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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7 ページ
... open set M of R* between the curves C and C. (b) The manifold is the open subset of R* obtained by removing the knots. when n = 2 or 3 these examples can be rather complicated and certainly not equivalent to Euclidean space in general ...
... open set M of R* between the curves C and C. (b) The manifold is the open subset of R* obtained by removing the knots. when n = 2 or 3 these examples can be rather complicated and certainly not equivalent to Euclidean space in general ...
9 ページ
... open sets, we may now suppose that it has a countable base of relatively compact open sets {V}; obviously M = U V. Normality follows from Lindelöf's theorem and metrizability is then a consequence of the Urysohn metrization theorem (see ...
... open sets, we may now suppose that it has a countable base of relatively compact open sets {V}; obviously M = U V. Normality follows from Lindelöf's theorem and metrizability is then a consequence of the Urysohn metrization theorem (see ...
11 ページ
... open sets has a locally finite refinement; more precisely, there is a covering (V, which (i) refines (U.} in the sense that each V, c U, for some o, and which (ii) is locally finite, that is, each pe M has a neighborhood W which ...
... open sets has a locally finite refinement; more precisely, there is a covering (V, which (i) refines (U.} in the sense that each V, c U, for some o, and which (ii) is locally finite, that is, each pe M has a neighborhood W which ...
16 ページ
... collection U of lines is open if it is the set of all lines through 0 which meet a given open set U. This example may be generalized as follows: Let M be the set of all r-planes through the origin in R", where n and r are fixed; for example ...
... collection U of lines is open if it is the set of all lines through 0 which meet a given open set U. This example may be generalized as follows: Let M be the set of all r-planes through the origin in R", where n and r are fixed; for example ...
21 ページ
... 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'..... a'). If &f/6x' is defined ...
... 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'..... a'). If &f/6x' is defined ...
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