A Cohomology (p+1) Form Canonically Associated with Certain Codimension-q Foliations on a Riemannian Manifold

Details A Cohomology (p+1) Form Canonically Associated with Certain Codimension-q Foliations on a Riemannian Manifold A Cohomology (p+1) Form Canonically Associated with Certain Codimension-q Foliations on a Riemannian Manifold

2005-08-09

arxiv.org/abs/math/0508164v2

Tokyo Journal of Mathematics, vol 29 (2006), No. 1, 247-270

Mathematics

Differential Geometry

20 pages

Scientific paper

Let $(M^{n},g)$ be a closed, connected, oriented, $C^{\infty}$, Riemannian, n-manifold with a transversely oriented foliation $\boldkey F$. We show that if $\lbrace X,Y \rbrace$ are basic vector fields, the leaf component of $[X,Y]$, $\Cal{V}[X,Y]$, has vanishing leaf divergence whenever ${\kappa}\wedge \chi_{\boldkey F}$ is a closed (possibly zero) de Rham cohomology (p+1)-form. Here ${\kappa}$ is the mean curvature one-form of the foliation ${\boldkey F}$ and ${\chi_{\boldkey F}}$ is its characteristic form. In the codimension-2 case, ${\kappa}\wedge \chi_{\boldkey F}$ is closed if and only if ${\kappa}$ is horizontally closed. In certain restricted cases, we give necessary and sufficient conditions for ${\kappa}\wedge{\chi_{\boldkey F}}$ to be harmonic. As an application, we give a characterization of when certain closed 3-manifolds are locally Riemannian products. We show that bundle-like foliations with totally umbilical leaves with leaf dimension greater than or equal to two on a constant curvature manifold, with non-integrable transversal distribution, and with Einstein-like transversal geometry are totally geodesic.

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