A conformal integral invariant on Riemannian foliations

TL;DR

This paper introduces a conformal integral invariant on Riemannian foliations that remains constant under certain metric changes and provides obstructions to the transverse Yamabe problem, especially in codimension 2.

Contribution

It establishes a new integral invariant for Riemannian foliations and explores its implications for the transverse Yamabe problem, highlighting differences across codimensions.

Findings

The integral of transverse scalar curvature is invariant under specific metric variations.

For codimension ≥ 3, the integral always vanishes for minimal Riemannian foliations.

Counterexamples in codimension 2 show the integral can be nonzero, indicating obstructions.

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, it is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension 2.