Lichnerowicz and Obata theorems for foliations

The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric (X, X) ≥ a (n - 1) |X|² for every X ∈ TM for some fixed a > 0, then the smallest positive eigenvalue λ of the Laplacian satisfies λ ≥ an. The Obata theorem states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvature a. In this paper, we prove that if M is a closed Riemannian manifold with a Riemannian foliation of codimension q, and if the normal Ricci curvature satisfies Ric^⊥ (X, X) ≥ a (q - 1) |X|² for every X in the normal bundle for some fixed a > O, then the smallest eigenvalue λ_B of the basic Laplacian satisfies λ_B ≥ aq. In addition, if equality occurs, then the leaf space is isometric to the space of orbits of a discrete subgroup of O (q) acting on the standard q-sphere of constant sectional curvature a. We also prove a result about bundle-like metrics on foliations: On any Riemannian foliation with bundle-like metric, there exists a bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric.