How are manifolds built from overlapping coordinate patches?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In differential geometry, a manifold is constructed by covering a topological space with a collection of open sets, each associated with a local coordinate system, often called a "patch" or "chart."

Because a manifold may have complex global properties that prevent a single coordinate system from describing the entire space, these patches are designed to overlap. Within these overlapping regions, the manifold is equipped with transition functions—or coordinate transformations—that relate the coordinates of one patch to those of another. These transformations must belong to a specified class (such as being differentiable) to ensure that the manifold maintains a consistent structure throughout. By gluing these patches together via these compatible transformations, mathematicians can model spaces that locally resemble Euclidean space while allowing for diverse global geometries.1