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Preliminaries on Manifolds 1 Manifolds
2 Differentiable Mappings and Submanifolds
3 Tangent Spaces
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1:1 immersion apply assume canonical chart choose coordinates codim codimension compact compute coordinate functions Corollary countable critical point curve defined Definition deformation denote dense df)p diffeomorphism differentiable dimension equations equivalence class Exercise exists a nbhd fact fiber given Hence homeomorphism immersed submanifold immersion with normal induced infinite order isomorphism Lemma Let f Let G Letf Lie group linear map locally Malgrange Preparation Theorem measure zero metric module Morse functions multijet nbhd ofO nondegenerate normal crossings Note obvious projection open nbhd open set open subset parameter group polynomial proof of Proposition Proof of Theorem prove ring satisfying singularities smooth function smooth manifolds smooth mapping stable mappings submanifold germs subspace Suppose tangent that/is Thom Transversality Theorem Thom-Boardman topology Transversality Theorem trivial tubular nbhd vector bundle vector field vector space well-defined Whitney xu x2