Some geometric consequences of the Weitzenböck formula on Riemannian almost-product manifolds; weak-harmonic distributions
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.
scientific article; zbMATH DE number 4026273