Symmetric submanifolds in symmetric spaces.
type: Item
instance of: scholarly article
title: Symmetric submanifolds in symmetric spaces.
zbMATH Open document ID: 1035.53079
DOI: 10.1016/S0926-2245(01)00069-9
author: Daria Osipova
published in: Differential Geometry and its Applications
publication date: 2002-08-14T00:00:00Z
review text: A submanifold \(N\) of a Riemannian manifold \(M\) is called a symmetric submanifold if for each point \(p\) in \(N\) there exists an involutive isometry of \(M\) which fixes \(p\), leaves \(N\) invariant and whose differential at \(p\) fixes the normal vectors of \(N\) at \(p\) and reflects the tangent vectors. (For \(M= E^n\), see \textit{D. Ferus}, [Math. Ann. 247, 81--93 (1980; Zbl 0446.53041)].) A submanifold \(N\) of \(M\) is called reflective if the geodesic reflection of \(M\) in \(N\) is a well defined global isometry [see \textit{D. S. P. Leung}, J. Differ. Geom. 14, 167--177, 179--185 (1979; Zbl 0453.53044).] Let \(M\) be an irreducible symmetric space of non-compact type with rank greater than one. Let \(T\) be a reflective submanifold of \(M\) such that the Euclidean factor in the de Rham decomposition of \(T\) is one-dimensional. Let \(o\in T\), denote by \(\gamma\) the exponential image of the one-dimensional component of \(T\) at \(o\) and assume that the geodesic \(\gamma\) is parametrized by arc length so that \(o= \gamma(0)\). Let \(Q\) be the reflective submanifold of \(M\) perpendicular to \(T\) at the point \(o\), and \(G_Q\) the Lie subgroup of the isometry group of \(M\), which is generated by the symmetries of \(Q\). It is proved that then \(G_Q\cdot o= Q\), and the orbit of \(G_Q\) through any point on \(\gamma\) different from \(o\) is a non-totally geodesic symmetric submanifold of \(M\). A list of such triples \((M,T,Q)\) is provided in a table.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1780870
zbMATH Keywords: symmetric submanifolds
zbMATH Keywords: reflective submanifolds
zbMATH Keywords: symmetric spaces of non-compact type
Wikidata QID: Q115338229
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1780870