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arbitrary basis bundle map called chapter closed cohomology compact compactly supported compute connected continuous map coordinate chart coordinate representation Corollary countable covector covering map defined definition denote derivative diffeomorphism differential forms dimension domain embedded submanifold equivalent Euclidean space Example Exercise exists Figure finite function f given GL(n global homotopy implies induced injective integral curve integral manifold inverse isomorphism left-invariant Lemma Let G Lie algebra Lie group Lie group homomorphism Lie subgroup linear map manifold with boundary map F matrix measure zero n-manifold neighborhood open set open subset oriented Problem Proof Proposition prove pushforward Rham Riemannian manifold satisfies smooth chart smooth coordinate smooth covering map smooth function smooth manifold smooth map smooth structure smooth vector field submersion subspace Suppose surjective symplectic tangent space tangent vector tensor field theorem topological manifold topological space trivial U C M vector bundle
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