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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... lemmas. (4.2) Lemma Let D be an arbitrary element of Ø(a). Then D is zero on any function fe C*(a) which is constant in a neighborhood of a. Proof Because the map D is linear, it is enough to show that if 1 denotes the constant function ...
... lemmas. (4.2) Lemma Let D be an arbitrary element of Ø(a). Then D is zero on any function fe C*(a) which is constant in a neighborhood of a. Proof Because the map D is linear, it is enough to show that if 1 denotes the constant function ...
35 ページ
... Lemma 42, D(f(a) = 0 and D(x' – a') = Dx = x'; and by Lemma 4.3, g'(a) = (6fföx'),. Therefore Df = }}_1 x'(Öfföx'), = X; f. Since f is an arbitrary element of C*(a), we have D = X. This completes the proof. | Theorem 4.1 allows us to ...
... Lemma 42, D(f(a) = 0 and D(x' – a') = Dx = x'; and by Lemma 4.3, g'(a) = (6fföx'),. Therefore Df = }}_1 x'(Öfföx'), = X; f. Since f is an arbitrary element of C*(a), we have D = X. This completes the proof. | Theorem 4.1 allows us to ...
42 ページ
... lemma, whose proof we leave as an exercise, which gives the transitivity property; symmetry and reflexivity are part of the definition. (6.3) Lemma Let U, V, W be open subsets of R", F: U → V, G: V → W mappings onto, and H = Go F: U ...
... lemma, whose proof we leave as an exercise, which gives the transitivity property; symmetry and reflexivity are part of the definition. (6.3) Lemma Let U, V, W be open subsets of R", F: U → V, G: V → W mappings onto, and H = Go F: U ...
44 ページ
... Lemma 6.3 combined with the use of Examples 6.1 and 6.2. Next we define the mapping G on the same domain by G(x) = x – F(x). Then, obviously, G(0) = 0 and DG(0) = 0. (Note: In the last equation the right-hand side is the 0 matrix.) (ii) ...
... Lemma 6.3 combined with the use of Examples 6.1 and 6.2. Next we define the mapping G on the same domain by G(x) = x – F(x). Then, obviously, G(0) = 0 and DG(0) = 0. (Note: In the last equation the right-hand side is the 0 matrix.) (ii) ...
46 ページ
... Lemma 6.3. Compute the Jacobian matrix for Examples 6.1 and 6.2. 4. Show that for transformations on a compact subset K of R" which satisfy the conditions of the contracting mapping theorem except that 0 < A * 1 (we allow A = 1), there ...
... Lemma 6.3. Compute the Jacobian matrix for Examples 6.1 and 6.2. 4. Show that for transformations on a compact subset K of R" which satisfy the conditions of the contracting mapping theorem except that 0 < A * 1 (we allow A = 1), there ...
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