Existence and Deformation of Geometric Stuctures on Manifolds

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。另外，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^&lt;**&gt;是一个具有非空边界的Kahler-Kleinian orbifold D^<2n>-LAMBDA/Gamma^&lt;**&gt;（相应地，是一个具有非空边界的Kleinian orbifold D^<n-1>-LAMBDA/Gamma^*），且H ^2（Gamma^&lt;**&gt; ; Z）=0。若pi_1（M）不是虚可解的，则（1）hol：SCR（U（1），M）-&gt; R（&lt;$**&gt;，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）-&gt; R（Gamma^**&gt;，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

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