Lower bounds for the eigenvalues of the basic Dirac operator on a Riemannian foliation

Abstract

On a foliated Riemannian manifold with a transverse spin structure, we give a lower bound for the square of the eigenvalues of the basic Dirac operator by the smallest eigenvalue of the basic Yamabe operator. We prove, in the limiting case, that the foliation is minimal, transversally Einsteinian with constant transversal scalar curvature. In particular, if the codimension of F is q=3,4,7 and 8, then F is transversally isometric to the action of discrete subgroup of O( q) acting on the q-sphere.