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Existence and Deformation of Geometric Stuctures on Manifolds

流形上几何结构的存在与变形

基本信息

批准号：
03640079
负责人：
KAMISHIAM Yoshinobu
金额：
$ 1.22万
依托单位：
Department of mathematics, Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1993
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03640079/
关键词：
Contact Structure Geometric structure Characteristic CR Vector Field Causality Lorentz Structure Pseudo-Hermitian Structure Deformation Curvaturelike Function LAMBDA Pseudo-Hermitian Structure Curvaturelike function Uniformization Sa

项目摘要

I.Lorentz Structure. We have sutdied Lorentz manifolds of constant curvature which admit causal Killing vector fielda. We relate Lorentz causal character of Killing vector fields to Lorentz 3-manifolds of constant curvature to obtain the following.Theorem A.(a) There exists no compacat Lorentz 3-manifold of constant positive curvature which admits a spacelike Killing vector field or a lightlike Killing vector field.(b) If a compact Lorentz flat 3-manifold admits a lighlike Killing vector field then it is an infranilmanifold.(c) If a compact Lorentz flat 3-manifold admits a spacelike Killing vector field and is not a euclidean space form, then it is an infrasolvmanifold but not an infranilmanifold.(d) A compact Lorentz 3-manifold of constant negative curvature admitting a timelike Killing vector field is a stnadard space form.(e) There exists no lightlike Killing vector field on a compact Lorentz 3-manifold of constant negative curvature.(f) If a compact Lorentz hyperbolic 3-manifold M … More admits a spacelike Killing vector field and the developing map is injective, then M is geodesically complete and a finite covering of M is either a homogeneous standard space form or a nonstandard space form.II.Standard Pseudo-Hermitian Structure. We have found a curvaturelike function LAMBDA on a strictly pseudoconvex pseudo-Hermitian manifold in order to study topological and geometric properties of those manifolds which admit characteristic CR vector fields. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. In contrast, we proved that aspherical CR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature LAMBDA.Moreover we shall classify those compact manifolds. We construct a model space (*, X) of standard pseudo-Hermitian structure of constant curvature LAMBDA.Here * is a finite dimensional Lie group and X is a homogeneous space from *. Then X is a connected simly connected complete standard pseudo-Hermitian manifold of constant LAMBDA and * is an (n+1)^2-dimensional Liegroup consisting of pseudo-Hermitian transformations of X onto itself. Then we have shown the following uniformization.Theorem B.Let M be a standard pseudo-Hermitian manifold of constant LAMBDA.Then M can be uniformized over X with respect to *. In addition, if M is compact, then(i) LAMBDA is a positive constant if and only if M is isomorphic to the spherical space form S^<2n+1>/F where F * U(n+1).(ii) LAMBDA=0 if and only if M is isomorphic to a Heisenberg infranilmanifold N/GAMMA, where GAMMA * N * U(n).(iii) LAMBDA is a negative constant if and only if M is isomorphic to a Lorentz stnadard space form H^^-^<, 2n>/GAMMA^^- (a complete Lorentz manifold of constant negative curvature), where GAMMA^^- * U^^-(n, 1).III.Deformation of CR-structures, Conformal structures. There is the natural homomorphism psi : Diff(S^1, M) -> Out(GAMMA). Note that Ker psi contains the subgroup Diff^0(S^1, M). Put G=Ker psi/Diff^0(S^1, M). We have obtained the following deformation.Theorem C.Let M be a closed S^1-invariant spherical CR-manifold of dimension 2n+1(resp.a closed S^1-invariant conformally flat n-manifold). Suppose that S^1 acts semifreely on M such that orbit space M^<**> is a Kahler-Kleinian orbifold D^<2n>-LAMBDA/GAMMA^<**> with nonempty boundary (resp.a Kleinian orbifold D^<n-1>-LAMBDA/GAMMA^* with nonempty boundary) and with H^2(GAMMA^<**> ; Z)=0. If pi_1(M) is not virtually solvable, then(1) hol : SCR(U(1), M) -> R(GAMMA^<**>, PU(n, 1))/PU(n, 1) X T^k is a covering map whose fiber is isomorphic to G.(2) hol : CO(SO(2), M) -> R(GAMMA^<**>, SO(n-1,1)^0/SO(n-1,1)^0 X T^k is a covering map whose fiber is isomorphic to G. Less

