A conformal integral invariant on Riemannian foliations

Abstract

Let $M$ be a closed manifold which admits a foliation structure $\mathcal{F}$ of codimension $q\ge 2$ and a bundle-like metric $g_0$. Let $[g_0{]}_B$ be the space of bundle-like metrics which differ from $g_0$ only along the horizontal directions by a multiple of a positive basic function. Assume $Y$ is a transverse conformal vector field and the mean curvature of the leaves of $(M,\mathcal{F},g_0)$ vanishes. We show that the integral ${\int }_{M}Y(R_{g^T}^T)d{\mu }_g$ is independent of the choice of $g\in [g_0{]}_B$, where $g^T$ is the transverse metric induced by $g$ and $R^T$ is the transverse scalar curvature. Moreover if $q\ge 3$, we have ${\int }_{M}Y(R_{g^T}^T)d{\mu }_g=0$ for any $g\in [g_0{]}_B$. However there exist codimension 2 minimal Riemannian foliations $(M,\mathcal{F},g)$ and transverse conformal vector fields $Y$ such that ${\int }_{M}Y(R_{g^T}^T)d{\mu }_g\ne 0$. Therefore, it is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension 2.

10 pages

References in corpus (1)

• Recent progress on the Yamabe problem