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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24 ページ
... formula. (1.6) Corollary Let U and g be as in Theorem 1.5. If |&g/6x' | < K on U, i = 1, 2, ..., n, then for any xe U, we have |g(x) = g(a) < KVn ||x – al. Proof Taking absolute values in the formula of Theorem 1.5 and using the Schwarz ...
... formula. (1.6) Corollary Let U and g be as in Theorem 1.5. If |&g/6x' | < K on U, i = 1, 2, ..., n, then for any xe U, we have |g(x) = g(a) < KVn ||x – al. Proof Taking absolute values in the formula of Theorem 1.5 and using the Schwarz ...
28 ページ
... formula gives DH(x) as a composite of functions which are at least continuous so that it must be continuous. This is equivalent to its entries being continuous which means that the coordinate functions of H, and thus H itself, are of ...
... formula gives DH(x) as a composite of functions which are at least continuous so that it must be continuous. This is equivalent to its entries being continuous which means that the coordinate functions of H, and thus H itself, are of ...
43 ページ
... formula for the derivatives of FT" at y: DF'(y) = (DF(x) , the term on the right denoting the inverse matrix to DF(x). This is one of the two basic theorems of analysis on which all of the theory in this book depends; the other is the ...
... formula for the derivatives of FT" at y: DF'(y) = (DF(x) , the term on the right denoting the inverse matrix to DF(x). This is one of the two basic theorems of analysis on which all of the theory in this book depends; the other is the ...
46 ページ
... formula above for them shows these entries to be given by composition of functions of class C" or greater and hence to be of class C" at least. This implies FT" is of class C***; so by induction FT' is of class C". This completes the ...
... formula above for them shows these entries to be given by composition of functions of class C" or greater and hence to be of class C" at least. This implies FT" is of class C***; so by induction FT' is of class C". This completes the ...
48 ページ
... requires that the rank be at most k. We compute D(F 9 GT") from the formula above for F o GT', giving D(F 9 GT')(x) = This is valid on U1, where F c GT' is. 48 II F U N CTI O N S OF S E V E R A L VARIA B L E S A N D M A PPINGS.
... requires that the rank be at most k. We compute D(F 9 GT") from the formula above for F o GT', giving D(F 9 GT')(x) = This is valid on U1, where F c GT' is. 48 II F U N CTI O N S OF S E V E R A L VARIA B L E S A N D M A PPINGS.
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