Geometry of reflective submanifolds in Riemannian symmetric spaces
type: Item
instance of: scholarly article
title: Geometry of reflective submanifolds in Riemannian symmetric spaces
zbMATH Open document ID: 1104.53048
author: Hiroyuki Tasaki
published in: Journal of the Mathematical Society of Japan
publication date: 2006-03-29T00:00:00Z
review text: Let \(M\) be a complete connected Riemannian manifold. A connected component of the fixed point set of an involutive isometry of \(M\) is called a reflective submanifold. Let \(\mathcal{R}(B)\) be the set of submanifolds conjugate to a given reflective submanifold \(B\) in a Riemannian symmetric space \(M\). The author proves that the set \(\mathcal{R}(B)\) has the structure of a symmetric space. In fact, if \(M\) is a Riemannian symmetric space of compact type, then so is \(\mathcal{R}(B)\). If \(M\) is a Riemannian symmetric space of noncompact type, \(\mathcal{R}(B)\) is a semi-Riemannian symmetric space of semisimple type. Using the structure of the symmetric space for a submanifold \(N\) in \(M\) the author establishes integral formulae which represent the integrals of the function \(C\rightarrow \text{vol}(N\cap C)\) on \(\mathcal{R}(B)\) by some extrinsic geometric quantities of \(N\).
reviewed by: Ana Pereira do Vale
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 5015998
zbMATH Keywords: reflective submanifold
zbMATH Keywords: symmetric space
zbMATH Keywords: Crofton formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1145287102
OpenAlex ID: W1979473319
Wikidata QID: Q115224913
DOI: 10.2969/JMSJ/1145287102
description: scientific article; zbMATH DE number 5015998