. D G ] 1 4 A ug 2 00 6 INVARIANT METRICS ON SO ( 4 ) WITH NONNEGATIVE CURVATURE

The starting point for constructing all known examples of compact manifolds with positive (or even quasi-positive) curvature is the fact that bi-invariant metrics on compact Lie groups are nonnegatively curved. In order to generalize this fundamental starting point, we address the question: given a compact Lie group G, classify the left-invariant metrics on G which have nonnegative curvature. The first two cases, G = SO(3) and G = U(2), were completely solved in [1]. For higher dimensional groups, new examples could potentially, via familiar quotient constructions, lead to new examples of quasi-positively curved spaces. To explore the remaining cases, we first formulate an infinitesimal version of the problem. That is, we consider a path h(t) of left-invariant metrics on G, with h(0) bi-invariant. The main work of this paper is towards classifying the possibilities for h′(0) such that the path appears (up to derivative information at t = 0) to remain nonnegatively curved. We derive restrictions on h′(0) for general G, and some specific to G = SO(4). These infinitesimal restrictions will be translated into global conclusions (about the set of all left-invariant metrics on G with nonnegative curvature) via power-series techniques in the follow-up paper [5]. The author is pleased to thank Craig Sutton, Emily Proctor, Zachary Madden and Nela Vukmirovic for their help with this project.