# Homotheties and Geodesics

A homothety is a smooth map between Riemannian manifolds $f:(M,g)\rightarrow (N,h)$ such that $f^*h=cg$ for some $c\ne 0$.

My questions is: how are the geodesics on these two spaces related? Can we say that the geodesics on $(N,h)$ are given by $\beta (t)= (f\circ \alpha)(t/\sqrt{c})$ where $\alpha$ is a geodesic on $(M,g)$?

• Welcome to Math.SE! Do you assume that $f$ is a diffeomorphism between $M$ and $N$? Or is a map such as $z\mapsto z^n$ on a circle also a homothety?
– user53153
Jan 9, 2013 at 0:02
• @Pavel: Thanks. Yes, I'm assuming it's a diffeomorphism. Do you have a reference for this result? Homotheties are not required to be diffeomorphisms. Jan 9, 2013 at 0:06
• I think it's just easier to prove than to find a reference for. The details depend on the definition of a geodesic that you are using, of course. For example, the property of being locally length-minimizing is invariant under a homothety.
– user53153
Jan 9, 2013 at 0:10
• @Pavel: OK, I'll take a look at your argument. Thanks. Jan 9, 2013 at 0:13