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1997 - 1998学年，我们研究了一种符合共形几何框架的复杂流形几何结构。回顾1。S.Bochner引入了kaihier流形上的基本曲率张量，称为Bochner曲率张量。这个张量也可以由酉群表示理论中的Weyl原理导出。当在共形几何框架中考虑厄米流形时，Tricerri和Vanhecke定义了厄米流形上的广义Bochner曲率张量。当厄米流形是Kahler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曲率张量是厄米流形上厄米度量的共形不变量。这里复杂的结构是固定的。因此，当厄米流形是局部共形KThler流形时，（广义）Bochner曲率张量与（原始）Boch…More ner曲率张量具有相同的公式。几年前，我们对具有消失Bochner张量的紧化Kahler流形进行了分类。一个Bochner曲率平坦厄米流形被定义为具有消失（广义）Bochner曲率张量的厄米流形。Bochner曲率平坦厄米几何是度量是Bochner曲率平坦的厄米几何。什么样的厄米几何允许波希纳曲率平坦的厄米几何是一个问题。不知道哪个紧化厄米流形支持一个波希纳曲率的法定度规。此外，紧致波希纳曲率平坦厄米流形的分类和紧致共形平坦流形的分类也远远不够有效。我们将注意力限制在局部共形Kahler流形上。然后我们得到了下面的分类定理，它与Kahler流形的分类定理不同。定理a：设（M, g， J）为2n维紧致局部共形Kahler流形（n>1）。如果Bochner曲率张量消失，则Al保形等价于下列局部保形Kahler流形之一：(1)复射影空间CP^n。(2)复欧几里得空间形式T^n_/F （F*U(n)）。(3)复双曲空间形式H^n_/GAMMA （GAMMA*PU (n, 1)）。(4)纤维空间H ^ m_ * __ <伽马> CP ^ < n - m >(γ* PU (m, 1) *聚氨酯(n - m + 1)、m = 1, 2,・,n - 1)(5)霍普夫歧管S ^ < 2 > * __ <伽马> S ^ 1 U (U) (F * * S ^ 1)。(1) F是一个有限群，GAMMA是一个自由、适当不连续作用的离散紧子群。（ii）上述(1)、(2)、(3)、(4)的流形为Kahler流形。当g从一开始就是Kahler度规时，（M, g， J）与上述(1),(2),(3),(4)的Kahler流形之一是全纯等距的（直到Kahler度规的常数标量倍）。回顾2。这一部分涉及（4n+3）维光滑流形上的几何结构。四元双曲空间的等距群以射影变换的形式传递于边界球上。它给出了一个几何形状（PSp(n+ 1,1), S^<4n+3>）。在这种几何结构上局部建模的（4n+3）流形称为球形伪四元数流形。结合Sasakian 3-结构讨论了球面伪四元数流形上的一类Carnot-Caratheodory结构。使用superrigidity四元的双曲组,我们证明了球面pseudo-quaternionic紧凑的几何刚度(4 n + 3)繁殖的基本组织同构的组或四元的双曲组。球面伪四元数结构是在球面S^<4n+3>局部建模的（4n+3）流形上的几何结构，其坐标变化位于洛伦兹群PSp（n+ 1,1）内。其中PSp（n+ 1,1）与四元双曲空间H^<n+1>_的等距群Iso（H^<n+1>_）同构。空间H^<n+1>_具有射影紧化，其边界为球面S^<4n+3>, PSp（n+ 1,1）在其上作为射影变换。这对（PSp(n+ 1,1), S^<4n+3>）被称为球形伪四元数几何。在这种几何结构上局部建模的（4n+3）流形称为球形伪四元流形。利用Corlette的马古利斯超刚性，我们证明了以下刚性：定理b .设M为紧致球面伪四元数（4n+3）流形，其基群pi(M)同构于PSp（M, 1）的离散一致子群，其中M为2* M *n。则M伪四元同构于双余集空间esp (M)*DELTASp(l)*Sp(n- M)*Sp（M, 1）·Sp(n- M +1)/GAMMA，其中M =2，·，n. Less

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)。