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In physics, and particularly in general relativity, we use the notion of manifold to describe space-time. In this way we have a space that locally looks like $\mathbb{R}^n$, a "flat" space.
Are there in mathematics not locally flat spaces? Assuming that the answer is yes, they have some specific name (like manifold) and they are actively studied? Is there a possible use in physics?
I think, you are confused by the terminology: A locally flat space usually means a smooth manifold equipped with a Riemannian (or pseudo-Riemannian) metric of zero curvature. (Pseudo) Riemannian manifolds of nonzero curvature do not have a special name for them. In general, it is best not to use the notions "space" or "looks like": You can say "homeomorphic" or "diffeomorphic" or "isometric", they all have different meanings. A "space" also means many different things in mathematics (topological space, vector space, metric space, etc.).
Of course, maybe you are genuinely interested in topological or metric spaces which are not topological manifolds or Riemannian manifolds. There are too many of these to list, open a textbook in "general" (or "point set") topology, and you will find a zoo of examples. Some of them share more and some share less properties with topological (or Riemannian) manifolds. For instance, there are integer homology manifolds which have local homology of manifolds, or Poincare duality spaces which satisfy a form of Poincare duality. Understanding which of these are homeomorphic (or homotopy-equivalent to) topological manifolds is an active area of research. Same on the metric side, Alexandrov spaces are metric spaces of curvature bounded below (this makes sense even if the space itself is not a manifold), they are fairly close to Riemannian manifolds and frequently appear in the study of such.
