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

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dc.creatorBaditoiu, Gabriel
dc.creatorEscobales Jr., Richard H.
dc.creatorIanus, Stere
dc.date2005-08-09
dc.date2006-03-09
dc.date.accessioned2026-07-07T06:42:45Z
dc.date.available2026-07-07T06:42:45Z
dc.descriptionLet $(M^{n},g)$ 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 $κ\wedge χ_{\boldkey F}$ is a closed (possibly zero) de Rham cohomology (p+1)-form. Here $κ$ is the mean curvature one-form of the foliation ${\boldkey F}$ and ${χ_{\boldkey F}}$ is its characteristic form. In the codimension-2 case, $κ\wedge χ_{\boldkey F}$ is closed if and only if $κ$ is horizontally closed. In certain restricted cases, we give necessary and sufficient conditions for $κ\wedge{χ_{\boldkey F}}$ 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.
dc.description20 pages
dc.identifierhttps://arxiv.org/abs/math/0508164
dc.identifierhttp://arxiv.org/abs/math/0508164
dc.identifierTokyo Journal of Mathematics, vol 29 (2006), No. 1, 247-270
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/102161
dc.subjectDifferential Geometry
dc.subject57R30
dc.titleA Cohomology (p+1) Form Canonically Associated with Certain Codimension-q Foliations on a Riemannian Manifold
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