Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

Abstract

A Riemannian manifold is said to be almost positively curved if the sets of points for which all 2-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented 2planes in R 7 admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group G 2 and the octonions, so do not obviously generalize to any other Grassmannians. arXiv:1707.07590v1 [math.DG]

Three new almost positively curved manifolds

“…From O'Neill's formula [9], a zero-curvature plane in E p,q with the Eschenburg metric must lift to a horizontal zero-curvature plane in SU (3). Using this fact and Proposition 2.6, Kerin [8] proved a characterization of points [A] ∈ E p,q having a zero-curvature with respect to the Eschenburg metric.…”

“…We now specialize to Eschenburg spaces E p,q with p = (0, 0, q 1 + q 2 + q 3 ) and q = (q 1 , q 2 , q 3 ) where the q i are relatively prime integers. In this case, the normalizer of U ⊆ G × G contains a U(2) × T 2 , where U(2) = K ⊆ SU(3), and T 2 is the maximal torus of diagonal matrices in SU (3). Noting that U(2) × T 2 ⊆ K × K, this implies that with respect to the Wilking metric on E p,q = SU(3)/S 1 , that left multiplication by U(2) and right multiplication by T 2 are isometries.…”

“…As a means to understand the difference between manifolds admitting non-negative curvature from those admitting positive curvature, much work has gone into understanding two classes of Riemannian manifolds which lie in between-the quasi-positively curved manifolds and the almost positively curved manifolds. Recall that a Riemannian manifold is said to be quasipositively curved if it is everywhere non-negatively curved and has a point p Examples with these weaker curvature notions are much more abundant than in the case of strictly positive curvature [30,26,21,8,9,10,20,19,14,15,23,29,11]. The main constructions are due to Wilking [30] and Tapp [26].…”

Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

“…), and G 2 /SO 4 are the only (simply connected) irreducible symmetric spaces of rank > 1 we know of to admit metrics of Ric 3 > 0; see Table 1 in Sect. 4. It is an open question whether there exists an irreducible symmetric space of rank > 1 which admits a metric of Ric 2 > 0, much less of sec > 0 (see [17] for the construction of a metric on G + 2 (R 7 ) of sec > 0 on an open dense subset).…”