Existence and Deformation of Geometric Stuctures on Manifolds

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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