ILLINOIS JOURNAL OF MATHEMATICS Volume 33, Number 4, Winter 1989 ON LIE GROUPS THAT ADMIT LEFT-INVARIANT LORENTZ METRICS OF CONSTANT SECTIONAL CURVATURE BY FRANK BARNET 1. Introduction We say that a non-abelian Lie group G is of special type if its Lie algebra, g, has the property that [x, y] is a linear combination of x and y for all x, y in g. In [3], J. Milnor proved that every left-invariant Riemannian metric on a Lie group of special type is of constant negative sectional curvature, and, in [4], K. Nomizu proved that every left-invariant Lorentz metric on such a Lie group is also of constant sectional curvature, but, depending on the choice of left-invariant Lorentz metric, the sign of the constant sectional curvature may be positive, negative, or zero. Lie groups of special type belong to a larger class, studied by E. Heintze in [2], of Lie groups that admit some left-invariant Riemannian metric of constant negative sectional curvature. A natural question is which Lie groups in the larger class admit left-invariant Lorentz metrics of constant sectional curvature and what are the possible signs of those curvatures. In this paper we answer this question completely by proving the following theorem. THEOREM 1.1. Let G be a Lie group that admits a left-invariant Riemannian metric of constant negative sectional curvature. Then: (i) G admits a lefi-invariant Lorentz metric of constant positive sectional curvature. (ii) G admits a left-invariant Lorentz metric of constant negative, or zero, sectional curvature if, and only if, contains a one-dimensional ideal. In [2], E. Heintze proved that ; contains an abelian ideal u of codimension 1 and that for any b not in u, ad(b)lu=,I+B, where B is a linear transformation which is skew-adjoint with respect to the inner product the left-invariant Riemannian metric induces on g, and where is non-zero. Lie groups of special type are precisely those for which B is identically zero. Together with Theorem 1.1, this result of Heintze gives us these two corol- laries. Received September 25, 1987. (C) 1989 by the Board of Trustees of the University of Illinois Manufactured in the United States of America 631