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Lower and upper bounds for the smallest positive eigenvalue of the Laplacean on a parallelizable Riemannian manifold are combined to obtain an explicit lower bound for the torsion of global orthonormal frame fields in terms of the diameter of the metric. INTRODUCTION AND RESULTS Consider a compact, connected n-dimensional Riemannian manifold M with metric tensor g. We assume that the tangent bundle TM is trivial. Then there exist parallelizations ow: TM -* Rn mapping g into the euclidean metric on R . Such co will be called orthonormal. Equivalently, the vector fields El defined by w(E,) = e, = ith standard basis vector of R0 form an orthonormal frame field on M. The torsion T of X is defined by T(X, Y) = co (dwo(X, Y)) for vector fields X and Y on M. T is the torsion tensor of the flat connection D on TM defined by DEI = 0. It is well known that a compact Riemannian manifold admits a torsion-free orthonormal parallelization if and only if it is isometric to a flat torus. In this note we study the relation between isometric invariants of a Riemannian manifold and the torsion of its orthonormal frame fields. Specifically, let K denote the maximum norm of T, i.e., K = max{flT(X, Y)fl: X, Y E TyM, IfXWI = IYl= 1, x E M}. The smallest positive eigenvalue of the Laplace operator of (M, g) on functions is denoted by A1 . Theorem 1. Let (M, g) be a compact connected Riemannian manifold and co: TM Rn an orthonormal parallelization. Iffor some p (1 () , then 2A < 100n8K 2. Example. Let M F r\G be a nilmanifold. Then there exist a sequence of Riemannian metrics { g, } and a corresponding sequence {c w } of orthonormal Received by the editors April 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

First eigenvalue of the Laplacean and torsion of parallelizable Riemannian manifolds