An Introduction to Differentiable Manifolds and Riemannian GeometryAcademic Press, 1986/04/21 - 429 ページ An Introduction to Differentiable Manifolds and Riemannian Geometry |
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1 ページ
... sections deal primarily with notational matters and the relation between Euclidean space, its model R", and real vector spaces. In Section 3 a precise definition of topological manifolds is given, and in the remaining sections this ...
... sections deal primarily with notational matters and the relation between Euclidean space, its model R", and real vector spaces. In Section 3 a precise definition of topological manifolds is given, and in the remaining sections this ...
19 ページ
... Section VIII.6.) The existence of such spaces was apparently already known to Gauss, who kept his knowledge to himself-presumably for fear of the profoundly disturbing philosophical and religious consequences such a discovery would ...
... Section VIII.6.) The existence of such spaces was apparently already known to Gauss, who kept his knowledge to himself-presumably for fear of the profoundly disturbing philosophical and religious consequences such a discovery would ...
20 ページ
... Section 4 in which T.(R") is defined in a way that admits generalization. Section 5 reviews the notion of vector field in R". Section 6 gives a detailed proof of the inverse function theorem. This theorem with its corollaries ...
... Section 4 in which T.(R") is defined in a way that admits generalization. Section 5 reviews the notion of vector field in R". Section 6 gives a detailed proof of the inverse function theorem. This theorem with its corollaries ...
21 ページ
... section that fis a function on an open set U c R". At each a e U, the partial derivative (ös/6x'), off with respect to x' is, of course, the following limit, if it exists: 1 j * \ . . 1 j rt [...] – lims" or " + h, ..., a”) f(a'..... a ...
... section that fis a function on an open set U c R". At each a e U, the partial derivative (ös/6x'), off with respect to x' is, of course, the following limit, if it exists: 1 j * \ . . 1 j rt [...] – lims" or " + h, ..., a”) f(a'..... a ...
22 ページ
... section and the following statements (1.1)–(1.3), whose proofs we leave as exercises, will clarify these concepts. (1.1) If f is differentiable at a, then it is continuous at a and all the partial derivatives (Öfföx'), exist. Moreover ...
... section and the following statements (1.1)–(1.3), whose proofs we leave as exercises, will clarify these concepts. (1.1) If f is differentiable at a, then it is continuous at a and all the partial derivatives (Öfföx'), exist. Moreover ...
目次
1 | |
20 | |
51 | |
Chapter IV Vector Fields on a Manifold | 106 |
Chapter V Tensors and Tensor Fields on Manifolds | 176 |
Chapter VI Integration on Manifolds | 229 |
Chapter VII Differentiation on Riemannian Manifolds | 297 |
Chapter VIII Curvature | 365 |
References | 417 |
Index | 423 |
多く使われている語句
algebra basis bi-invariant C*-vector field compact completes the proof components connected coordinate frames coordinate neighborhood Corollary corresponding countable covariant tensor covering curve p(t defined definition denote derivative diffeomorphism differentiable manifold dimension domain of integration element equations equivalent Euclidean space example Exercise exists fact finite fixed point formula functions geodesic geometry given Gl(n hence homeomorphism homotopy identity imbedding inner product integral curve isometry isomorphism Lemma Let F Lie group G linear map mapping F matrix notation obtain one-parameter subgroup one-to-one open set open subset oriented orthogonal orthonormal parameter plane properly discontinuously properties prove rank real numbers regular submanifold Remark Riemannian manifold Riemannian metric Section Show structure submanifold subspace suppose surface symmetric tangent space tangent vector tensor field Theorem Let topology uniquely determined vector field vector space zero