An invariant formula for orthogonal distributions
type: Item
instance of: scholarly article
title: An invariant formula for orthogonal distributions
zbMATH Open document ID: 0631.53023
DOI: 10.1007/BF00054609
author: Patrick Coulton
published in: Annals of Global Analysis and Geometry
publication date: 1988-01-01T00:00:00Z
review text: We give an integral formula relating certain curvatures of a foliation to the Ricci curvature of the ambient space. In particular, if M is a space form, we have the integral formula: \[ (1/vol(M))\int _{M}(2K_ 2(x)+S(x))dM=(m-1)c, \] where \(\dim (M)=m\), \(K_ 2\) is the second curvature polynomial of the foliation (it is assumed that the orthogonal distribution to the foliation is integrable), S represents the intrinsic Ricci curvature averaged over the tangent space of the foliation, and c represents the sectional curvature of the space form. In addition, we show that if \(S^ 2\times S^ 2\) has a foliation with minimal leaves, then it is either the trivial cross product foliation or the orthogonal distribution is nowhere integrable. Similar results are included.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4026270
zbMATH Keywords: minimal submanifold
zbMATH Keywords: compact space forms
zbMATH Keywords: integral formula
zbMATH Keywords: foliation
MaRDI profile type: Publication
cites work: Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions
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cites work: Visibility and parallelotopes in projective space
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description: scientific article; zbMATH DE number 4026270