CoInvestigator(Kenkyūbuntansha) 
前橋 敏之 熊本大学, 理学部, 教授 (90032804)
岡 幸正 熊本大学, 理学部, 助教授 (50089140)
梅村 浩 熊本大学, 理学部, 教授 (40022678)
吉田 知行 熊本大学, 理学部, 教授 (30002265)
OKA Yukimasa Kumamoto Universisty, Associate Professor
YOSHIDA Tomoyuki Kumamoto Universisty, Professor

Budget Amount *help 
¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1993 : ¥300,000 (Direct Cost : ¥300,000)
Fiscal Year 1992 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1991 : ¥1,000,000 (Direct Cost : ¥1,000,000)

Research Abstract 
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 3manifolds of constant curvature to obtain the following. Theorem A. (a) There exists no compacat Lorentz 3manifold of constant positive curvature which admits a spacelike Killing vector field or a lightlike Killing vector field. (b) If a compact Lorentz flat 3manifold admits a lighlike Killing vector field then it is an infranilmanifold. (c) If a compact Lorentz flat 3manifold 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 3manifold 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 3manifold of constant negative curvature. (f) If a compact Lorentz hyperbolic 3manifold M
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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 PseudoHermitian Structure. We have found a curvaturelike function LAMBDA on a strictly pseudoconvex pseudoHermitian 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 pseudoHermitian manifolds of constant curvature LAMBDA.Moreover we shall classify those compact manifolds. We construct a model space (*, X) of standard pseudoHermitian 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 pseudoHermitian manifold of constant LAMBDA and * is an (n+1)^2dimensional Liegroup consisting of pseudoHermitian transformations of X onto itself. Then we have shown the following uniformization. Theorem B.Let M be a standard pseudoHermitian 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 CRstructures, 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^1invariant spherical CRmanifold of dimension 2n+1(resp.a closed S^1invariant conformally flat nmanifold). Suppose that S^1 acts semifreely on M such that orbit space M^<**> is a KahlerKleinian orbifold D^<2n>LAMBDA/GAMMA^<**> with nonempty boundary (resp.a Kleinian orbifold D^<n1>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(n1,1)^0/SO(n1,1)^0 X T^k is a covering map whose fiber is isomorphic to G. Less
