Today I was reading about manifolds and I found this illustrative image about them.

“[…] To find global information, the being would have the walk around both surfaces and be very careful to check angles and distances. If the English: Created by Jason Hise with Maya and M... being were nearsighted and could not check distances and angles, then its examination of the local vicinity, or neighborhood, would fail to detect any local distortions due to the curvature of the surface. It might then conclude that the surrounding space was Euclidean, or flat, in nature. This is what we mean when we say that a differential manifold looks locally like $latex \boldsymbol{R}^p$.

Our two-dimensional being might well consult an atlas to find its way around the geography of these two-dimensional worlds. We are used to seeing the surface of the Earth displayed in an atlas. However, we know that because the Earth is a sphere, we cannot get all points plotted on a single page or chart without tearing the picture and destroying the natural continuity between Illustration of spherical geometry where the a...neighboring points. Just as a portion of the surface of the Earth can be described by a chart, so a portion of a differential manifold is described by a chart, here understood in a mathematical sense. Just as a single page of an atlas cannot cover the  entire surface of the Earth without disrupting continuity, so a single chart cannot usually cover the entire region of a differential manifold. The mathematical charts used to describe a manifold must also be collected together into an atlas. Of course, such charts, if they cover the manifold, will overlap in places. Thus they are not arbitrarily related, but must, in a certain sense, describe the same smoothness on the region of overlap. If the same town appeared on two different pages of a geographical atlas, we would expect the local descriptions on the two pages to be compatible, even if not identical. […]”

From the book: “The statistical theory of shape” by Christopher G. Small


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