Positive biorthogonal curvature on S^2 x S^2

We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, is strictly positive; and this can be done with an arbitrarily small lower bound on the distance between the planes considered. Although they have positive Ricci curvature, these metrics do not have nonnegative sectional curvature. Such metrics also have positive biorthogonal curvature, meaning that the average of sectional curvatures of any two orthogonal planes is positive.


Introduction
Let (M, g) be a 4-dimensional Riemannian manifold. For each plane σ ⊂ T p M at a point p ∈ M , denote by σ ⊥ the orthogonal plane to σ, i.e., σ ⊕ σ ⊥ = T p M is a g-orthogonal direct sum. Define the biorthogonal (sectional) curvature of σ as the average of the sectional curvatures of σ and σ ⊥ , i.e., sec ⊥ g (σ) := 1 2 sec g (σ) + sec g (σ ⊥ ) . The Hopf Conjecture, that asks if S 2 ×S 2 admits a metric with sec > 0, remains one of the most intriguing open problems in Riemannian geometry. With the standard product metric g 0 , at every point p ∈ S 2 × S 2 there exists σ ⊂ T p M with sec ⊥ g0 (σ) = 0. Namely, any mixed plane σ at p (i.e., spanned by vectors of the form (X, 0) and (0, Y )) is such that σ ⊥ is also a mixed plane, hence sec g0 (σ) = sec g0 (σ ⊥ ) = 0. A natural question in this context is if the weaker condition sec ⊥ > 0 can be satisfied in S 2 × S 2 [2].
The goal of this note is to give a positive answer, also covering a stronger curvature positivity condition, that can be defined in any dimension. Namely, choose a distance (inducing the standard topology) on the Grassmannian bundle Gr 2 T M of planes tangent to M , and for each θ > 0 and σ ⊂ T p M , let sec θ g (σ) := min σ ′ ⊂TpM dist(σ,σ ′ )≥θ 1 2 sec g (σ) + sec g (σ ′ ) .
Theorem. For every θ > 0, there exist Riemannian metrics g θ on S 2 × S 2 with sec θ g θ > 0, arbitrarily close to the standard product metric g 0 in the C k -topology, k ≥ 0. In particular, S 2 × S 2 admits metrics of positive biorthogonal curvature.
The condition sec θ g > 0 means that at every point p ∈ M , the average of sectional curvatures of any two planes σ 1 , σ 2 ⊂ T p M that are at least θ > 0 apart from each other is positive. One can intuitively think of θ as a lower bound for the "angle" between the planes considered. Notice that if θ 1 < θ 2 , then sec θ1 g > 0 clearly implies sec θ2 g > 0. Furthermore, for every metric g on M , there exists θ g > 0 such that if sec θ g > 0 for some 0 < θ ≤ θ g , then Ric g > 0, see Proposition 4.1. In particular, for 4-manifolds, if θ ≤ min p∈M, σ⊂TpM dist(σ, σ ⊥ ), then sec θ g > 0 implies sec ⊥ g > 0. The construction of g θ is so that these metrics converge to a limit metric g 0 as θ → 0 (possibly different from the product metric), in the C k -topology, for any k ≥ 0. This convergence easily implies that, for θ > 0 sufficiently small, the metrics g θ have positive Ricci curvature (Proposition 4.1) and positive biorthogonal curvature. In particular, the above theorem shows that a natural interpolating condition between Ric > 0 and sec > 0 is satisfied on S 2 × S 2 .
