Structural equations and an integral formula for foliated manifolds
type: Item
instance of: scholarly article
title: Structural equations and an integral formula for foliated manifolds
zbMATH Open document ID: 0578.53025
DOI: 10.1007/BF00149274
author: Akhil Ranjan
published in: Geometriae Dedicata
publication date: 1986-01-01T00:00:00Z
review text: The following formula has been proved and applied: \[ \sum_{i,j}K(v_ i,x_ j)=div F-\sum_{j}tr(A^{v_ j})^ 2+div H- | T| \quad^ 2+| F|^ 2+| H|^ 2. \] Here, K is the sectional curvature of a Riemannian manifold M, \(\{v_ j\}\) and \(\{x_ i\}\) are orthogonal frames of an integrable subbundle \({\mathcal V}\subset TM\) and of its orthogonal complement \({\mathcal H}\), T and A are the second fundamental forms of \({\mathcal V}\) and \({\mathcal H}\), \(F=\sum A_{x_ i}x_ i\) and \(H=\sum T_{v_ j}v_ j.\) \{The reviewer [''An integral formula for a Riemannian manifold with two orthogonal complementary distributions'', Colloq. Math. (to appear)] obtained a very similar formula without assumptions of integrability of \({\mathcal V}.\}\)
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 3925736
zbMATH Keywords: sectional curvature
zbMATH Keywords: orthogonal frames of an integrable subbundle
zbMATH Keywords: second fundamental forms
reviewed by: Paweł Walczak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00149274
OpenAlex ID: W2058777656
description: scientific article; zbMATH DE number 3925736