Existence of Riemannian metrics with positive biorthogonal curvature on simply connected 5-manifolds
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Archiv der Mathematik
Existence of Riemannian metrics with positive biorthogonal curvature on simply connected 5-manifolds Boris Stupovski and Rafael Torres
Abstract. Using the recent work of Bettiol, we show that a first-order conformal deformation of Wilking’s metric of almost-positive sectional curvature on S 2 × S 3 yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel–Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes. Mathematics Subject Classification. Primary 53C20, 53C21, Secondary 53B21. Keywords. Positive biorthogonal curvature, Biorthogonal curvature, 5manifolds.
1. Introduction and main results. Let (M, g) be a compact Riemannian nmanifold and let secg be the sectional curvature of the metric. We often abuse notation and denote the Riemannian metric by (M, g) as well. For each 2-plane σ ∈ Gr2 (Tp M ) = {X ∧ Y ∈ Λ2 Tp M : ||X ∧ Y ||2 = 1},
(1.1)
⊥
let σ ⊂ Tp M be its orthogonal complement. That is, there is a g-orthogonal direct sum decomposition σ ⊕ σ ⊥ = Tp M at a point p ∈ M . Definition 1. The biorthogonal curvature of a 2-plane σ ∈ Gr2 (Tp M ) is sec⊥ g (σ) :=
min
1
σ ∈Gr2 (Tp M ) 2 σ ⊂σ ⊥
(secg (σ) + secg (σ ))
(1.2)
(cf. [3, Section 5.4]). We say that (M, g) has positive biorthogonal curvature sec⊥ g > 0 if (1.2) is positive for every σ ∈ Gr2 (Tp M ) at every point p ∈ M .
B. Stupovski and R. Torres
Arch. Math.
A stronger curvature condition is the following. Choose a distance on the Grassmanian bundle Gr2 (T M ) that induces the standard topology. Definition 2. The distance curvature of a 2-plane σ ⊂ Tp M is 1 secθg (σ) := min (secg (σ) + secg (σ )) σ ∈Gr2 (Tp M ) 2
(1.3)
dis(σ,σ )≥θ
for each θ > 0 (cf. [3, Section 5.2]). We say that (M, g θ ) has positive distance curvature secgθ > 0 if for every θ > 0, there is a Riemannian metric (M, g θ ) for which (1.3) is positive for every σ ∈ Gr2 (Tp M ) at every point p ∈ M . Bettiol [4] classified up to homeomorphism closed simply connected 4manifolds that admit a Riemannian metric of positive biorthogonal curvature by constructing metrics of positive distance curvature on S 2 × S 2 [2, Theorem, Proposition 5.1], [3, Theorem 6.1], and showing that positive biorthogonal curvature is a property that is closed under connected sums [3, Proposition 7.11], [4, Proposition 3.1]. In this paper, we extend Bettiol’s results to dimension five. More precisely, we build upon Bettiol’s work and show that an application of a first-order conformal deformation to Wilking’s metric (S 2 × S 3 , gW ) of almost-positive sectional curvature [11] yields the main result of this note. Theorem A. For every θ > 0, there is a Riemannian metric (S 2 × S 3 , g θ ) such that (a) secθgθ > 0; (b) there is a limit metric g 0 su
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