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| 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| for every X in the normal bundle for some fixed a > 0, then the smallest eigenvalue λB of the basic Laplacian satisfies λB ≥ 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.