Isometric imbeddings of Riemannian manifolds and their rigidity

黎曼流形的等距嵌入及其刚性

• 批准号: 16540070
• 负责人: AGAOKA Yoshio
• 金额: $ 1.54万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540070/
• 关键词: Riemannian manifold; submanifold; symmetric space; Gauss equation; rigidity; canonical imbedding; curvature; class number; 等長埋め込み; エルミート対称空間; 標準等長埋め込み; 第二基本形式; シンプレクティック群; 第2基本形式; 射影空間; 擬平坦数

S.Kobayashi has constructed canonical isometric imbeddings of Riemannian symmetric spaces. Among these spaces we showed that the canonical isometric imbeddings of the Cayley projective plane P^2(Cay), the quaternion projective plane P^2(H), the symplectic group Sp(n), and the Hermitian symmetric space Sp(n)/U(n) are rigid in the local sense. (This work is collaborated with E.Kaneda.) We already know that the canonical isometric imbeddings for these spaces give the least dimensional isometric imbeddings into the Euclidean spaces even in the local standpoint. The above results show that these imbeddings possess the essential uniqueness, which give the crucial results of local isometric imbeddings for these spaces. This theorem is proved by showing the essential uniqueness of solutions of the Gauss equation in a given codimension. The proof heavily depends on the character for each space, and it seems impossible to treat these spaces in a unified way. But the rigidity of the canonical isometric imbedding seems to hold for a wider class of spaces, and to show this conjecture is our next task.As another result, we showed that the class number of the complex projective space P^n(C) and the quaternion projective space P^n(H) is larger than or equal to 2n-2, and 4n-3, respectively. This result improves our previous estimate on the class number for these spaces. (This work is also collaborated with E.Kaneda.) To prove this result, we examine extensively the maximal pseudo-abelian subspaces, and show that the Gauss equation does not admit a solution in a given codimension. But the gap between the known upper bound estimate and the above lower bound estimate of the class number for these two spaces is quite large, and we must fill this gap in our next study.

S.小林构造了黎曼对称空间的标准等距嵌入。在这些空间中，我们证明了Cayley射影平面P^2（Cay）、四元数射影平面P^2（H）、辛群Sp（n）和埃尔米特对称空间Sp（n）/U（n）的正则等距嵌入在局部意义下是刚性的。(This与E. Kaneda合作）。我们已经知道，这些空间的正则等距嵌入给出了欧氏空间的最小维等距嵌入，即使在局部观点下也是如此。上述结果表明，这些嵌入具有本质唯一性，从而给出了这类空间的局部等距嵌入的关键结果。这个定理是通过证明高斯方程在给定余维中解的本质唯一性来证明的。证明很大程度上取决于每个空间的特征，似乎不可能以统一的方式对待这些空间。但正则等距嵌入的刚性似乎适用于更广泛的空间类，证明这一猜想是我们的下一个任务.作为另一个结果，我们证明了复射影空间P^n（C）和四元数射影空间P^n（H）的类数分别大于或等于2n-2和4 n-3.这个结果改进了我们以前对这些空间的类数的估计。(This工作也与E. Kaneda合作。为了证明这个结果，我们广泛地研究了极大伪阿贝尔子空间，并证明了高斯方程在给定的余维中不存在解。但是这两个空间的类数的已知上界估计和上述下界估计之间的差距是相当大的，我们必须在下一步的研究中填补这一空白。

项目成果

Rigidity of the canonical isometric imbedding of the Cayley projective plane P^2(Cay)

凯莱射影平面 P^2(Cay) 正则等距嵌入的刚性

• 发表时间: 2005
• 期刊: Hokkaido Mathematical Journal 34巻 2号
• 作者: J.Itch;K.Kiyohara;H.Tamura;N.Tanaka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka
• 通讯作者: Y.Agaoka

A lower bound for the class number of P^(n) and P^n(H)

P^(n) 和 P^n(H) 的类别数的下界

• 发表时间: 2006
• 期刊: Hokkaido Mathematical Journal Vol.35, No.4
• 作者: J.Itch;K.Kiyohara;H.Tamura;N.Tanaka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka
• 通讯作者: Y.Agaoka

Local isometric imbeddings of Riemannian symmetric spaces and their rigidity

黎曼对称空间的局部等距嵌入及其刚性

• 发表时间: 2008
• 期刊: Sugaku Expositions Vol.21,No.1
• 作者: Y. Agaoka;E. Kaneda
• 通讯作者: E. Kaneda

A lower bound for the class number of P^n(C) and PAn(H)

P^n(C) 和 PAn(H) 类别数的下界

• 发表时间: 2006
• 期刊: Hokkaido Mathematical Journal Vol.35 No.4
• 作者: J.Itch;K.Kiyohara;H.Tamura;N.Tanaka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka
• 通讯作者: Y.Agaoka

Rigidity of the canonical isometric imbedding of the Hermitian symmetric space Sp(n)/U(n)

埃尔米特对称空间 Sp(n)/U(n) 正则等距嵌入的刚性

• 期刊: Hokkaido Mathematical Journal に掲載決定
• 作者: J.Itch;K.Kiyohara;H.Tamura;N.Tanaka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;Y.Agaoka;阿賀岡芳夫;Y.Agaoka;Y.Agaoka;阿賀岡 芳夫;Y.Agaoka;Y.Agaoka
• 通讯作者: Y.Agaoka