A conformal integral invariant on Riemannian foliations

Abstract:

Let $M$ be a closed manifold which admits a foliation structure $\mathcal {F}$ of codimension $q\geq 2$ 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 $\int _MY(R^T_{g^T})d\mu _g$ 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 $q\geq 3$, we have $\int _MY(R^T_{g^T})d\mu _g=0$ 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 $\int _MY(R^T_{g^T})d\mu _g\neq 0$. Therefore, $\int _MY(R^T_{g^T})d\mu _g$ is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension $2$.