The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry"， an English version can be found here.
When we talk about the physical "space", we model it in terms of manifold, and usually to do some geometry one starts with definitions of Riemannian manifolds, on which one can talk about length, angle, area, volume, etc. This, of course, models our intuition of the space we are in pretty well and it seems rather reasonable to believe the space we are in carries a Riemannian metric. However, Riemann predates all the physics break through in the 20th century, which includes general relativity and quantum mechanics. The former is like a major triumph of Riemannian geometry but I don't know if the latter has anything to do with it.
My question is the following: with these physics breakthrough,
has the hypotheses which lie at the bases of geometry been reconsidered?
In other words, Riemannian geometry seems to model geometry on the scale of daily-life objects (i.e. not too small or too large) pretty well, and it seems to model the space pretty well on a large scale, but what about the microscopic level? Is it still a good model of the physical "space" on that scale?
In case people want to label this as "not a real question", let me quote Riemann's own question at the end of his paper ("plan of inquiry", copied from the English version):
III. Application to Space.
§ 1. System of facts which suffice to determine the measure-relations of space assumed in geometry.
§ 2. How far is the validity of these empirical determinations probable beyond the limits of observation towards the infinitely great?
§ 3. How far towards the infinitely small? Connection of this question with the interpretation of nature.
Question 2 and 3 on his list is the thing I want to ask here. (I don't understand what 1 means here.) .