locimthm_132sp17.pdf - Math 132 Topology II Smooth...

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Math 132 - Topology II: Smooth Manifolds. Spring 2017. Local Immersion Theorem George Melvin In this note we will state and prove the Local Immersion Theorem, following Guillemin- Pollack’s Differential Topology . Lemma (Linear algebra exercise) . Let A be a l × k matrix, where k l , and rank A = k . Then, there exists invertible k × k matrix P and an invertible l × l matrix Q such that I k 0 = QAP. Show that the existence of Q,P as above is equivalent to the following: if L A R k R l is the linear map L A ( x ) = Ax , then there exists isomorphisms T R k R k and S R l R l such that the matrix of S L A T R k R l with respect to the standard bases is I k 0 . Theorem (Local Immersion Theorem) . Let X R N , Y R M be smooth manifolds, f X Y a smooth map. Suppose that f is a local immersion at x X . Then, there exists local parameterisations R k V 1 U 1 X φ 1 ψ 1 R l V 2 U 2 Y φ 2 ψ 2 with x U 1 , and y = f ( x ) U 2 , such that ( ψ 1 f φ 1 )( x 1 ,...,x k ) = ( x
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