# Jeffrey LeeTexas Tech University

## Lichnerowicz and Obata theorems for Riemannian foliations (joint work with Ken Richardson

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The standard Lichnerowicz comparison theorem (1958) states that, given an n-dimensional, closed Riemannian manifold with Ricci curvature satisfying Ric(X,X) \ge k(n-1)||X||2 for every X in TM for some fixed k>0, then the smallest positive eigenvalue l of the Laplacian on M satisfies l \ge nk. The Obata theorem (1962) states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvature k. In this paper, we prove that if M is a Riemannian manifold with a Riemannian foliation of codimension q, and if the normal Ricci curvature (ie curvature is restricted to and contracted on the normal bundle of the foliation) satisfies Ricn(X,X) \ge k(q-1)||X||2 for every X in the normal bundle for some fixed k>0, then the smallest positive eigenvalue lB of the basic Laplacian on M satisfies lB \ge qk. In addition, if equality occurs, then the leaf space of the foliation is isometric to the space of orbits of a discrete subgroup of O(q) acting on the standard q-sphere of constant curvature k. We also prove a result about bundle-like metrics: on any Riemannian foliation with bundle-like metric, there exists another 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