We stress that sec ⊥ > 0 alone does not imply Ric > 0, as illustrated by S 1 × S 3 with the standard product metric. This metric clearly has sec ⊥ > 0, but since S 1 × S 3 has infinite fundamental group, it does not support any metrics with Ric > 0. Thus, in order to have a condition of this type that is stronger than Ric > 0, it is crucial that sec θ > 0 can be satisfied no matter how small θ > 0, and that the corresponding metrics converge. In general, sec ⊥ > 0 only implies positive scalar curvature, which poses some topological restrictions on 4-manifolds (e.g., vanishing of all the Seiberg-Witten invariants), but these restrictions are by far not as strong as the ones implied by sec > 0 or Ric > 0. In particular, although sec ⊥ > 0 is comparatively flexible, generic smooth 4-manifolds do not support metrics with this property. Another indication of this relative flexibility of sec ⊥ > 0 is that CP 2 #CP 2 also admits metrics with this property (Proposition 5.1); while, similarly to S 2 × S 2 , it remains an open question whether it admits a metric with sec > 0. Our metrics g θ with sec θ > 0 on S 2 × S 2 can be chosen invariant under the antipodal action of Z 2 ⊕ Z 2 . Thus, for all θ > 0, the quotient RP 2 × RP 2 also admits metrics with sec θ > 0, arbitrarily close to the standard product metric. This illustrates a remarkable difference between sec θ > 0 (in particular, sec ⊥ > 0) and sec > 0 since, by Synge's Theorem, RP 2 × RP 2 cannot have a metric with sec > 0. It is, however, somewhat expected that obstructions of Synge type do not detect these average curvature conditions, since even finiteness of the fundamental group goes unnoticed. We also remark that (S 2 × S 2 , g θ ) has many points with planes of zero curvature (and even negative curvature), however any two such planes are always within distance θ from one another in the Grassmannian of planes tangent at that point. In this way, θ corresponds to a measure of how big the regions formed by planes with nonpositive curvature can be in the Grassmannian. It would be interesting to know if metrics with sec θ > 0 on S 2 × S 2 can also be constructed while keeping sec ≥ 0, as this could give a quantitative insight on the possibility of existence of quasipositively curved metrics.
The techniques used to construct all of the above metrics are (smooth) deformations. Metric deformations to improve curvature have a long history, stemming from Berger and his students in the 1970's to the recent construction proposed by Petersen and Wilhelm [7,8] of a positively curved exotic sphere. Of particular importance in the present note are techniques developed by Müter [6] and Strake [11,12], respectively regarding Cheeger deformations and deformations positive of first-order. The Cheeger deformation is a method to attempt to increase curvature on nonnegatively curved manifolds with symmetries, by shrinking the metric in the direction of orbits of a large isometry group. This technique was introduced by Cheeger [1], inspired by the construction of Berger metrics on odddimensional spheres, where the round metric is shrunk in the direction of the Hopf fibers. Müter [6] carried out a systematic study of Cheeger deformations in his PhD thesis under W. Meyer, establishing ground for a much better understanding of these deformed metrics. Strake [12], another PhD student of W. Meyer during the same period, studied metric deformations of nonnegatively curved metrics for which the first variation of the sectional curvature of any zero curvature plane is positive. These deformations are called positive of first-order, and if the manifold is compact, they yield actual positively curved metrics. They also observed that, in this infinitesimal sense, Cheeger deformations are nonnegative of first-order.
Our deformation process from the product metric g 0 to a metric with sec θ > 0 has two steps, in which the above techniques are combined. The first is a Cheeger deformation, described in detail by Müter [6,13]. More precisely, we consider the cohomogeneity one diagonal SO(3)-action on S 2 × S 2 and shrink g 0 in the direction of the orbits. This deformation gives a family of metrics g t , t > 0, with sec gt ≥ 0 and much fewer planes of zero curvature than g 0 . Namely, (S 2 × S 2 , g t ) has a circle's worth of zero curvature planes on points that lie on the diagonal or the anti-diagonal ±∆S 2 = {(p, ±p) : p ∈ S 2 } ⊂ S 2 × S 2 , and a unique zero curvature plane at any other point. This means that sec θ ≥ 0, and equality holds only for some planes whose base point is in one of the submanifolds ±∆S 2 (Proposition 2.3). Next, for fixed t > 0, set g := g t . The second step is to employ a first-order local conformal deformation g s = g + s h, where h = φ g, and φ is supported in a tubular neighborhood of ±∆S 2 . Given the geometry of (S 2 × S 2 , g), we construct φ such that the first derivative with respect to s of the average of two g s -sectional curvatures is positive (Proposition 3.2). The function φ is proportional to the squared g-distance to ±∆S 2 , multiplied by a cutoff function. The strategy for such a construction is adapted from Strake [11,12]. Finally, a standard compactness argument (Proposition 3.3) implies that sec θ gs > 0 for all sufficiently small s > 0, proving the desired result. This paper is organized as follows. In Section 2, we review basic aspects of Cheeger deformations, following Müter [6,13]. We describe the metric on S 2 × S 2 obtained by a Cheeger deformation with respect to the diagonal SO(3)-action in terms of sec θ . In Section 3, we analyze the effects of a first-order deformation and construct the variation starting from the Cheeger deformed metric that proves the above Theorem. Some remarks on the geometry of the constructed metrics are given in Section 4. Finally, we briefly discuss 4-manifolds with positive biorthogonal curvature (including the construction for CP 2 #CP 2 ) in Section 5.
