Submanifolds In Symmetric Spaces Research Articles

The constancy of principal curvatures of curvature-adapted submanifolds in symmetric spaces

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under a certain condition, and derive the constancy of the principal curvatures of such submanifolds. As a result, we derive that such submanifolds are isoparametric.

Research of submanifolds in symmetric spaces by using the complexification and the infinite dimensional geometry

Research of submanifolds in symmetric spaces by using the complexification and the infinite dimensional geometry

`Spindles' in symmetric spaces

We study families of submanifolds in symmetric spaces of compact type arising as exponential images of s -orbits of variable radii. If the s -orbit is symmetric such submanifolds are the most important examples of adapted submanifolds, i.e. of submanifolds of symmetric spaces with curvature invariant tangent and normal spaces.

On focal points of submanifolds in symmetric spaces

The first purpose of this paper is to give a smart proof of the Morse index theorem for squared distance function of submanifolds in s a symmetric space. The second purpose is to classify focal points into strong ones and weak ones and to give a class of submanifolds in symmetric spaces all of whose focal points (other than conjugate points) are strong ones. The third purpose is to construct examples of submanifolds in symmetric spaces admitting weak ones.

Symmetric submanifolds in symmetric spaces

In this paper we construct new examples of symmetric non-totally geodesic submanifolds in irreducible symmetric spaces of non-compact type and of rank⩾2. These symmetric spaces are characterized by the fact that they contain a reflective submanifold with one-dimensional Euclidean factor; they are listed at the end of the paper.

Stability on minimal submanifolds in symmetric spaces

We determine the stabilityof totally geodesic submanifolds in a compact symmetric space, which are called polars and meridians (see 2.1). These subspaces were introduced by Chen and Nagano ([CN-1]) and we have many interesting results after that ([CN-2], [N-l], [N-2], [NS-1], [NS-2] and [NS-3]). Recently, several results have been obtained about the stabilityof totally geodesic submanifolds in compact symmetric spaces. Ohnita gave the formula for the index, the nullityand the Killing nullity of a totally geodesic submanifold in a compact symmetric space in [0], in which he also proved that the Helgason sphere in a compact symmetric space is stable. Tasaki proved that the Helgason sphere in a compact Lie group is homologically volume-minimizing in its real homology classin [Ts-1]. He used the calibrationtheory. And there are studies about the stability of certain closed subgroups in a compact Lie group by Mashimo and Tasaki ([MT-1] and [MT-2]). Mashimo determined all the unstable Cartan embeddings of compact symmetric spaces in [M]. And there is a result about the stabilityof symmetric i?-spaces in Hermitian symmetric spaces and totally complex submanifolds in quaternionic Kahler symmetric spaces of classical type by Takeuchi ([Tk-2]). Recently Nagano and the author have obtained a result on a relationshipbetween the stabilityof minimal submanifolds and that of /^-harmonic maps ([NS-3]). In the present paper we study the stability of all the polars and meridians in every compact symmetric space by using Ohnita's method in Section 3. We will also study the stabilityof totally complex totallygeodesic submanifolds in quaternionic Kahler symmetric spaces of exceptional type in Section 4. I should like to express my gratitude to Professor T. Nagano for his useful advice and kindly support. This is the author's doctoral dissertation submitted to Sophia University in October, 1992.

Polar sets as nondegenerate critical submanifolds in symmetric spaces

Polar sets as nondegenerate critical submanifolds in symmetric spaces

Classification of totally umbilical submanifolds in symmetric spaces

AbstractA submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.