Existence and Deformation of Geometric Stuctures on Manifolds

I.Lorentz结构。我们研究了常曲率洛伦兹流形，它具有因果消灭向量场。我们将消灭向量场的洛伦兹因果性质与常曲率的洛伦兹3流形联系起来，得到如下结果。定理a (a)不存在允许类空间杀伤向量场或类光杀伤向量场存在的常正曲率紧致洛伦兹3-流形。(b)如果紧化洛伦兹平面3流形存在类光的kill向量场，则它是一个基础流形。(c)如果紧致洛伦兹平面3-流形存在类空间的kill向量场，且不是欧几里得空间形式，则它是次流形而不是次流形。(d)具有类时杀戮向量场的常负曲率紧致洛伦兹3流形是标准空间形式。(e)常负曲率的紧化洛伦兹3流形上不存在类光的杀戮向量场。(f)如果紧致洛伦兹双曲3流形M…More允许类空间的杀戮向量场，且展开映射是内射的，则M是测地完备的，M的有限覆盖要么是齐次标准空间形式，要么是非标准空间形式。为了研究具有特征CR向量场的严格伪凸伪厄米流形的拓扑和几何性质，我们在严格伪凸伪厄米流形上找到了一个类曲率函数。众所周知，共形平坦流形包含一类常曲率的黎曼流形。相反，我们证明了非球面CR流形包含一类常曲率λ的标准伪厄米流形。此外，我们将对这些紧化流形进行分类。构造了一个常曲率λ的标准伪厄米特结构的模型空间（*，X）。这里*是一个有限维李群，X是*的齐次空间。那么X是一个连通的单连通的完全标准伪厄米流形，它是常数，*是一个（n+1）^2维的李氏群，由X对自身的伪厄米变换组成。然后我们展示了下面的均匀化。定理b设M是常数的标准伪厄米流形。那么M就可以在X上对*统一化。另外，如果M是紧的，则(i) λ是一个正常数当且仅当M同构于球面空间形式S^<2n+1>/F，其中F * U（n+1）。（ii） λ =0当且仅当M同构于Heisenberg基础流形N/GAMMA，其中GAMMA * N * U(N)。（iii） λ是一个负常数当且仅当M同构于一个洛伦兹标准空间形式H^^-^<, 2n>/GAMMA^^-（一个恒定负曲率的完全洛伦兹流形），其中GAMMA^^- * U^^-(n, 1)。cr结构变形，保形结构。存在自然同态psi: Diff(S^1， M) -> Out（GAMMA）。注意，Ker包含子群Diff^0（S^1， M）。令G= k /Diff^0（S^1， M）我们得到了如下的变形。定理c：设M为2n+1维的S^1不变球形cr流形。一个封闭的S^1不变共形平面n流形)。假设S^1半自由作用于M，使得轨道空间M^<**>是一个Kahler-Kleinian轨道D^<2n>-LAMBDA/GAMMA^<**>，具有非空边界。具有非空边界的Kleinian轨道D^<n-1>-LAMBDA/GAMMA^*)和H^2(GAMMA^<**>； Z)=0。如果pi_1(M)不可解，则(1)hol: SCR(U(1)， M) -> R(GAMMA^<**>， PU(n, 1))/PU(n, 1) X T^k是一个光纤与g同构的覆盖图。(2)hol: CO(SO(2)， M) -> R(GAMMA^<**>， SO(n-1,1)^0/SO(n-1,1)^0 X T^k是一个光纤与G. Less同构的覆盖图

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