Difference between revisions of "Differential Geometry II - 2020"
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(Created page with "==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-inva...") |
(→Niko, 6.8.2020) |
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* Levi-Civita Connection: | * Levi-Civita Connection: | ||
Def: torsion-free and compatible with metric | Def: torsion-free and compatible with metric | ||
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Koszul-Formula | Koszul-Formula | ||
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Uniqueness of L-C-Connection | Uniqueness of L-C-Connection | ||
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(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) | (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 | * Rauch comparison theorem | ||
Def: Jacobi Field (and geometric interpretation) | Def: Jacobi Field (and geometric interpretation) | ||
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Thm 3.18: Statement | Thm 3.18: Statement | ||
* Gauss equations | * Gauss equations | ||
Def: 2nd FF \(h(X,Y)\) and \(h_N(X,Y)\) (Mind: what are relations of LC-connections and metric in \(M\) and \(\bar{M}\) | Def: 2nd FF \(h(X,Y)\) and \(h_N(X,Y)\) (Mind: what are relations of LC-connections and metric in \(M\) and \(\bar{M}\) | ||
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Proof: Show symmetry of (\h\) | Proof: Show symmetry of (\h\) | ||
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State Gauss equations | State Gauss equations | ||
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State in terms of \(sec\) instead of \(R\) | State in terms of \(sec\) instead of \(R\) | ||
* Hadamard, Cartan | * Hadamard, Cartan | ||
State theorem 4.12 | State theorem 4.12 | ||
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Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script) | Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script) | ||
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Prove claim. | Prove claim. | ||
Revision as of 09:18, 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
- Gauss equations
Def: 2nd FF \(h(X,Y)\) and \(h_N(X,Y)\) (Mind: what are relations of LC-connections and metric in \(M\) and \(\bar{M}\)
Proof: Show symmetry of (\h\)
State Gauss equations
State in terms of \(sec\) instead of \(R\)
- Hadamard, Cartan
State theorem 4.12
Claim in proof: \(sec\leq0\) implies no conjugate points. (This was an exercise in script)
Prove claim.
