Lichnerowicz inequality on foliated manifold with a parallel 2-form

Abstract

In this paper, we prove that if $(M,g)$ is a closed orientable Riemannian manifold with a ”transversely oriented harmonic $g$-Riemannian foliation of codimension $q$ on $M$ and if there exists a parallel basic 2-form on $M$ and a positive constant $k$ such that the transversal Ricci curvature satisfies $Ric_{\nabla}(Z,Z)\geq k(q-1)|Z|^2$ for every transverse vector field $Z$, then the smallest nonzero eigenvalue $\lambda_B$ of the basic Laplacian $\Delta_B$ satisfies $\lambda_B\geq 2k(q-1).$