Examples 5.2 (c) and (d)

(c) If M is any manifold embedded in Es, then we have seen above that M inherits the structure of a Riemannian metric from a given inner product on Es. In particular, if M is any 3-dimensional manifold embedded in E4 with the metric shown above, then M inherits such a inner product.

(d) As a particular example of (c), let us calculate the metric of the two-sphere M = S2, with radius r, using polar coordinates x1 = , x2 = . To find the coordinates of g** we need to calculate the inner product of the basis vectors /x1, /x2 in the ambient space Es. We saw in Section 3 that the ambient coordinates of /xi are given by

where

Thus,

This gives

so that

 

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