Examples 5.2 (c) and (d)
(c) If M is any manifold embedded in E_{s}, then we have seen above that M inherits the structure of a Riemannian metric from a given inner product on E_{s}. In particular, if M is any 3dimensional manifold embedded in E_{4} 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 twosphere M = S^{2}, with radius r, using polar coordinates x^{1} = , x^{2} = . To find the coordinates of g_{**} we need to calculate the inner product of the basis vectors /x^{1}, /x^{2} in the ambient space E_{s}. We saw in Section 3 that the ambient coordinates of /x^{i} are given by
j th coordinate  =  x^{i} 
where
Thus,
x^{1}  =  r(cos(x^{1})cos(x^{2}), cos(x^{1})sin(x^{2}), sin(x^{1})) 
x^{2}  =  r(sin(x^{1})sin(x^{2}), sin(x^{1})cos(x^{2}), 0) 
This gives
so that
g_{**}  = 

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