# Not locally flat space

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?