Manifold-related Question I'm a little new to this area of mathematics. I understand that manifolds are topological spaces that locally resemble Euclidean space near each point. I also understand that manifolds are very common.
In reference to computer science, the manifold hypothesis is the idea that high-dimensional natural data lie on low-dimensional manifolds.
Are manifold so commonly found as to suggest this idea? There are certainly spaces that are not manifolds. Are these found very infrequently in nature? This hypothesis would suggest anything that we can record would lie on a manifold.
Also is there a term for a topological space that locally resembles a non-euclidean space (like a hyperbolic, spherical, or projective space) near each point? For example, can't spaces such as these still potentially be useful? What about spaces that do not locally resemble euclidean spaces in only a few points? Are there terms for these spaces? Are these spaces not commonly found in nature, in any practical setting, or in mathematical problems?
 A: Manifolds are extremely useful ideas because most reasonable 'shapes' with some idea of 'homogeneity' are often manifolds. What I mean by homogeneity is that the dimension of the data doesn't change as you move from one point to another. What makes them so useful is that often you can model them by equations (if only locally). Since manifolds are topological objects, you can perturb them (bend, stretch, wiggle etc) without affecting what they are. This makes them suited to data because it gives them great resistance to noise. As a result, even if your data doesn't look like exactly a sphere, it may still be equivalent to one, allowing you to deduce interesting facts. 
Even though there is a large variety of shapes out there, even if globally such shapes aren't manifolds, often small sections of them can be thought of as manifolds. For example, a sphere with a line through both poles isn't a manifold, but away from the poles, you can still see that they look locally Euclidean. We have a classifications of manifolds and so once you identify the manifold, you might be able to classify it, and hence learn some important facts about the data you're looking at.
When we say a space resembles Euclidean space, we don't mean that it has a Euclidean metric, we mean it has the same topology (open-set structure) as Euclidean space. As a result, you can locally look like hyperbolic space or spherical space and still be a manifold because you also look like Euclidean space.  A prime example is that projective spaces are manifolds (indeed $\mathbb{R}P^1$ is the circle and $\mathbb{C}P^1$ is the sphere, while others are harder to visualize because they don't fit nicely in 3-dimensions), and that the sphere (which is a spherical space) is also a manifold.
