Jacobi fields, Riccati equation and Riemannian foliations
type: Item
instance of: scholarly article
title: Jacobi fields, Riccati equation and Riemannian foliations
zbMATH Open document ID: 0866.53017
author: Philippe Tondeur
author: Lieven Vanhecke
published in: Illinois Journal of Mathematics
publication date: 1997-07-10T00:00:00Z
review text: If \((M,{\mathcal F})\) is a foliated Riemannian manifold and \(\gamma\) is a geodesic orthogonal to the leaves of \(\mathcal F\), then a vector field \(V\) along \(\gamma\) is \(\mathcal F\)-Jacobi when the orthogonal to \(\mathcal F\) component of \(V'+R(\dot\gamma,V)\dot\gamma\), \(R\) being the curvature tensor on \(M\) and \(V'\) being the covariant derivative of \(V\) in the direction of \(\dot\gamma\), vanishes. The Jacobi equation defining \(\mathcal F\)-Jacobi fields can be expressed in terms of the curvature and the integrability tensor of the normal bundle of \(\mathcal F\). This leads to a Riccati type equation for the shape operator \(S\) of \(\mathcal F\) along \(\gamma\). This equation is studied in detail and applied to Riemannian and harmonic foliations under some curvature restrictions on \(M\).
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53B21
zbMATH DE Number: 926191
zbMATH Keywords: Jacobi field
zbMATH Keywords: foliation
zbMATH Keywords: Riccati equation
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: geodesic
reviewed by: Paweł Walczak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 926191