Isometric imbeddings of Riemannian manifolds and their rigidity

S.Kobayashi构造了黎曼对称空间的规范等距嵌入。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Hermite对称空间Sp(N)/U(N)的典范等距嵌入在局部意义下是刚性的。(这项工作是与E.Kaneda合作的。)我们已经知道，这些空间的正则等距嵌入给出了欧几里德空间的最低维等距嵌入，即使在局部观点上也是如此。上述结果表明，这些嵌入具有本质唯一性，这给出了这些空间的局部等距嵌入的重要结果。通过证明高斯方程在给定余维上解的本质唯一性，证明了这一定理。证据在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。另外，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4n-3。这一结果改进了我们先前对这些空间的类数的估计。(这项工作也是与E.Kaneda合作的。)为了证明这一结果，我们广泛地考察了极大伪阿贝尔子空间，并证明了高斯方程在给定的余维内不允许有解。但这两个空间的类数的已知上界估计与上述下界估计之间的差距相当大，我们必须在下一步的研究中填补这一差距。

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.