Acknowledgement. It is a pleasure to thank Fernando Galaz-García, Karsten Grove, Paolo Piccione and Wolfgang Ziller for valuable comments and suggestions during the elaboration of this paper. We also express our sincere gratitude to the referee for the careful reading of the manuscript and constructive criticism.

First step: Cheeger deformation
Although the techniques used in this section are mostly available elsewhere in the literature, see [6,13,14], we briefly recall a few basic aspects as a service to the reader. For convenience, we use the same notation as the above references.
2.1. Cheeger deformation. Let (M, g) be a Riemannian manifold and G a compact Lie group that acts on M by isometries. The Cheeger deformation of g is a 1-parameter family g t , t ≥ 0, of G-invariant metrics on M , defined as follows. Let Q be a bi-invariant metric on G, and endow M × G with the product metric g + 1 t Q. Consider the submersion and define g t as the metric on M that turns ρ into a Riemannian submersion. The family of metrics g t extends smoothly across t = 0, with g 0 = g, thus providing a deformation of such metric. Since sec Q ≥ 0, it follows immediately from the Gray-O'Neill formula that, if sec g0 ≥ 0, then also sec gt ≥ 0, t ≥ 0. As we will see, many planes with zero curvature with g 0 usually gain positive curvature with g t . For each p ∈ M , denote by G p the isotropy group at p and by g p its Lie algebra. Fix the Q-orthogonal splitting g = g p ⊕ m p , and identify m p with the tangent space T p G(p) to the G-orbit through p via action fields. More precisely, we identify Notice that the dimensions of V p and H p may vary with p ∈ M , hence these are not distributions.
Let P t : m p → m p be the Q-symmetric automorphism that relates the metrics Q and g t , i.e., such that It is an easy computation that P t is determined by P 0 in the following way: Thus, if we let C t : T p M → T p M be the g-symmetric automorphism that relates g and g t , i.e., such that we then get where X V and X H are the vertical and horizontal components of X respectively. This reveals how the geometry of g t changes with t, since if P 0 has eigenvalues λ i , then C t has eigenvalues 1 1+tλi corresponding to the vertical directions and eigenvalues 1 in the horizontal directions. In other words, as t grows, the metric g t shrinks in the direction of the orbits and remains the same in the orthogonal directions.

Curvature evolution.
Let us now analyze how the curvature changes under this deformation. Henceforth, we assume the initial metric g 0 has sec g0 ≥ 0. As explained above, this implies sec gt ≥ 0 for all t ≥ 0. Given X ∈ T p M , denote by X m the unique vector in m p such that (X m ) * p = X V p . Also, given a plane σ = span{X, Y }, we write As explained by Ziller [13], the crucial observation of Müter is that, to analyze the evolution of sec gt , it is much more convenient to study sec gt (C −1 t (σ)) rather than sec gt (σ). In more recent literature, the 1-parameter family of bundle automorphisms induced by C −1 t in the Grassmannian bundle Gr 2 T M of planes on M is being called Cheeger reparametrization, see [7,8]. The following result of Müter [6, Satz 3.10] (see also [13,Cor 1.4]) summarizes how the curvature of g t evolves.
In other words, up to the Cheeger reparametrization, zero curvature planes with nondegenerate vertical projection have positive curvature with g t , for all t > 0.
