m at h . D G ] 2 3 Ju n 20 08 HOMOGENEOUS METRICS WITH NONNEGATIVE CURVATURE

Given compact Lie groups H ⊂ G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H . Such an enlarging is possible if (K, H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics. LetH ⊂ G be compact Lie groups, with Lie algebras h ⊂ g, and let g0 be a bi-invariant metric on G. The space G/H with the induced normal homogeneous metric, denoted (G, g0)/H, has nonnegative sectional curvature. Little is known about which other G-invariant metrics on G/H have nonnegative sectional curvature, except in certain cases. In all cases where G/H admits a G-invariant metric of positive curvature, the problem has been studied along with the determination of which G-invariant metric has the best pinching constant; see [9],[10],[11]. When H is trivial, this problem was solved for G = SO(3) and U(2) in [1], and partial results for G = SO(4) were obtained in [5]. Henceforth, we identify G-invariant metrics on G/H with AdH-invariant inner products on p = the g0-orthogonal complement of h in g. In Section 1, it is an easy application of Cheeger’s method to prove that the solution space is star-shaped. That is, if g is a G-invariant metric on G/H with nonnegative curvature, then the inverse-linear path, g(t), from g(0) = g0|p to g(1) = g is through nonnegatively curved Ginvariant metrics. Here, a path of inner products on p is called “inverse-linear” if the inverses of the associated path of symmetric matrices form a straight line. This observation reduces our problem to an infinitesimal one: first classify the directions, g′(0), one can move away from the normal homogeneous metric such that the inverse-linear path g(t) appears (up to derivative information at t = 0) to remain nonnegatively curved. Then, for each candidate direction, check how far nonnegative curvature is maintained along that path. In Section 2, we derive curvature variation formulas necessary to implement this strategy, inspired by power series derived by Müter for curvature along an inverse-linear path [8]. In Section 3, we consider an intermediate subgroup K between H and G, with subalgebra k, so we have inclusions h ⊂ k ⊂ g. Write p = m ⊕ s, where m is the orthogonal compliment of h in k and s is the orthogonal compliment of k in g. The inverse-linear path of G-invariant ∗Supported by the Schwerpunktprogramm Differentialgeometrie of the Deutsche Forschungsgesellschaft. 1 2 LORENZ SCHWACHHÖFER, KRISTOPHER TAPP metrics on G/H which gradually enlarges k is described as follows for all A,B ∈ p: (0.1) gt(A,B) = ( 1 1− t ) · g0(A, B) + g0(A, B), where superscripts denote g0-orthogonal projections onto the corresponding spaces. This variation scales the fibers of the Riemannian submersion (G, g0)/H → (G, g0)/K. For t < 0, these fibers are shrunk, and gt has nonnegative curvature because it can be redescribed as a submersion metric obtained by a Cheeger deformation: (G/H, gt) = ((G/H, g0)× (K, (−1/t) · g0))/K. For t > 0, these fibers are enlarged, and the situation is more complicated. We will prove: