Geodesics can be defined on manifolds using Riemannian exponential maps. I am looking for conditions where matrix exponential maps be used to define geodesics on matrix Lie groups.
2026-03-26 00:52:55.1774486375
For a matrix Lie group when would matrix log and exponential be same as Riemannian log and exponential?
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When the Lie Group is compact and connected, then the Riemannian Exponential coincides with the Lie Exponential, so you can use it to find Geodesics.
This also holds for a Lie Group which admits a biinvariant Metric (which is always tha case if the Group is compact).
Edit: When the Group is compact then you also have to choose a biinvariant metric (but it would not make sense to talk about geodesics without a metric anyways...)