Let $(M,g)$ be a connected Riemannian manifold.
I have been working through the problems my advisor gave, and it mentioned the concept called "geodesic distance." Although I plan to ask my advisor what it means, I tried to guess the meaning in the meantime. My guess is that for $p,q\in M$, the geodesic distance between $p$ and $q$ is
$d_g(p,q) = \inf\{\int_{0}^{1}|\gamma'(t)|dt|\gamma:[0,1]\to M,\gamma(0)=p,\gamma(1)=q \textrm{ is a geodesic}\}$
However, Riemannian geometry has the concept of so-called Riemannian distance, defined by
$d_r(p,q) = \inf\{\int_{0}^{1}|\gamma'(t)|dt|\gamma:[0,1]\to M,\gamma(0)=p,\gamma(1)=q \textrm{ is a piecewise } C^1 \textrm{ curve}\}$
My question is, do these distances agree in general?
I think if $M$ is complete, they should be equal using Hopf-Rinow. But I don't know about the general case. I feel like it does not agree, but I cannot come up with a counterexample.
Best Answer
If $M$ is not complete, then there need not exist any geodesic between two points at all, so that $d_g(p,q)=\infty$. For instance, if $M$ is $\mathbb{R}^n$ with a point removed, there is no geodesic from $p$ to $q$ if the removed point is on the line segment between them.
I don't know of any situation where $d_g$ is of interest and it does not coincide with $d_r$; I would assume "geodesic distance" always refers to $d_r$ even if the name is a bit inappropriate if $M$ is incomplete.