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Chapter III Differentiable Manifolds and Submanifolds
Chapter IV Vector Fields on a Manifold
Chapter V Tensors and Tensor Fields on Manifolds
Chapter VI Integration on Manifolds
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action algebra arbitrary basis bi-invariant bilinear form compact completes the proof components connected coordinate frames coordinate neighborhood Corollary corresponding countable covering curve p(t deﬁned deﬁnition denote derivative diffeomorphism differentiable manifold dimension domain of integration element equations equivalent Euclidean space example Exercise exists fact ﬁnd ﬁnite ﬁrst ﬁxed point formula geodesic geometry give given Gl(n hence homeomorphism homotopy identiﬁed identity imbedding implies inner product integral curve isometry isomorphism Lemma Let F Let G Lie group G linear map mapping F matrix nonsingular notation obtain one-parameter subgroup one-to-one open set open subset oriented orthogonal orthonormal parameter plane positive deﬁnite properly discontinuously properties prove rank real numbers regular submanifold Remark Riemannian manifold Riemannian metric satisﬁes Section Show structure submanifold subspace suppose surface symmetric tangent space tangent vector Theorem Let topology uniquely determined vector ﬁeld vector space zero