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Consider a homogeneous space $G/\text{Stab}_p \cong M$ where $G$ is a compact Lie group active transitively on $M$ (a compact manifold).

If $F$ is a Finsler Metric on $G$ which pushes forward unambiguously through the map $\phi: G \rightarrow M$ given by $\phi(g) = g \circ p$ then can one know that the geodesics of the push forward metric are the push forward of the geodesics on $G$?

I'd really appreciate a reference for the case where $F$ is a Riemannian metric also.

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