Submanifolds In Riemannian Symmetric Spaces Research Articles

Codimension reduction in symmetric spaces

In this paper we give a short geometric proof of a generalization of a well-known result about reduction of codimension for submanifolds of Riemannian symmetric spaces.

Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians

In this article, the totally geodesic submanifolds in the complex 2 2 -Grassmannian G 2 ( C n + 2 ) G_2(\mathbb {C}^{n+2}) and in the quaternionic 2 2 -Grassmannian G 2 ( H n + 2 ) G_2(\mathbb {H}^{n+2}) are classified. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2 2 published by Chen and Nagano (1978) is incomplete. For example, G 2 ( H n + 2 ) G_2(\mathbb {H}^{n+2}) with n ≥ 5 n \geq 5 contains totally geodesic submanifolds isometric to a H P 2 \mathbb {H}P^2 , its metric scaled such that the minimal sectional curvature is 1 5 \tfrac 15 ; they are maximal in G 2 ( H 7 ) G_2(\mathbb {H}^7) . G 2 ( C n + 2 ) G_2(\mathbb {C}^{n+2}) with n ≥ 4 n \geq 4 contains totally geodesic submanifolds which are isometric to a C P 2 \mathbb {C}P^2 contained in the H P 2 \mathbb {H}P^2 mentioned above; they are maximal in G 2 ( C 6 ) G_2(\mathbb {C}^6) . Neither submanifolds are mentioned by Chen and Nagano.

Totally geodesic submanifolds of the complex quadric

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Q m : = SO ( m + 2 ) / ( SO ( 2 ) × SO ( m ) ) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Q m by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Q m are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]: The first type is constituted by manifolds isometric to C P 1 × R P 1 ; their existence follows from the fact that Q 2 is (via the Segre embedding) holomorphically isometric to C P 1 × C P 1 . The second type consists of 2-spheres of radius 1 2 10 which are neither complex nor totally real in Q m .

Geometry of reflective submanifolds in Riemannian symmetric spaces

The set ℛ(B) of submanifolds conjugate to a given reflective submanifold B in a Riemannian symmetric space M has a structure of symmetric space. Using this structure, for a submanifold N in M we establish integral formulae which represent the integrals of the functions C ↦ vol ( N ∩ C ) on ℛ ( B ) by some extrinsic geometric amounts of N.

Symmetric submanifolds associated with irreducible symmetric R-spaces

We construct examples of symmetric submanifolds in Riemannian symmetric spaces of noncompact type and obtain the classification of symmetric submanifolds in irreducible Riemannian symmetric spaces of noncompact type and rank greater than one. This finishes the classification problem of symmetric submanifolds in Riemannian symmetric spaces completely.

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.

Totally geodesic hypersurfaces of naturally reductive homogeneous spaces

Totally geodesic submanifolds of Riemannian symmetric spaces have been well investigated and it has been shown that they have beautiful and fruitful properties. In particular, due to the (M+,M_)-theory by B.Y. Chen and T. Nagano [1] this subject has made great progress. Naturally reductive homogeneous spaces are known as a natural generalization of Riemannian symmetric spaces. K. Tojo [6] investigated totally geodesic submanifolds of naturally reductive homogeneous spaces and obtained a necessary and sufficient condition of their existence. We will recall his result in section 3. Moreover he implicitly made the following conjecture.

On the Classification of Reflective Submanifolds of Riemannian Symmetric Spaces

On the Classification of Reflective Submanifolds of Riemannian Symmetric Spaces

The reflection principle for minimal submanifolds of Riemannian symmetric spaces

The reflection principle for minimal submanifolds of Riemannian symmetric spaces