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Physical Mathematics by Kevin Cahill

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11

Tensors and local symmetries

11.1 Points and coordinates

A point on a curved surface or in a curved space also is a point in a higher-dimensional flat space called an embedding space. For instance, a point on a sphere also is a point in three-dimensional euclidean space and in four-dimensional space-time. One always can add extra dimensions, but it’s simpler to use as few as possible, three in the case of a sphere.

On a sufficiently small scale, any reasonably smooth space locally looks like n-dimensional euclidean space. Such a space is called a manifold. Incidentally, according to Whitney’s embedding theorem, every n-dimensional connected, smooth manifold can be embedded in 2n-dimensional euclidean space 2n. So the embedding space for such ...

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