Lichnerowicz and Obata theorems for foliations

Jeffrey Lee and Ken Richardson

Vol. 206 (2002), No. 2, 339–357

DOI: 10.2140/pjm.2002.206.339

Abstract

The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric ≥ a² 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^⊥ ≥ a² for every X in the normal bundle for some fixed a > 0, 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 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.

Milestones

Received: 5 December 2000

Published: 1 October 2002