Topological method in Differential Geometry and Conformal theory

微分几何和共形理论中的拓扑方法

• 批准号: 09640121
• 负责人: KAMISHIMA Yoshinobu
• 金额: $ 1.86万
• 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 Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640121/
• 关键词: Locally conformal Kahler structure; Integrability; Global deformation; Pseudo-quaternionic structure; Superrigidity; Complex Surfaces; Quaternionic Hyperbolic geometry; Sasakian Structure; Carnot-Caratheodory; 剛耐性; spherical pseudo-quaternion

Throughout the academic year 1997 to 1998, we have studied a geometric structure on complex manifolds which fits into the framework of conformal geometry.Review 1. S.Bochner introduced the basic curvature tensor on Kahier manifolds, called Bochner curvature tensor. This tensor can be also derived from the Weyl's principle on unitary group representation theory. When Hermitian manifolds are taken into account in the framework of conformal geometry, Tricerri and Vanhecke have defined a generalized Bochner curvature tensor on Hermitian manifolds. This curvature tensor coincides with the original Bochner curvature tensor when a Hermitian manifold is Kahler. They observed that the generalized Bochner curvature tensor is a conformal invariant on Hermitian metrics on a Hermitian manifold. Here the complex structure is fixed. As a consequence, when a Hermitian manifold is a locally conformal KThler manifold, the (generalized) Bochner curvature tensor has the same formula as the (original) Boch … More ner curvature tensor.Several years ago, we have classified compact Kahler manifolds with vanishing Bochner tensor. A Bochner curvature flat Hermitian manifold is defined to be a Hermitian manifold with vanishing (generalized) Bochner curvature tensor. Bochner curvature flat Hermitian geometry is Hermitian geometry whose metrics are Bochner curvature flat. It is a problem what kind of Hermitian geometry admits a Bochner curvature flat Hermitian geometry. It is unknown which compact Hermitian manifold supports a Bochner curvature fiat metric. In addition, the classification of compact Bochner curvature flat Hermitian manifolds is far from valid as well as the classification of compact conformally flat manifolds. We restrict our attension to the class of locally conformal Kahler manifolds. Then we have arrived at the following classification theorem, which is different from that of the Kahler manifolds.Theorem A.Let (M, g, J) be a 2n-dimensional compact locally conformal Kahler manifold (n>1). If the Bochner curvature tensor vanishes, then Al is conformally equivalent to one of the following locally conformal Kahler manifolds :(1) The complex projective space CP^n.(2) A complex euclidean space form T^n_/F (F*U(n)).(3) A complex hyperbolic space form H^n_/GAMMA (GAMMA*PU (n, 1)).(4) A fiber space H^m_*__<GAMMA>CP^<n-m>(GAMMA*PU(m, 1)*PU(n-m+1), m=1, 2, ・, n-1)(5) A Hopf manifold S^<2-1>*__<GAMMA>S^1(F*U(u)*S^1).(i) F is a finite group and GAMMA is a discrete cocompact subgroup which acts freely and properly discontinuously.(ii) The manifolds of the above (1), (2), (3), (4) are Kahler manifolds. When g is a Kahler metric from the beginning, (M, g, J) is holomorphically isometric (up to a constant scalar multip1e of the Kahler metric) to one of the above Kahler manifolds of (1), (2), (3), (4).Review 2. This part concerns a geometric structure on (4n+3)-dimensional smooth manifolds. The isometry group of quaternionic hyperbolic space acts transitively on the boundary sphere as projective transformations. It gives a geometry (PSp(n+1, 1), S^<4n+3>). A(4n+3)-manifold locally modelled on this geometry is said to be a spherical pseudo-quaternionic manifold. We discuss a Carnot-Caratheodory structure on spherical pseudo-quaternionic manifolds in connection with the Sasakian 3-structure. Using superrigidity in quaternionic hyperbolic group, we have proved the geometric rigidity of compact spherical pseudo-quaternionic (4n+3)-manifolds when the fundamental group is isomorphic to either an amenable group or a quaternionic hyperbolic group. A spherical pseudo-quaternionic structure is a geometric structure on a (4n+3)-manifold locally modelled on the sphere S^<4n+3> with coordinate changes lying in the Lorentz group PSp(n+1, 1). Here PSp(n+1, 1) is isomorphic to the isometry group Iso(H^<n+1>_) of the quaternionic hyperbolic space H^<n+l>_. The space H^<n+1>_ has the projective compactification whose boundary is the sphere S^<4n+3> on which PSp(n+1, 1) acts as projective transformations. The pair (PSp(n+1, 1), S^<4n+3>) is said to be spherical pseudo-quaternionic geometry. A (4n+3)-manifold locally modelled on this geometry is said to be a spherical pseudo-quaterniortic manifold. By using the Margulis' superrigidity by Corlette, we proved that the following rigidity.Theorem B.Let M be a compact spherical pseudo-quaternionic (4n+3)-manifold whose fundamental group pi(M) is isomorphic to a discrete uniform subgroup of PSp(m, 1) for some m where 2*m*n. Then M is pseudo-quaternionically isomorphic to the double coset spaceSp(m)*DELTASp(l)*Sp(n-m)*Sp(m, 1)・Sp(n-m+1)/GAMMA where m=2, ・, n. Less

