Study on Submanifold Theory of Compact Riemannian Symmetric Spaces

本文研究紧致单连通黎曼对称空间的子流形理论。我们研究它通过使用格拉斯曼几何，介绍了R.哈维和H. B.劳森在他们的考虑校准几何。特别是我们研究格拉斯曼几何的轨道型。这里讨论的主要问题有以下三个：（1）存在性问题，（2）分类，（3）子流形理论的应用。对于每一个问题，我们都得到了如下结果和预见：（1）一般地，给定一个轨道型Grassmann几何，该几何是否允许相伴子流形的存在性问题等价于定义在周围对称空间的等距群上的某个一阶偏微分方程组的局部可解性。在这个等价条件下，结合子流形的几何性质，即所谓的第二基本形式，可以用一阶偏微分方程组的解来刻画.对于曲线的轨道Grassmann几何和真实的超曲面的轨道Grassmann几何，肯定地解决了存在性问题，对于强曲率不变型的轨道Grassmann几何，阐明了相应子流形的几何结构. (2)在轨道型格拉斯曼几何中，有一类重要的几何，称为全测地型几何。上述强曲率不变型的格拉斯曼几何构成了全测地型几何的一个子类。我们已经完成了这一子类的分类，通过使用有限的图，如Dynkin的图。我们假设这种方法对于一般的全测地线型也是有用的。(3)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。我们假设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.