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Tagged with riemannian-geometry lie-groups
155 questions
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The orthogonal group of a riemannian metric
Let the inner product of the vectors X and Y on a given four dimensional manifold (EDIT: make this R4) be defined as (X*Y) = gikXiYk; using the summation convention for repeated indicies.
Let A be a ...
- 251
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Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
- 251
6
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3
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Action of the group of isometries on a manifold
Hi guys,
I am able to prove that any symmetric manifold is complete (Consider a local geodesic and use the symmetry to flip it, effectively doubling the length of the geodesic, ad infinitum). I want ...
- 516
2
votes
1
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Frobenius Theorem
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$
Then I suppose the following properties hold for M,
There exists a metric ...
- 3,541
6
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4
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Killing fields on homogeneous spaces
Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space.
Then how are the Killing fields on $G/H$ the projection of the right-invariant vector fields ...
- 3,541