Convexity of reflective submanifolds in symmetric \(R\)-spaces
type: Item
instance of: scholarly article
title: Convexity of reflective submanifolds in symmetric \(R\)-spaces
zbMATH Open document ID: 1263.53046
author: Peter Quast
author: Makiko S. Tanaka
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2013-03-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmj/1356038981
review text: A reflective submanifold of a Riemannian manifold \(P\) is a connected component of the fixed point set of an involutive isometry of \(P\). In this paper, the authors prove a theorem stating that a reflective submanifold \(M\) of a symmetric \(R\)-space \(P\) is convex: any shortest geodesic arc in \(M\) is still shortest in the ambient space \(P\). Moreover, the authors provide explicit counterexamples to this statement for symmetric ambient spaces which are not symmetric \(R\)-spaces, namely for a flat \(2\)-torus with non-rectangular rhombic lattice (the long diagonal is reflective, but not convex), and the space \({\mathrm{Ad}}({\mathrm{SU}}(3))\), i.e., the quotient of \({\mathrm{SU}}(3)\) by its center. Symmetric \(R\)-spaces are defined as particular orbits of \(s\)-representations, but the proof of the theorem uses the following intrinsic characterization, proven by \textit{O. Loos} [Math. Z. 189, 211--226 (1985; Zbl 0583.53044)]: Given a maximal Abelian subspace \({\mathfrak t}\) of the Lie triple system associated to \(P\), the unit lattice of \(P\) is defined as \(\{X\in {\mathfrak t}\mid {\mathrm{Exp}}(X)=o\}\), where \(o\) is the chosen basepoint of \(P\); for a symmetric \(R\)-space, this unit lattice permits a basis over \({\mathbb Z}\) which is orthogonal. This fact is used to show that the tangent cut locus of the reflective submanifold \(M\) at \(o\in M\) is given by the intersection of \(T_oM\) with the tangent cut locus of \(P\) at \(o\), which is equivalent to the assertion of the theorem.
reviewed by: Oliver Goertsches
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 6141568
zbMATH Keywords: symmetric space
zbMATH Keywords: symmetric R-space
zbMATH Keywords: reflective submanifold
zbMATH Keywords: convexity
MaRDI profile type: Publication
OpenAlex ID: W1978527159
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DOI: 10.2748/TMJ/1356038981
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description: scientific article; zbMATH DE number 6141568