Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by $$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, dt\,,$$ where the infinum is taken over all $C^1$-curves connecting $x$ and $y$.

I'm interested in the basic properties of the usual constructions which solely depend on the metric and not on $g$. So what can be said about the smoothness of geodesics, exponential map,... if one forgets $g$ and only consider $(M, d)$ as a smooth metric manifold.

Clearly one can define geodesics only with respect to $d$ and assuming that the space is locally uniquely geodesic one can define the exponential map $\text{exp}$ as a map from (a subset of) local geodesics $\mathcal{G}$ to a neighborhood of $M$. This is basic topology of metric spaces, but I could not find any exposition discussing the smoothness of this constructions if $M$ is a manifold. So for example: Can $\text{exp}$ be considered as a map from the tangent bundle to the manifold (thus is the map $\gamma \rightarrow \dot\gamma(0)$ a bijection of $\mathcal{G}$ with some open subset of $TM$) and is it smooth? Does there exists a (adapted) definition of Jacobi-Fields?

(I'm not so interested in results which first recover the Riemannian metric $g$ [I think there is an old paper of Palais discussing this issue] and then run along the basic route to define geodesics, exponential map and Jacobi-Fields. The reason is, that I have a generalization of Riemannian geometry in mind where the standard procedure is definitely not possible.)