Eigenvalue estimates for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form

On a compact Riemannian manifold M with a transverse spin foliation F of codimension q >= 3, if M admits a non-trivial basic harmonic 1-form omega, then any eigenvalue lambda of the basic Dirac operator satisfies the inequality lambda(2) >= q-1/(4(Q-2)) inf(M) (sigma(del) + vertical bar kappa vertical bar(2)), where sigma(del) is the transversal scalar Curvature and kappa is the mean curvature form of F. I it the limiting case, omega is minimal and co is parallel. (c) 2006 Elsevier B.V. All rights reserved.