Almost Nonnegative Curvature on Some Fake 6- And 14-Dimensional Projective Spaces

Abstract

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake RP 6 's of Hirsch and Milnor and on the analogous fake RP 14 's.One of the great unsolved problems of Riemannian geometry is to determine the structure of collapse with a lower curvature bound. An apparently simpler, but still intractable problem, is to determine which closed manifolds collapse to a point with a lower curvature bound. Such manifolds are called almost nonnegatively curved. Here we construct almost nonnegative curvature on some fake RP 6 s and RP 14 s.Theorem A. The Hirsch-Milnor fake RP 6 s and the analogous fake RP 14 s admit Riemannian metrics that simultaneously have almost nonnegative sectional curvature and positive Ricci curvature.Remark. By considering cohomogeneity one actions on Brieskorn varieties, Schwachhöfer and Tuschmann observed in [15] that in each odd dimension of the form, 4k + 1, there are at least 4 k oriented diffeomorphism types of homotopy RP 4k+1 s that admit metrics that simultaneously have positive Ricci curvature and almost nonnegative sectional curvature.The Hirsch-Milnor fake RP 6 s are quotients of free involutions on the images of embeddings ι of the standard 6-sphere, S 6 , into some of the Milnor exotic 7-spheres, Σ 7 k ([12], [14]). Our proof begins with the observation that the SO (3)-actions that Davis constructed on the Σ 7 k s in [5] leave these Hirsch-Milnor S 6 s invariant and commute with the Hirsch-Milnor free involution. Next we compare the Hirsch-Milnor/Davis (SO (3) × Z 2 )-action on ι (S 6 ) ⊂ Σ 7 k with a very similar linear action of (SO (3) × Z 2 ) on S 6 ⊂ R 7 and apply the following lifting result of Searle and the second author. Theorem B. (See Proposition 8.1 and Theorems B and C in [17]) Let (M e , G) and (M s , G) be smooth, compact, n-dimensional G-manifolds with G a compact Lie group. Suppose that the orbit spaces M e /G and M s /G are equivalent, and M s /G has almost nonnegative curvature.Then M e admits a G-invariant family of metrics that has almost nonnegative sectional curvature. Moreover, if the principal orbits of (M e , G) have finite fundamental group and the quotient of the principal orbits of M s has Ricci curvature ≥ 1, then every metric in the almost nonnegatively curved family on M e can be chosen to also have positive Ricci curvature. Date: October 15, 2015. 1991 Mathematics Subject Classification. 53C20.