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.)
Best Answer
If you forget about the Riemannian metric, you should also forget about the tangent bundle and manifold structure. Then you end up with metric spaces of inner type or Aleksandrov spaces, where you can find a continuous curves with arc-length the distance. Look at "M. Gromov: Metric structures for Riemannian and Non-Riemannian Spaces, Birkhaeuser 1999".
Added in edit: Interesting remarks by Thomas Richard and Igor Belegradek. Answering a remark by the OP: You can define a "geodesic structure" by requiring an exponential mapping with some properties. If you differentiate this you end up with the geodesic spray, a certain vector field on $TM$. If you differentiate this again and flip coordinates, you get a vector field on $TTM$ whose integral curves project to Jacobi fields, velocity fields of geodesics, etc. See 22.6-22.9 of here, and also this paper. But this is not the route of low regularity.