In the nineteenth century, there was only one known geometry of space, which Euclid had described two millennia before. Synthetic geometers were picking at the Euclid's parallel postulate, looking for ways to deduce it from his other postulates, when they realized that perfectly reasonable non-Euclidean geometries do actually exist. These are the elliptic and hyperbolic geometries of Gauss, Lobachevsky, Bolyai and so forth. Each is characterized by a single length scale, the curvature radius. Analytic geometers, including Gauss again and especially Riemann, then realized that actually there were infinitely many different geometries whose curvatures vary from one point to another. Curvature, and spatial geometry in general, is like a physical field: it can vary from place to place.
Riemann was the first to point out the immediate implication of his realization of the infinite variety of geometries space might have. Namely, the geometry of the physical space we live in is a question _for_experiment_. We may no longer blithely assume it to be Euclidean for simplicity, as Newton did. Einstein forty years later or so identified the physical effect of spatial curvature with the gravitational field.
I would argue that Tesla's assertion that space has no properties is itself hopelessly metaphysical. If one fixes spatial geometry out of aesthetics, and denies from the beginning that it can be dynamical, how can one possibly probe the question scientifically? Plus, general relativity works so very well to describe things we actually observe. Curved spacetime, to the best of our current scientific knowledge, is simply a fact.