The metric on a manifoldM that we considered so far comes from Smirnov’s theorem (see Theorem 2.1.5), and also from the fact that M is embeddable in some Euclidean space. Of these, the second metric is more important for us, because the first metric has nothing to do with smooth structure, it may be obtained for any nice topological manifold. In this chapter we shall obtain another metric on M which gives the same topology of M as a manifold.
KeywordsRIEMANNIAN Manifold Tangent Vector Open Neighbourhood Smooth Curve Smooth Vector
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