The mixed scalar curvature flow on a fiber bundle

Details The mixed scalar curvature flow on a fiber bundle The mixed scalar curvature flow on a fiber bundle

2012-03-28

arxiv.org/abs/1203.6361v1

Mathematics

Differential Geometry

13 pages

Scientific paper

Let $(M, g)$ be a closed Riemannian manifold, and $\pi: M\to B$ a fiber bundle with compact fiber. We study conformal flow of the metric restricted to the orthogonal distribution $D$ with the speed proportional to the mixed scalar curvature, while the fibers are totally geodesic. For a twisted product we show that the mean curvature vector $H$ of $D$ satisfies the Burgers equation, while the warping function obeys the heat equation. In this case the metrics $g_t$ converge to the product. For general $D$, we modify the flow using certain measure of "non-umbilicity" and the integrability tensor of $D$, while the fibers are totally geodesic. Then $H$ (assumed to be potential along fibers) satisfies the forced Burgers equation, and $g_t$ converges to a metric $\bar g$, for which $H$ depends only on the $D$-conformal class of initial metric. If the "non-umbilicity" of fibers is constant in a sense, then the mixed scalar curvature is quasi-positive for $\bar g$, and $D$ is harmonic.

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