2.3. The case of S 2 × S 2 . Consider S 2 × S 2 endowed with the standard product metric g 0 and the diagonal SO(3)-action: This is a cohomogeneity one isometric action with orbit space a closed interval, so there are codimension one principal orbits (corresponding to interior points of the interval) and two singular orbits (corresponding to the endpoints), see [14]. These singular orbits are the diagonal and anti-diagonal submanifolds: The principal isotropy G p , p ∈ ±∆S 2 , is trivial, since it consists of orientationpreserving isometries of R 3 that fix two linearly independent directions. The singular isotropies are formed by orientation-preserving isometries of R 3 that fix one direction, hence are isomorphic to SO(2). Thus, the group diagram of this action Following Müter [6], we identify the Lie algebra of SO(3) with R 3 by: Considering (so(3), Q) endowed with the standard bi-invariant metric, the above is an isometric identification with Euclidean space (R 3 , ·, · ). In this way, since the Lie exponential in SO (3) is given by matrix exponentiation, the action field induced by Z ∈ so (3) is: So, if x, y ∈ R 3 are such that x, p 1 = y, p 2 = 0, then for all z ∈ R 3 , For general x, y ∈ R 3 , analogously to (2.7), we have: From (2.2), the above is equal to P 0 X, Y , so we get an explicit formula for P 0 : m p → m p in our example: In particular, it follows that the subspace {p 1 , p 2 } ⊥ ⊂ m p is invariant under P 0 and hence under P t and C t , see (2.3) and (2.5).
Remark 2.4. For n ≥ 3, although there exists an analogous cohomogeneity one SO(n+1)-action on S n ×S n , the corresponding Cheeger deformation fails to produce so many positively curved planes. This is due to the fact that SO(n + 1), n ≥ 3, is not positively curved, cf. Corollary 2.2. As a result, this step in the construction of our metrics with sec θ > 0 only works on S n × S n if n = 2.
3. Second step: first-order local conformal deformation As seen above, the Cheeger deformed metrics g t , t > 0, have sec θ gt ≥ 0 and equality holds only for certain planes (of the form (2.8)) at ±∆S 2 . In order to get these planes to also have sec θ > 0, we now carry out a (local) first-order conformal deformation, inspired by results of Strake [11]. More precisely, choose g to be a Cheeger deformed metric g t for any t > 0 and consider the new 1-parameter family where h is some symmetric (0, 2)-tensor to be defined, and ε > 0 is small enough so that g s is still a Riemannian metric. Given the above geometry of the Cheeger deformed metric g, we will choose h such that p ∈ ±∆S 2 and sec g (σ 1 ) = sec g (σ 2 ) = 0.
The crucial observation that makes this possible is that these planes are never tangent to ±∆S 2 . Our choice will be such that h is supported only near ±∆S 2 and is pointwise proportional to g, justifying the terminology. We start by recalling the first variation of sec gs (σ), see Strake [11, Sec 3.a].
Proposition 3.1. Let (M, g) be a Riemannian manifold with sec g ≥ 0 and X, Y ∈ T p M be g-orthonormal vectors that span a g-flat plane σ ⊂ T p M . Consider a first-order variation g s = g + s h. Then the first variation of sec gs (σ) is given by Now, observe that if N ⊂ M is an embedded submanifold, the squared distance function ψ(p) = dist g (p, N ) 2 is smooth in a sufficiently small tubular neighborhood of N . The gradient of ψ at p vanishes if p ∈ N , and points in the outward radial direction if p ∈ N . The Hessian of ψ at p ∈ N is given by: Hess ψ (X, X) = 2g(X ⊥ , X ⊥ ) = 2 X ⊥ 2 g , X ∈ T p M, where X = X ⊤ +X ⊥ ∈ T p N ⊕(T p N ) ⊥ is the g-orthogonal decomposition in tangent and normal parts to N . Proposition 3.2. Consider the metrics g s on S 2 × S 2 , given by (3.1). There exists a smooth function φ : S 2 × S 2 → R, supported in a neighborhood of ±∆S 2 , such that if h = φ g, then (3.2) holds.
In order to conclude the proof of the Theorem in the Introduction, we quote the following elementary fact.

Remarks on the construction
4.1. First-order deformations and the Hopf conjecture. The above firstorder deformation g s works to get sec θ > 0 on all of M = S 2 × S 2 because the only points p ∈ M that have planes σ 1 , σ 2 ⊂ T p M with f (0, σ 1 , σ 2 ) = 0 are contained in the submanifolds ±∆S 2 , which admit a relatively compact neighborhood and where ∂f ∂s (0, σ 1 , σ 2 ) > 0. The same cannot be done for the sectional curvature because at every point there is a plane σ with sec g (σ) = 0. The only type of first-order deformation that would give sec gs > 0 would be one with d ds sec gs (σ) s=0 > 0 for all σ with sec(σ) = 0. It was proved by Strake [11,Prop. 4.3] that such a deformation does not exist on (S 2 × S 2 , g), due to the presence of totally geodesic flat tori.

