A remark on left invariant metrics on compact Lie groups

The investigation of manifolds with non-negative sectional curvature is one of the classical fields of study in global Riemannian geometry. While there are few known obstruction for a closed manifold to admit metrics of non-negative sectional curvature, there are relatively few known examples and general construction methods of such manifolds (see [Z] for a detailed survey). In this context, it is particularly interesting to investigate left invariant metrics on a compact connected Lie group G with Lie algebra g. These metrics are obtained by left translation of an inner product on g. If this metric is biinvariant then its sectional curvature is non-negative, and it is known that the set of inner products on g whose corresponding left invariant metric on G has non-negative sectional curvature is a connected cone; indeed, each such inner product can be connected to a biinvariant one by a canonical path ([T]). In the present article, it is shown that the stretching of the biinvariant metric in the direction of a subalgebra of g almost always produces some negative sectional curvature of the corresponding left invariant metric on G. In fact, the following theorem answers a question raised in [Z, Problem 1, p.9].