I'm wondering if someone could please point me out to a reference (or the actual identity) where the following is shown.
Let $\gamma:I\to (\mathbb{R}^n,d=d_{euclidean})$ be a geodesic in $n$-Euclidean space and denote by $d_p:\mathbb{R}^n\to \mathbb{R}$ the distance function to a fixed point $p\in\mathbb{R}^n$. Then it's easy to show that, for the function $f=d_p\circ \gamma:I\to \mathbb{R}$, the following holds (when everything is smooth): $$f''f=1-f'^2 .$$
Does an analogous simple identity hold when considering the simply connected spaces of constant curvature $k$ rather than Euclidean space? (Of course, this should follow by finding some nice coordinates and computing to get the identity, but I figured I could ask and save myself some time).
Thanks for any help!
This is false in other spaces $M$ of constant curvature. The reason is that $f$ is a 1-Lipschitz function (true in any Riemannian manifold) and, hence, $f'^2\le 1$. Hence, your identity implies that $f''\ge 0$. However, if $M$ has positive curvature, there are pairs $(p, \gamma)$ such that $f$ is not convex (this is a good exercise which requires no computations, just pure thought). Hence, your identity cannot hold for the sphere. Using real-analyticity for distance function as a function of the curvature, one concludes that the identity cannot hold for hyperbolic spaces either.