4.2.
Other compact subsets. Notice that condition (3.2) does not contain any information on the compact subset K θ , which is the domain considered for the function f (s, σ 1 , σ 2 ) = 1 2 sec gs (σ 1 ) + sec gs (σ 2 ) . This means that the same argument above could be applied to obtain positivity of the average of sectional curvatures of planes that satisfy some other conditions codified in the form of a compact subset Replacing the domain of f by K, provided that K does not intersect the diagonal (i.e., the subset ∆ = {(p, σ, σ) : σ ⊂ T p M }), we get from Proposition 3.3 that f (s, σ 1 , σ 2 ) > 0 for s > 0 small enough and all (σ 1 , σ 2 ) ∈ K. We must require that K be away from the diagonal, otherwise f (0, σ 1 , σ 2 ) would also have zeros on points outside the singular orbits ±∆S 2 , and there is no firstorder variation that accounts for ∂f ∂s (0, σ 1 , σ 2 ) > 0 at all such points. Notice also that for every K with the required properties above, there exists θ > 0 such that K ⊂ K θ , so all other possibilities are accounted for by using the domains K θ .

4.3.
Ricci curvature. Since we know that g θ can be constructed arbitrarily C kclose (for any k ≥ 0) to the product metric g 0 , it automatically follows that such metrics can be chosen with positive Ricci curvature. Nevertheless, existence of metrics with Ric > 0 can be directly deduced from the existence of metrics with sec θ > 0 for arbitrarily small θ > 0, that converge to a limit metric as θ → 0, in the C k -topology, k ≥ 0, as we shall now prove. This abstract property is hence stronger 2 than Ric > 0 for compact manifolds (and, of course, weaker than sec > 0), regardless of the dimension of M . In this way, the Theorem in the Introduction shows that a natural interpolating condition between Ric > 0 and sec > 0 is satisfied on S 2 × S 2 .
Proposition 4.1. Let M be a compact n-dimensional manifold such that for every θ > 0 there exists a metric g θ with sec θ g θ > 0. Assume that there exists a metric g 0 on M such that g θ → g 0 in the C 0 -topology, as θ → 0. Then Ric g θ > 0 for θ > 0 sufficiently small; in particular, if g θ → g 0 also in the C 2 -topology, then Ric g 0 ≥ 0. 3 Proof. For any metric g on M , define: The above defines a positive number, that depends continuously on the metric g, such that if sec θ g > 0 for some 0 < θ ≤ θ g , then Ric g > 0. In fact, Ric g (v) > 0 for any direction v, since this is a sum of (n − 1) sectional curvatures whose pairwise average is positive, because sec θ g > 0. Given the continuous family G := {g θ : θ ∈ [0, 1]}, let θ * := min{θ g : g ∈ G}. It then follows that θ * > 0 and hence for any 0 < θ ≤ θ * , we have Ric g θ > 0.
Remark 4.2. An immediate consequence of the above is that, although S 1 × S 3 has a metric with sec ⊥ > 0, it cannot satisfy sec θ > 0 for all θ > 0 with metrics that do not diverge (otherwise, it would have a metric with Ric > 0).

4.4.
Negative sectional curvatures. Although the first step in our deformation preserves sec ≥ 0 from the product metric, the second step does not. In fact, for all θ > 0 there are planes σ in (S 2 × S 2 , g θ ) with sec g θ (σ) < 0. This follows from an obstruction to positive first-order deformations observed by Strake [11,Sec. 4]. Namely, all zero planes in the Cheeger deformed metric g = g t from Section 2 are tangent to a totally geodesic flat torus, see Müter [6,Satz 4.26]. Pick one such torus i : T 2 ֒→ (S 2 × S 2 , g), that intersects ±∆S 2 . The first-order deformation g s = g + s h on S 2 × S 2 induces a first-order deformation i * g s on T 2 . As observed by Strake [11,Lemma 4.1], since i(T 2 ) is totally geodesic, the first variation for the sectional curvature on T 2 coincides with the ambient variation: .
