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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.)

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Your question is interesting but a bit vague. A central point is how does your distance $d$ relate to the smooth structure of $M$ ? The thing is that given $(M,d)$ you can always twist $d$ by a non smooth homeomorphism of $M$ and you'll get something which is isometric to your original $(M,d)$, however geodesics won't be smooth anymore. The only condition I can think about is to require that $d^2$ is smooth in a neighborhood of the diagonal of $M\times M$, but I have no idea if this gives an interesting theory. –  Thomas Richard Jul 12 '13 at 13:01
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Even though you dismiss "the basic route to define geodesics..." it could be useful to look at results dealing with Riemannian metrics of low regularity, see e.g. math.uni-bonn.de/people/lytchak/Finsler.ps [Lytchak-Yaman "On Hoelder continuous Riemannian and Finsler metrics", Trans. Amer. Math. Soc., Vol. 358 (2006), 2917-2926], and arxiv.org/abs/1306.4776 ["The exponential map of a $C^{1,1}$-metric", Kunzinger-Steinbauer-Stojkovic] –  Igor Belegradek Jul 12 '13 at 13:49
    
@T. Richard: Thanks for your comment. In fact you pointed out the interesting and important 'subquestion' about the relation of $d$ with the smooth structure. I think one can not assume $d$ to be smooth (because locally it is isometric to the non-smooth euclidean metric). But on the other hand the metrics induced by Riemannian metrics result in smooth geodesics and exponetial maps. I was hoping, there is a characterization of these metrics and one can than directly proof the smoothness of exp ect. –  Tobias Diez Jul 12 '13 at 14:43
    
Other examples to have in mind: (strictly convex, reversible) Finsler metrics; sub-Riemannian manifold (e.g. Heisenberg group), which provide interesting metrics on smooth manifold, but have specific features. For example, in sub-Riemannian geometry you can't have an exponential map with Riemannian properties, as all geodesics have initial velocity in a strict subspace of $T_xM$. –  Benoît Kloeckner Jul 13 '13 at 7:00
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up vote 8 down vote accepted

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.

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"A first course in metric geometry" by Burago, Burago and Ivanov is also a nice reference on these topics. However, I am not sure this answer the question completely. For instance, one can say (really roughly) that the beginning of Finsler geometry is like the beginning of Riemannian geometry except that you forget you had a scalar product on $TM$ and just work with the norm. I'm not sure it is a good example. My point is that there may be a class of "smooth" distances on $M$ which yields an interesting theory... –  Thomas Richard Jul 12 '13 at 13:28
    
A question about your first paragraph: Why shouldn't one consider a smooth manifold endowed with a metric and ask under what conditions the exponential map is smooth? Clearly for metrics induced by a Riemannian metric this is true, but I was hoping that there is a bigger class and/or a proof which does not construct this Riemannian metric as the first step. –  Tobias Diez Jul 12 '13 at 14:54
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