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The Tangent Bundle
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algebra basis boundary bundle called Chapter choose classical clearly closed compact complete components compute Conclude condition connected Consequently consider constant contained continuous coordinate system COROLLARY corresponding cover curve define definition denote derivative diffeomorphism differentiable element equal equation equivalence exact example fact fibre finite follows function function f geodesic given gives hence homeomorphic identify identity implies integral isomorphism k-form left invariant Lemma length Lie group linear locally manifold matrix means neighborhood Notice obtain open set orientation positive Problem Proof Proposition prove result Riemannian metric satisfies sequence Show structure submanifold subset suffices Suppose takes tangent vector tensor Theorem true union unique vector field vector space write