Mathematics – Differential Geometry
Scientific paper
2011-09-09
Mathematics
Differential Geometry
14 pages
Scientific paper
Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then we introduce the Extrinsic Geometric Flow depending on the mean curvature vector field of the distribution, and show existence/uniquenes and convergence of a solution as $t\to\infty$, when the complementary distribution is integrable with compact leaves. We apply the method to the problem of prescribing mean curvature vector field of a foliation, and give examples for harmonic and umbilical foliations and for the double-twisted product metrics, including the codimension-one case.
Rovenski Vladimir
Wolak Robert
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