Isometric imbeddings of Riemannian manifolds and their rigidity

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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.