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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6 ページ
... open subset U" of R", n fixed. We say that a space with this property is locally Euclidean of dimension n, and in order to stay as close as possible to Euclidean spaces, we will consider spaces called manifolds, defined as follows. (3.1) ...
... open subset U" of R", n fixed. We say that a space with this property is locally Euclidean of dimension n, and in order to stay as close as possible to Euclidean spaces, we will consider spaces called manifolds, defined as follows. (3.1) ...
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 ball B,(x) of radius e in R". We denote this homeomorphism by q), and we suppose q.(p) = x. Then it is clear ... set U of E” is homeomorphic to some open set U" of E" with m + n'? The answer is no, but the proof is difficult and ...
... open ball B,(x) of radius e in R". We denote this homeomorphism by q), and we suppose q.(p) = x. Then it is clear ... set U of E” is homeomorphic to some open set U" of E" with m + n'? The answer is no, but the proof is difficult and ...
10 ページ
... open subset U" of R". Letting q : U → U" be this correspondence, we call the pair U, q, a coordinate neighborhood and the numbers x*(q), ..., x"(a), given by p(q) = (x'(q), ..., x"(q)), the coordinates of qe M. We have assumed that ...
... open subset U" of R". Letting q : U → U" be this correspondence, we call the pair U, q, a coordinate neighborhood and the numbers x*(q), ..., x"(a), given by p(q) = (x'(q), ..., x"(q)), the coordinates of qe M. We have assumed that ...
11 ページ
... open sets has a locally finite refinement; more precisely, there is a ... subset U" of R" except the set of points (x', ..., x" ', 0), which obviously ... open sets which has the property that each p e M is contained in an open set U with ...
... open sets has a locally finite refinement; more precisely, there is a ... subset U" of R" except the set of points (x', ..., x" ', 0), which obviously ... open sets which has the property that each p e M is contained in an open set U with ...
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