Some geometric consequences of the Weitzenböck formula on Riemannian almost-product manifolds; weak-harmonic distributions
type: Item
instance of: scholarly article
title: Some geometric consequences of the Weitzenböck formula on Riemannian almost-product manifolds; weak-harmonic distributions
zbMATH Open document ID: 0631.53026
author: A. H. Rocamora
published in: Illinois Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: By using the Weitzenböck formula, we get a linear relation among linear invariants of Riemannian almost-product manifolds, from which we deduce some geometric consequences. We also define and study weak-harmonic distributions. The vertical distribution \({\mathcal V}\), of a Riemannian almost-product manifold is said to be weak-harmonic if the canonical projection h: T\({\mathcal M}\to {\mathcal H}\) from the tangent bundle onto the horizontal bundle is an \({\mathcal H}\)-valued 1-form orthogonal to \(\Delta ^{{\mathcal H}}h\), \(\Delta ^{{\mathcal H}}\) being the Laplace operator induced by a connection on \({\mathcal H}\) (which is a generalization of a basic connection). We prove that some of the main results of \textit{F. W. Kamber} and \textit{Ph. Tondeur} [Ill. J. Math. 28, 458-471 (1984; Zbl 0529.53027)] on harmonic foliations remain valid for weak-harmonic distributions. On the other hand, these results are direct consequences from the relation among the invariants mentioned above. Furthermore, we show new results about weak-harmonicity.
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 4026273
zbMATH Keywords: almost-product manifolds
zbMATH Keywords: harmonic distributions
zbMATH Keywords: Laplace operator
zbMATH Keywords: harmonic foliations
MaRDI profile type: Publication
Recommended article: Q3645458
Recommended article: On compact Riemannian manifolds with harmonic Weyl curvature
Recommended article: On weakly harmonic maps from Finsler to Riemannian manifolds
Recommended article: Q4416460
Recommended article: Q4602642
Recommended article: Q5437735
Recommended article: Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds
Recommended article: Campesino a Campesino: Voices from Latin America's Farmer to Farmer Movement for Sustainable Agriculture - By Eric Holt-Gimenez
Recommended article: On weakly conformally symmetric pseudo-Riemannian manifolds
Recommended article: Q5426268
description: scientific article; zbMATH DE number 4026273