Optimal pinching constants of odd dimensional homogeneous spaces

Abstract

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B13=SU(5)/(Sp(2)×ℤ2S1) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W7 1,1=SU(3)/S1 1,1. We prove that the optimal pinching constants are precisely in both cases. So far B13 and W7 1,1 were only known to admit Riemannian metrics with pinching constants.¶We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W7 k,l =SU(3)/S1 k,l . Our computations cover the cone of invariant T2-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.¶In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W7 k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B13 is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ3/ℤ2 and inherits an Sp(2)-invariant Riemannian metric, and the other is W7 1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W7 1,1 and the Berger space B13.