A conformal integral invariant on Riemannian foliations

Abstract

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