My question is about computing the distance between two points in a Riemannian manifold.
Suppose that $(M,g)$ is compact so that it is geodesically complete and geodesically convex.
Let $X\in\Gamma(TM)$ be a vector field. Fix a point $p\in M$. Let $\gamma:\mathbb{R}\to M$ denote the unique geodesic with initial velocity $X_p$. That is, $$(\exp)_p(X)=\gamma(1) \ \ \text{ and } \ \ (\exp)_p(tX)=\gamma(t).$$
My question is, why does $$d\left((\exp)_p(X),(\exp)_p(tX)\right)=|1-t||X_p| \ ?$$
Here $d:M\times M\to \mathbb{R}$ denotes the Riemann distance function. That is
$$d(p,q)=\inf\left\{\int|\rho^\prime(t)|dt \ ; \rho \text{ is an admissible curve between $p$ and $q$}\right\}$$
Since geodesics are length minimizing wouldn't $\gamma$ be the unique curve between $(\exp)_p(X)$ and $(\exp)_p(tX)$ which minimizes the distance function? But the integral of $|\gamma^\prime(t)|$ isn't $|1-t||X_p|$ ?
Any help is very much appreciated.
Geodesics are only locally length-minimizing, i.e., only for $t$ sufficiently close to $1$ will $\gamma$ necessarily be the geodesic of shortest length joining $\gamma(1)$ and $\gamma(t)$. All that can be said in general is that the distance in question is at most $\left|1-t \right| \, |X_p|$.
To elaborate, there may be multiple geodesics joining $\gamma(1)$ and $\gamma(t)$; $\gamma$ is one such geodesic, but it may not be the geodesic of shortest length. A good example to keep in mind is the round sphere $S^2$: between any two distinct (non-antipodal) points $p$ and $q$ on the sphere, there is a unique great circle passing through $p$ and $q$, but there are two geodesics joining $p$ and $q$, each corresponding to one of the two ways to go around the great circle.
Edit (to explain $|1-t| \, |X_p|$): The point is that the velocity vector of a geodesic defined via the exponential map has constant length along the geodesic, i.e., for all $s$, $|\gamma'(s)| = |\gamma'(0)| = |X_p|$. This is a consequence of Gauss's lemma, which says that the exponential map is a "radial isometry". Assuming that $t<1$, the length of $\gamma(s)$ between $s=1$ and $s=t$ is $$ \int_t^1 |\gamma'(s)| \, ds = \int_t^1 |X_p| \, ds = (1-t)|X_p|$$ (and similarly if $t > 1$).