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Unformatted text preview: 1.3. TANGENT VECTORS AND MAPPINGS 19 Theorem 1.3.2 Let n and r be two positive integers such that n > r. Let M be an n-dimensional manifold and N be an r-dimensional man- ifold. Let q N be a point in N such that the inverse image W = F- 1 ( q ) , is nonempty. Suppose that for each point p W the di ff er- ential F * of the map F is surjective, that is, has the maximal rank rank F * ( p ) = r . Then W is an ( n- r ) dimensional submanifold of M. Example. Morse Map. (Height function F : M R for trousers surface M in R 3 ). Let p M and v T p M be a tangent vector to M at p . Then F * : T p M R = T F ( p ) M is the projection of v to z-axis defined by F * ( v 1 , v 2 , v 3 ) = v 3 . Let p M and z = F ( p ). F- 1 ( z ) is an embedded submanifold if F * is onto, that is, T p M is not horizontal. If T p M is horizontal, then F * = (hence, not onto)....
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