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We map each point and a small enough neighbourhood of it to , so we can talk about the manifold points in terms of coordinates.
Does not require any further structure besides a consistent topological map. Notably, does not require metric nor an addition operation to make a vector space.
Manifolds are cool. Especially differentiable manifolds which we can do calculus on.
A notable example of a Non-Euclidean geometry manifold is the space of generalized coordinates of a Lagrangian. For example, in a problem such as the double pendulum, some of those generalized coordinates could be angles, which wrap around and thus are not euclidean.