## Differentiable manifolds: Notes Math 354 A, Spring 1952 |

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affine connection algebra bilinear called compact components coordinate neighborhood coordinate system Corollary corresponding covariant curvature tensor definition denote derivation determined differentiable manifold differentiable structure differential forms differential system dimension dual bases element Euclidean exists exterior differential exterior polynomials fact field of type finite number following theorem form of degree formula function f functions of class geodesic geometrical given group of holonomy Hence homeomorphism hyperbolic implies induces integral interior product isomorphism Lemma linear differential forms linear map linear subspace linearly independent map f map g Moreover multiplication normal coordinates one-one open subset parametrized curve partition of unity permutation properties prove regularly imbedded respectively Riemann manifold satisfying the conditions scalar function scalar product segment spherical structure of class Suppose tangent space tangent vectors tensor field tensor product Theorem 2.1 tion topology torsion un(q vector field vector space whore zero