Invariants On the Geometric Manifolds with Group Actions

具有群作用的几何流形上的不变量

• 批准号: 14340026
• 负责人: KAMISHIMA Yoshinobu
• 金额: $ 3.97万
• 依托单位: Tokyo Metropolitan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340026/
• 关键词: Pseudo-conformal auaternionic structure; Gemetric structure; Quaternionic CRstructure; Pseudo-conformal auaternionic CR structure; Uniformization; Conformal structure; Vanishing; Obstruction; Quaternionic Kahler structure; Contact Structure; CR-

We have studied a geometric structure on a (4n+3)-dimensional smooth manifold M which is an integrable, nondegenerate codimension 3 subbundle D on M whose fiber supports the structure of 4n-dimensional quaternionic vector space. We call it a psesuo-conformal quterninonic structure. This structure has a refinement which is said to be a psesuo-conformal quterninonic CR structure. The structure is thought of as a generalization of the quaternionic CR structure. In order to study this geometric structure on M, we single out an sp(1)-valued 1-form ω locally on a neighborhood U of M such that Ker ω = D|U. We shall construct the invariants on the pair (M, ω) whose vanishing implies that M is uniformized with respect to a finite dimensional flat quaternionic CR geometry. In fact we have proved the standard psesuo-conformal quterninonic structure on the spahere S^<4n+3> coincides with the standard pseudo-quaternionic CR structure on S^<4n+3> The invariants obtained on a (4n+3)-manifold M have the same formula as the curvature tensor of quaternionic (indefinite) Kaehler manifolds. From this viewpoint, we shall exhibit a quaternionic analogue of Chern-Moser's CR structure. As to the global existence of the 1-form ω on a (4n+3)-manifold M is related to the Pontrjagin classes. We have shown the relation that 2p_1(M)=(n+2)p_1(L). In particular, if 2p_1(M)=0, then there exists a global 1-form co on M which represents a pseudo-conformal quaternionic structure D. As a consequence, there exists a hyperoomplex structure {I, J, K} on D.

我们研究了 (4n+3) 维光滑流形 M 上的几何结构，它是 M 上的可积非简并余维 3 子束 D，其纤维支持 4n 维四元向量空间的结构，我们称之为伪共形。该结构有一个改进，被认为是伪共形四元 CR 结构。四元 CR 结构的推广 为了研究 M 上的这种几何结构，我们在 M 的邻域 U 上局部选取一个 sp(1) 值 1 形式 ω，使得 Ker ω = D|U。构造对 (M, ω) 的不变量，其消失意味着 M 相对于有限维平面四元数 CR 几何是均匀的。事实上，我们已经证明了标准的伪共形四元数。空间 S^<4n+3> 上的结构与 S^<4n+3> 上的标准伪四元 CR 结构一致 在 (4n+3)-流形 M 上获得的不变量具有与曲率张量相同的公式从这个角度来看，我们将展示 Chern-Moser 的 CR 结构的四元数类似物。 (4n+3)-流形 M 上的 1-形式 ω 与 Pontrjagin 类相关 我们已经证明了 2p_1(M)=(n+2)p_1(L) 的关系。 =0，则 M 上存在全局 1 型 co，代表伪共角四元结构 D。因此，M 上存在超复杂结构 {I, J, K} D .

项目成果

CR manifolds and transformation groups

CR 流形和变换群

• 发表时间: 2003
• 期刊: Selected Topics in CR Geometry (Ed. by S.Dragomir), Quaderni di Matematica. 1329
• 作者: A.Amarzaya;M.Guest;M.Guest;T.Tsuboi;Toshitake Kohno;Takashi Tsuboi;河野俊丈;河野俊丈;Yoshinobu Kamishima;Yoshinobu Kamishima;Kouji Fujiwara(共著);Yoshinobu Kamishima (L.Ornea);Kouji Fujiwara;神島芳宣;藤原耕二;神島芳宣;神島芳宣
• 通讯作者: 神島芳宣

Note on realization of cusp cross-sections of complex hyperbolic orbifolds

关于复杂双曲轨道折叠尖点横截面实现的注意事项

• 发表时间: 2003
• 期刊: 数理研講究録(S.Kamiya (ed.)), Perspectives of Hyperbolic Spaces II 数理解析研究所 1387
• 作者: A.Amarzaya;M.Guest;M.Guest;T.Tsuboi;Toshitake Kohno;Takashi Tsuboi;河野俊丈;河野俊丈;Yoshinobu Kamishima;Yoshinobu Kamishima;Kouji Fujiwara(共著);Yoshinobu Kamishima (L.Ornea);Kouji Fujiwara;神島芳宣;藤原耕二;神島芳宣
• 通讯作者: 神島芳宣

Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds

(4n 3) 维流形上的四元和对四元 CR 结构

• 发表时间: 2005
• 期刊: Central European J. of Mathematics 2(5)
• 作者: 神島芳宣

Bochner flat structures from the viewpoint of spherical, Heisenberg CR-geometry

从球面、海森堡 CR 几何角度看博赫纳平面结构

• 发表时间: 2003
• 期刊: Geometry and Analysis on CR-manifolds Inst. of Math.Academia Sinica 12
• 作者: M.Hino;Akihiko Inoue;Y.Daido;A.Moriwaki;Toshiyuki Koto;神島芳宣
• 通讯作者: 神島芳宣

神島芳宣: "Three-dimensional Lie group actions on compact (4n+3)-dimensional geometric manifolds"D.flerential geometry and application. 1-26 (2004)

Yoshinobu Kamishima：“紧致 (4n+3) 维几何流形上的三维李群作用”D.flerential 几何和应用。 1-26 (2004)。