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Manifolds.exp_lie != Manifolds.expeven at identity, -
Trivialized Jl == Jr[VanGoor Mahony 2024] - if the metric is biinvariant, the Cartan-Schouten-0 connection == Levi-Civita
- e.g. for
M=SO(n), exp(M,) = exp_lie(M,)
- e.g. for
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for Lie Groups, is a push forward (using Jacobian of function Manifolds.translate) equivalent to the parallel transport. And, how does this relate to left-invariance
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Manifolds.translate_diff vs Chirikjian Vol.2 ~p30 Jacobian (see about basis and generating Jacobian). See also Forney Jacobian and push-forward vs basis. Differential of group translate is vector transport. 8-o https://math.stackexchange.com/questions/2066207/how-can-one-define-parallel-transport-on-lie-groups
- Differential vs Jacobian only one has direction
- if you transport (maybe push forward) two vectors from TpM to TqM, and they no longer match up, then there is torsion along the path/geodesic? Does this relate to commutability
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Compare manifold approach vs vector field approach.
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Left invariant vector transport is invariant (i.e. identity)
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is dLa (a map), i.e. the differential/derivative, is this also the Jacobian
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when doing parallel transport do you need the following:
- Riemannian you need a metric to show the curvature?
- for Lie Group you need an affine connection (i.e. covariant derivative, exp_lie, group operation / one parameter subgroup to get geodesic curve along which to transport)