Grassmann geometries on compact symmetric spaces of general type
type: Item
instance of: scholarly article
title: Grassmann geometries on compact symmetric spaces of general type
zbMATH Open document ID: 0911.53033
author: Hiroo Naitoh
published in: Journal of the Mathematical Society of Japan
publication date: 1998-09-16T00:00:00Z
review text: Let \(M\) be an \(m\)-dimensional smooth manifold and \(s\) a positive integer such that \(1\leq s\leq m\). Let \(G^s(TM)\) be the Grassmann bundle over \(M\) with the Grassmann manifold of \(s\)-planes as each fibre. Given a subset \(\Sigma\) of \(G^s(TM)\), an \(s\)-dimensional connected submanifold \(S\) of \(M\) is called a \(\Sigma\)-submanifold if all the tangent spaces of \(S\) belong to \(\Sigma\). The collection of \(\Sigma\)-submanifolds constitutes a so-called \(\Sigma\)-geometry, a Grassmann geometry introduced by \textit{R. Harvey} and \textit{H. B. Lawson} [Acta Math. 148, 47-157 (1982; Zbl 0584.53021)]. In this paper we assume that \(M\) is a compact simply connected Riemannian symmetric space and \(\Sigma\) is a \(G\)-orbit satisfying the following condition, where \(G\) denotes the isometry group of \(M\) and acts canonically on \(G^s(TM)\): any element \(V\in\Sigma\) and its orthogonal complement \(V^{\bot}\) are preserved by the curvature tensor of \(M\). We consider such a \(\Sigma\)-geometry. A \(\Sigma\)-geometry always has a unique compact totally geodesic \(\Sigma\)-submanifold. The aim of this paper is to complete the classification of all \(\Sigma\)-geometries which admit non-totally geodesic \(\Sigma\)-submanifolds. As an application, we have the classification of symmetric submanifolds of \(M\).
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 1199730
zbMATH Keywords: symmetric space
zbMATH Keywords: Grassmann geometry
zbMATH Keywords: symmetric submanifold
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05030557
OpenAlex ID: W2103883449
DOI: 10.2969/JMSJ/05030557
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description: scientific article; zbMATH DE number 1199730