Integral formulae on foliated symmetric spaces
type: Item
instance of: scholarly article
title: Integral formulae on foliated symmetric spaces
zbMATH Open document ID: 1246.53039
author: Vladimir Yu. Rovenskij
author: Paweł Walczak
published in: Mathematische Annalen
publication date: 2012-02-10T00:00:00Z
review text: Let \(M\) be a compact Riemannian manifold endowed with a foliation \({\mathcal F}\) of codimension 1 and a unit vector field \(N\) normal to \({\mathcal F}\). The paper under review provides an infinite family of formulas relating integrals of elementary symmetric functions of principal curvatures of the leaves to similar integrals of some algebraic invariants obtained from the Weingarten operator of the foliation and the curvature operator in the direction of \(N\). Formulas of this type are eventually obtained even for some foliations with singularities, in particular for singular Riemannian foliations. Earlier results can be recovered as special cases, for instance those by \textit{F. Brito}, \textit{R. Langevin} and \textit{H. Rosenberg} [J. Differ. Geom. 16, 19--50 (1981; Zbl 0472.53049)], \textit{A. Ranjan} [Geom. Dedicata 20, 85--91 (1986; Zbl 0578.53025)], and \textit{P. G. Walczak} [Colloq. Math. 58, No. 2, 243--252 (1990; Zbl 0766.53024)]. For the sake of simplicity, the main calculations in the present paper are carried out for locally symmetric spaces.
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 6005030
zbMATH Keywords: Weingarten operator
zbMATH Keywords: Jacobi operator
zbMATH Keywords: singular Riemannian foliation
MaRDI profile type: Publication
full work available at URL: s00208-011-0637-4
OpenAlex ID: W2049712309
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DOI: 10.1007/S00208-011-0637-4
description: scientific article; zbMATH DE number 6005030