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."

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.