在整个学年1997年至1998年，我们已经研究了几何结构的复流形，适合共形几何的框架。S.Bochner引入了Kaplanche流形上的基本曲率张量，称为Bochner曲率张量。这个张量也可以从酉群表示理论的外尔原理导出。当在共形几何的框架下考虑Hermitian流形时，Tricerri和Vanhecke定义了Hermitian流形上的广义Bochner曲率张量。当埃尔米特流形是Kahler流形时，这个曲率张量与原来的Bochner曲率张量重合。他们观察到广义Bochner曲率张量是埃尔米特流形上埃尔米特度量的共形不变量。在这里，复杂的结构是固定的。因此，当Hermitian流形是局部共形KThler流形时，（广义）Bochner曲率张量具有与（原始）Boch曲率张量相同的公式。 ...更多信息 几年前，我们已经把Bochner张量为零的紧致Kahler流形分类为一类。定义Bochner曲率平坦的Hermitian流形为具有零（广义）Bochner曲率张量的Hermitian流形。Bochner曲率平坦的Hermitian几何是度量是Bochner曲率平坦的Hermitian几何。什么样的厄米特几何允许Bochner曲率平坦的厄米特几何是一个问题。不知道哪一个紧致厄米特流形支持Bochner曲率平坦度量。此外，紧Bochner曲率平坦的Hermitian流形的分类远不如紧共形平坦流形的分类那样有效。我们把注意力限制在局部共形Kahler流形上。定理A.设（M，g，J）是一个2n维紧致局部共形Kahler流形（n>1）。若Bochner曲率张量为零，则Al共形等价于下列局部共形Kahler流形之一：（1）复射影空间CP^n。(2)复欧氏空间形式T^n_/F（F*U（n））。(3)复双曲空间形式H^n_/<$MA（<$MA *PU（n，1））. (4)一个纤维空间H^m_*__<GAMMA>CP^<n-m>（&lt;$MA *PU（m，1）*PU（n-m+1），m=1，2，·，n-1）（5）一个Hopf流形S^<2-1>*_<GAMMA>S ^1（F*U（u）* S ^1）. (i)F是一个有限群，而Gamma是一个自由且适当不连续作用的离散余紧子群。(ii)上述（1）、（2）、（3）、（4）的流形是Kahler流形。当g从始至终是Kahler度量时，（M，g，J）与（1），（2），（3），（4）中的Kahler流形之一全纯等距（直到Kahler度量的常数标量倍）。这一部分讨论了（4 n +3）维光滑流形上的一个几何结构。四元数双曲空间的等距群作为射影变换传递地作用在边界球面上。它给出了一个几何（PSp（n+1，1），S^&lt;4 n +3&gt;）。一个在这个几何上局部建模的（4 n +3）-流形被称为球面伪四元数流形。我们讨论了球面伪四元数流形上的一个与Sasakian 3-结构有关的Carnot-Caratheodory结构。利用四元数双曲群的超刚性，证明了当基本群同构于一个顺从群或一个四元数双曲群时，紧致球面伪四元数（4 n +3）-流形的几何刚性.球面伪四元数结构（英语：Spherical pseudo-quaternionic structure）是一种在球面S^&lt;4 n +3&gt;上局部建模的（4 n +3）-流形上的几何结构，其坐标变化位于洛伦兹群PSp（n+1，1）中。这里PSp（n+1，1）同构于四元数双曲空间H^&lt;n +1&gt;_的等距群Iso（H^&lt;n +1&gt;_）。空间H^&lt;n+1&gt;_具有以球面S^&lt;4 n +3&gt;为边界的射影紧化，在球面S^&lt;4 n +3&gt;上PSp（n+1，1）充当射影变换。对（PSp（n+1，1），S^&lt;4 n +3&gt;）被称为球面伪四元数几何。一个在这个几何上局部建模的（4 n +3）-流形被称为球面伪四元数流形。利用Corlette的Margulis超刚性，证明了以下刚性：定理B.设M是紧致球面伪四元数（4 n +3）-流形，其基本群pi（M）同构于PSp（m，1）的离散一致子群，其中m为2*m*n。则M伪四元数同构于双陪集空间Sp（m）* Δ Sp（1）*Sp（n-m）* Sp（m，1）·Sp（n-m+1）/Gamma其中m=2，·，n.少

项目成果

神島,芳宣: "Locally conformally Kahlerian structures and uniformization" Proceedings of the first Brazil-USA Workshop in Campinas Walter de Gruyter & Co.174-190 (1997)

Kamishima, Yoshinobu：“局部共形卡勒结构和统一化”坎皮纳斯第一届巴西-美国研讨会论文集 Walter de Gruyter & Co.174-190 (1997)

"Geometric superrigidity of (4n+3) -manifolds with quaternionic hyperbolic fundamental groups" Surikenkokyuroku. 1022 (Analysis of Discrete groups). 142-152 (1998)

“具有四元双曲基本群的 (4n 3) 流形的几何超刚性”Surikenkokyuroku。

神島 芳信: "Locally conformal Kahler manifolds with a family of constant curvature tensors" Kumamoto Journal. 11. 19-41 (1998)

Yoshinobu Kamishima：“具有常曲率张量族的局部共形卡勒流形”Kumamoto Journal 11. 19-41 (1998)。

""Geometric superrigidity of spherical pseudo-quaternionic manifolds, "Proceedings of GARC Workshop, Seoul National University" (to appear). (1999)

““球形伪四元流形的几何超刚性”，“首尔国立大学 GARC 研讨会论文集”（即将出版）。

神島,芳宣: "Locally conformal Kahler manifolds with a family of constant curvature tensors" Kumamoto J. Math.11. 19-41 (1998)

Kamishima, Yoshinobu：“具有常曲率张量族的局部共形卡勒流形”Kumamoto J. Math.19-41 (1998)。