on an upper half plane; product metrics; wrapped products |
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| Oct 5 |
Lecture
10: Geodesics; how metric and torsion determines connection |
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| Oct 12 |
Lecture
11 and 12: Levi-Civita connection; connection coefficients in orthonormal and holonomic frames; arc length; geodesics as curves locally minimalizing arc length; geodesics in pseudo-riemannian setting; energy functional |
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| Oct 19 |
Lecture
13 and 14: Metric connections as connections which preserve scalar product under the parallel transport; Riemann tensor and its symmetries; symmetries of curvature tensor of general connection: the role of the metricity and vanishing torsion conditions; vanishing of the Riemann tensor as neccessary and sufficient condition for an existence of a local coordinate system in which the metric is flat; decomposition of the Riemann tensor onto SO(n)-irreducibles: Weyl, Ricci, Ricci sclar; conformal significance of the Weyl tensor; examples in low dimensions |
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| Oct 26 |
Lecture
15 and 16: Cannonical metrics on quadrics in flat (pseudo)-Riemannian manifolds; their curvature; Eisntein manifolds; Einstein field equations; examples of DeSitter and antiDeSitter spaces; isometries; Killing equations; full solution to the system of Killing equations in terms of flat metrics; isometry groups of maximal dimension; spaces of constant curvature; construction of all local metrics that have constant curvature in n-diemnsions; |
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| Nov 2 |
Lecture
17 and 18: Sectional curvature; spaces of constant sectional curvature; Spherical symmetry; 4-dimensional Lorentzian case: stationary vs static spaces; Schwarzschild metric; Homogeneity of geodesics; exponentila map; normal coordinates; normal ball; example of these concepts in case of Lobachevski metric; |
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| Nov 9 |
Lecture
19 and 20: Jacobi fields (exactly as in the relevant chapter of Do Carmo) |
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| Nov 16 |
Lecture
21: Local isometric embedding of hypersurfaces in R^(n+1); Gauss Codazzi equations |
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| Nov 23 |
Lecture
22: Local isometric embedding in R^(n+k) of codimension k; Gauss-Codazzi-Ricci equations; |
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| Nov 30 |
Lecture
23: Local isometric embedding of a Riemannian manifold (M^n,g) in a Riemannian manifold (M^(n+k), G); Gauss-Codazzi-Ricci equations |
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| Dec 7 |
Lecture
24 and 25: Gauss-Kronecker curvature; mean curvature; isoparametric hypersurfaces in space forms, with particular emphasis on hypersurfaces in spheres; minimal surfaces; Enepper-Weierstrass formula for minimal surfaces in R^3 |
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| Thank you for the attention! Pawel Nurowski | |||