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## Main Question or Discussion Point

Hello.

Suppose that [itex]\sigma: (M, g) \to (N, h)[/itex] is an isometric diffeomorphism between two Riemannian manifolds

Because [itex]\sigma[/itex] preserves distances, and geodesics are locally length minimizing, it is intuitively clear that [itex]\sigma_* \gamma = \sigma \circ \gamma[/itex] is a geodesic on

In particular, I don't see which characterization of geodesics is the most convenient (I suppose it is the locally length minimizing property; but I don't really see how to express that formally).

Any help is greatly appreciated.

Suppose that [itex]\sigma: (M, g) \to (N, h)[/itex] is an isometric diffeomorphism between two Riemannian manifolds

*M*and*N*and let [itex]\gamma: [0, 1] \to M[/itex] be a geodesic on*M*.Because [itex]\sigma[/itex] preserves distances, and geodesics are locally length minimizing, it is intuitively clear that [itex]\sigma_* \gamma = \sigma \circ \gamma[/itex] is a geodesic on

*N*, but I'm having some trouble proving this.In particular, I don't see which characterization of geodesics is the most convenient (I suppose it is the locally length minimizing property; but I don't really see how to express that formally).

Any help is greatly appreciated.