Over a complete Riemannian manifold $(M,g)$, in a neighborhood of $p \in M$, the local distance can be approximated as follows: $\forall v,u \text{ unit vectors in } T_pM, \text{ and small } s, t$
$$ d ( Exp_p (tv) , Exp_p (su) ) = \sqrt{t^2 + s^2 - 2 st \cos \theta} \left( 1 - \kappa_p (v,u) \cdot O ( t s ) + O (t^2 s^2) \right) , $$
where $\theta$ is the angle between $v$ and $u$ (in $T_pM$), and $\kappa_p (v,u)$ is the sectional curvature at $p$, along $v$ and $u$.
Is there a convenient textbook reference (with proof) for this?
Edit: Some nice discussions on this have happened here: Square of the distance function on a Riemannian manifold.