一、洛伦兹结构本文研究了常曲率Lorentz流形中的因果Killing向量场。我们将Killing向量场的Lorentz因果特征与常曲率的Lorentz三维流形联系起来，得到如下定理A。(a)在具有常正曲率的Lorentz 3-流形上不存在允许类空Killing向量场或类光Killing向量场的流形。(b)如果紧致Lorentz平坦3-流形存在类光Killing向量场，则它是一个infranilmanifold。(c)如果紧致Lorentz平坦3-流形存在类空Killing向量场且不是欧氏空间形式，则它是下解流形而不是下流形。(d)具有常负曲率的紧致Lorentz三维流形是一种标准空间形式，它允许类时Killing向量场存在。(e)在常负曲率的紧致Lorentz三维流形上不存在类光Killing向量场。(f)如果紧Lorentz双曲三维流形M ...更多信息若M是测地线完备的，且M的有限覆盖是齐次标准空间形式或非标准空间形式，则M是测地线完备的.II.标准伪厄米特结构.为了研究具有特征CR向量场的流形的拓扑和几何性质，我们在严格伪凸伪Hermitian流形上找到了一个类曲率函数LAMBDA.众所周知，共形平坦流形包含一类常曲率黎曼流形。与此相反，我们证明了非球面CR流形包含一类常曲率LAMBDA的标准伪厄米流形，并对这些紧致流形进行了分类。构造了一个具有常曲率LAMBDA的标准伪厄米特结构的模型空间（*，X），这里 * 是有限维李群，X是由 * 构成的齐次空间.则X是一个连通的、简单连通的、完备的、具有常数LAMBDA的标准伪厄米特流形，* 是一个由X到自身的伪厄米特变换构成的（n+1）^2维李群。定理B.设M是一个标准的具有常数LAMBDA的伪厄米特流形，则M可以关于 * 在X上一致化。此外，如果M是紧的，则（i）LAMBDA是正常数当且仅当M同构于球面空间形式S^<2n+1>/F，其中F * U（n+1）。(ii)LAMBDA=0当且仅当M同构于一个Heisenberg infranilmanifold N/<$MA，其中<$MA * N * U（n）。(iii)LAMBDA是一个负常数当且仅当M同构于一个Lorentz标准空间形式H^^-U ^^（n，1），其中Gamma^^-U^^-（n，1）。III. CR-结构的变形，共形结构。存在自然同态psi：Diff（S^1，M）-> Out（Gamma）。注意Ker psi包含子群Diff^0（S^1，M）。设G=Ker psi/Diff^0（S^1，M）。定理C.设M是2n+1维的闭S^1不变球面CR-流形（或闭S^1不变共形平坦n-流形）。设S^1半自由地作用在M上，使得轨道空间M^<**>是一个具有非空边界的Kahler-Kleinian orbifold D^<2n>-LAMBDA/Gamma^<**>（相应地，是一个具有非空边界的Kleinian orbifold D^<n-1>-LAMBDA/Gamma^*），且H ^2（Gamma^<**> ; Z）=0。若pi_1（M）不是虚可解的，则（1）hol：SCR（U（1），M）-> R（<$**>，PU（n，1））/PU（n，1）XT ^k是一个覆盖映射，其纤维同构于G. (2)H：CO（SO（2），M）-> R（Gamma^**>，SO（n-1，1）^0/SO（n-1，1）^0 × T^k是一个覆盖映射，其纤维同构于G。少