TY - JOUR

T1 - Lichnerowicz and Obata theorems for foliations

AU - Lee, Jeffrey

AU - Richardson, Ken

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2002/10

Y1 - 2002/10

N2 - 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 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.

AB - 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 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.

UR - http://www.scopus.com/inward/record.url?scp=0036804809&partnerID=8YFLogxK

U2 - 10.2140/pjm.2002.206.339

DO - 10.2140/pjm.2002.206.339

M3 - Article

AN - SCOPUS:0036804809

VL - 206

SP - 339

EP - 357

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -