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Welcome to the ApproxManifoldProducts.jl wiki!

Working Assumptions

• Manifolds.exp_lie != Manifolds.exp even 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,)

To Confirm:

• 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

• 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

• Compare manifold approach vs vector field approach.

• Left invariant vector transport is invariant (i.e. identity)

• is dLa (a map), i.e. the differential/derivative, is this also the Jacobian