Difference between revisions of "Differential Geometry II - 2020"
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==Niko, 6.8.2020== | ==Niko, 6.8.2020== | ||
| − | + | Levi-Civita Connection | |
| − | Def: torsion-free and compatible with metric | + | * Def: torsion-free and compatible with metric |
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| − | * | + | * Koszul-Formula |
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| − | * | + | * Uniqueness of L-C-Connection |
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| − | * Hadamard, Cartan | + | * (Mentioned case of Lie-group with bi-invariant Riem. metric and left-invariant vector fields, but I didn't remember and we just skipped that) |
| − | State | + | |
| − | Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script) | + | Rauch comparison theorem |
| − | Prove claim. | + | * Def: Jacobi Field (and geometric interpretation) |
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| + | * Thm 3.18: Statement | ||
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| + | Preissmann | ||
| + | * State and prove thm 4.20 | ||
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| + | Gauss equations | ||
| + | * Def: 2nd FF \(h(X,Y)\) and \(h_N(X,Y)\) (Mind: How are the LC-connections \(D,\bar{D}\) and metrics \(g,\bar{g}\) of \(M\) and \(\bar{M}\) related) | ||
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| + | * Proof: Show symmetry of (\h\) | ||
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| + | * State Gauss equations | ||
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| + | * State in terms of \(sec\) instead of \(R\) | ||
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| + | Hadamard, Cartan | ||
| + | * State thm 4.12 | ||
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| + | * Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script) | ||
| + | |||
| + | * Prove claim. | ||
Latest revision as of 09:23, 10 August 2020
Niko, 6.8.2020
Levi-Civita Connection
- Def: torsion-free and compatible with metric
- Koszul-Formula
- Uniqueness of L-C-Connection
- (Mentioned case of Lie-group with bi-invariant Riem. metric and left-invariant vector fields, but I didn't remember and we just skipped that)
Rauch comparison theorem
- Def: Jacobi Field (and geometric interpretation)
- Thm 3.18: Statement
Preissmann
- State and prove thm 4.20
Gauss equations
- Def: 2nd FF \(h(X,Y)\) and \(h_N(X,Y)\) (Mind: How are the LC-connections \(D,\bar{D}\) and metrics \(g,\bar{g}\) of \(M\) and \(\bar{M}\) related)
- Proof: Show symmetry of (\h\)
- State Gauss equations
- State in terms of \(sec\) instead of \(R\)
Hadamard, Cartan
- State thm 4.12
- Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script)
- Prove claim.
