Submanifolds of symmetric spaces

研究了对称空间中的子流形，它被认为是三维欧氏空间中规则曲面的推广。首先我们考虑对称空间M^^〜以及所有此类子流形的空间M中满足某些良好条件的子流形F。然后一般我们可以构造M上的纤维丛E和从E到M的自然映射，使得每个纤维都双射映射到F。从这些对象出发，我们可以从M^^中的子流形M构造M^^中的广义直纹子流形M。在此背景下，基本问题是研究M在M^^中极小与M在M中极小的关系。另一方面，通过与M.奥尔特加在格拉纳达。最后与水津熏合作证明了2-球面乘积中的极小拉格朗日曲面的基本定理。

We investigated submanifolds in symmetric spaces, which are considered as a generalization of rules surfaces in 3-dimensional Euclidean space. First we consider submanifolds F which satisfies some good condition in symmetric space M^^〜 and also the space M of such all submanifolds. Then generally we can construct fibre bundle E over M and natural map from E to M^^〜 such that each fiber is mapped bijectively to F. From these object we can construct generalized ruled submanifolds M in M^^〜 from submanifolds Σ in M. In this context, fundamental problem is to study relationship of which M is minimal in M^^〜 and Σ in M. We gave some answers to this problem in some geometrically important cases.On the other hand, we investigated congruence of Frenet curves in complex quadrics by using isoparametric functions on the unit sphere in the tangent space by the joint work with M. Ortega at Granada. Finally we proved fundamental theorem for minimal Lagrangian surfaces in the product of 2-spheres by the joint work with Kaoru Suizu.