Study on Submanifold Theory of Compact Riemannian Symmetric Spaces

本文研究紧单连通黎曼对称空间的子流形理论。我们利用R.Harvey和H.B.Lawson在考虑校准几何时引入的Grassmann几何来研究它。特别地，我们研究了轨道类型的Grassmann几何。这里主要讨论三个主题：(1)存在问题，(2)分类，(3)子流形理论的应用。对于每个主题，我们得到了以下结果和预知：(1)一般而言，给定一个轨道型Grassmann几何，该几何是否允许伴随的子流形的存在性问题等价于定义在环境对称空间的等距群上的一阶偏微分方程组的局部可解性。此外，在这个等价条件下，用一阶偏微分方程组的解刻划了结合子流形的几何性质，称为第二基本形式.对于曲线的轨道Grassmann几何和实超曲面的轨道Grassmann几何，肯定地解决了存在性问题，并且对于强曲率不变型的轨道Grassmann几何，阐明了伴生子流形的几何结构.(2)在轨道型Grassmann几何中，存在一类重要的全测地线型几何.如上所述的强曲率不变型Grassmann几何构成了全测地线类的一个子类。我们利用动态图等有限图完成了对这一子类的分类。作为Grassmann几何的应用，我们得到了对称子流形的分类和Gauss映射的推广。我们认为这些概念对于对称空间的子流形理论也是非常有用的。

This investigation is on the submanifold theory of compact simply connected Riemannian symmetric spaces. We study it by using a Grassmann geometry, introduced by R.Harvey and H.B.Lawson in their consideration of calibrated geometry. Particularly we study a Grassmann geometry of orbital type. The main subjects treated here are the following three : (1) the existence problem, (2) the classification, and (3) applications for submanifold theory. For each subject, we have obtained the following results and foreknowledges :(1) Generally, given a Grassmann geometry of orbital type, the existence problem whether the geometry admits associated submanifolds or not is equivalent to the local solvability of a certain system of 1st order PDE's defined on the isometry group of the ambiant symmetric space. Moreover, under this equivalence, the geometrical property of an associateed submanifold, what is called the 2nd fundamental form, is charac- terized in terms of a solution of the system of 1st order PDE's. Also, for the orbital Grassmann geometries of curves and the ones of real hypersurfaces, the existence problem has been solved affirmatively, and for the orbital Grassmann geometries of strongly curvature-invariant type the geometric structure of associated submanifolds has been clarified.(2) Among Grassmann geometries of orbital type, there exists an important class, what is called of totally geodesic type. The Grassmann geometries of strongly curvature-invariant type, described above, constitute a subclass of the class of totally geodesic type. We have completed the classification of this subclass, by using such finite diagrams as Dynkin's diagrams. We suppose that this method also is useful even for the cases of general totally geodesic type.(3) As applications of Grassmann geometry, we have the classification of symmetric submanifolds and the generalization of Gauss mappings. We suppose that these notions are also very useful for the submanifold theory of symmetic spaces.