TITLE: Shortest-path Distances Determining the Metric?
QUESTION [14 upvotes]: The metric of a Riemannian manifold determines the shortest
distance between any two points.

I assume the reverse holds?  That is, if you are given the
shortest distance d(x,y) between every pair of points
of a manifold M, the metric for M is determined?
I am mainly interested in compact, connected, closed 2-manifolds,
but the most general answer would be appreciated.
(Apologies in advance for my naivete.)
Assuming I am correct above, is there some natural subset
S of M such that knowing d(x,y) between every
pair of points of S uniquely determines the metric (up to isometry)?
For example, suppose S is a simple closed geodesic on
M?  Perhaps some assumptions on M are necessary:
genus zero, convex, ... ?

REPLY [4 votes]: Nice paper, I was not aware of it. There is this paper: 
I.G. Nikolaev, A metric characterization of riemannian spaces, Siberian Advances in Mathematics, 1999, v. 9, N4, 1-58 
see the mathscinet page: 
http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Nikolaev&s5=metric%20characterization&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq
Citing from the Abstract: "We present a metric characterization of Riemannian spaces: a locally compact metric space with intrinsic metric in which geodesics are locally extendable, 
and which has Holder-continuous curvature as a metric space, is isometric with a C^2-Riemannian manifold."
Edit [by J.O'Rourke]. Following Willie's advice (in the comments), here is the MathSciNet link.