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The Tangent Bundle
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basis called Chapter choose classical clearly closed Cm function Cm manifold Cm structure compact support components compute condition connected consider continuous coordinate system coordinate system x,U COROLLARY countable covariant critical point define denote diagram diffeomorphism differential equations disjoint union element equivalence classes exact example fact fibre finite formula function f functor geodesic hence Hk(M homeomorphic homotopic identity implies inner product integral curve integral manifold isomorphism k-form l-form left invariant Lemma Let f Lie algebra Lie group linear transformation map f matrix metric space Mobius strip n-dimensional n-manifold neighborhood notation obtain one-one open set open subsets order isomorphic orientation preserving oriented manifold Problem Proposition prove Qk(V Riemannian metric satisfies sequence Show singular submanifold subspace Suppose tangent bundle tangent vector tensor field topology trivial unique vector bundle vector field vector space well-ordered set