In fact, this follows directly by differentiating the Gauss equation of i(T 2 ) ⊂ (S 2 × S 2 , g s ) at s = 0. Let i(p) be a point where i(T 2 ) intersects ±∆S 2 . Then if σ = T p T 2 , di(σ) is such that sec θ g (di(σ)) = 0, so the construction in Section 3 is such that (4.2) is positive. By the Gauss-Bonnet Theorem, A(s) = T 2 sec i * gs vol i * gs = 2πχ(T 2 ) vanishes identically, so that Since the above integrand is positive at i(p) ∈ ±∆S 2 , it must also be negative somewhere. Together with (4.2) and the fact that i(T 2 ) ⊂ (S 2 × S 2 , g) is totally geodesic and flat, this means that g s , s > 0, must have some negative sectional curvature.

4.5.
Limiting case. Since θ > 0 can be chosen arbitrarily small for our construction, a natural question is what happens to g θ as θ → 0. By the above observations, the metric g s in (3.1) has some negative sectional curvature as soon as s > 0. This implies that as θ → 0, the interval 0 < s < s θ for which g s has sec θ > 0 shrinks until it disappears when θ = 0, since s θ must also go to zero. In fact, if there was a uniform lower bound 0 < s * ≤ s θ for all θ > 0, then the metrics g s , 0 < s < s * , would be such that the average sectional curvatures of any two distinct planes at the same point is positive, which in particular implies sec gs ≥ 0, 0 < s < s * , contradicting Subsection 4.4. This is also reflected by the fact that the domain K must be chosen compact in order for Proposition 3.3 to hold, hence one cannot simply take K to be the complement of the diagonal, see also Subsection 4.2.
4.6. Finite quotient. Our construction of metrics g θ with sec θ > 0 on S 2 × S 2 can be made invariant under the antipodal action of Z 2 ⊕ Z 2 , so that they induce metrics with sec θ > 0 on RP 2 ×RP 2 . In particular, RP 2 ×RP 2 admits metrics with sec ⊥ > 0. Since such metrics come from a local isometric covering (S 2 × S 2 , g θ ) → RP 2 × RP 2 , they also do not have sec ≥ 0 due to the above observations. The first step in the construction gives rise to metrics invariant under Z 2 ⊕ Z 2 , since it is a Cheeger deformation with respect to the SO(3)-action, which commutes with the Z 2 ⊕ Z 2 -action. As a side note, it was observed by Müter [6,Satz 4.27] that the induced metric on RP 2 × RP 2 at this stage is such that all its zero curvature planes are tangent to totally geodesic flat tori. The second and final step of our construction can also be made so that the resulting metrics are Z 2 ⊕ Z 2invariant. Namely, this property is equivalent to the function φ : S 2 × S 2 → R in Proposition 3.2 being Z 2 ⊕ Z 2 -invariant, which can be achieved by defining the cutoff functions χ ± in a symmetric way. 4.7. Biorthogonal pinching and isotropic curvature. The biorthogonal curvature of a manifold (M, g) is said to be (weakly) 1 /4-pinched if there exists a positive function δ such that δ 4 ≤ sec ⊥ g (σ) ≤ δ for all σ. This notion can be extended to any dimensions by requiring that the average of any two mutually orthogonal planes is 1 /4-pinched. As observed by Seaman [10], this pinching condition implies that the manifold has nonnegative isotropic curvature. It was later proved by Seaman [9], and independently by Micallef and Wang [5], that if an even dimensional compact orientable manifold (M, g) with b 2 (M ) = 0 has nonnegative isotropic curvature and positive biorthogonal curvature at one point, then (M, g) is Kähler, b 2 (M ) = 1 and M is simply-connected. Consequently, our metrics of positive biorthogonal curvature on S 2 × S 2 cannot satisfy the biorthogonal 1 /4-pinching condition, since b 2 (S 2 ×S 2 ) = 2. Moreover, it also follows that such metrics do not have nonnegative isotropic curvature. 4.8. Modified Yamabe invariant. As observed by Costa [2], the minimum of the biorthogonal curvature at each point is a modified scalar curvature, with correspond- where the supremum is taken over all metrics g on M . It is observed that if a metric g ∈ [g 0 ] is conformal to the standard product metric on S 2 × S 2 , then Y ⊥ 1 (S 2 × S 2 , g) ≤ 0. In particular, no metric conformal to g 0 can have positive biorthogonal curvature. However, as a direct consequence of the Theorem in the Introduction, we have that Y ⊥ 1 (S 2 × S 2 ) > 0, see [2, Thm 3 (1)].

Other 4-manifolds with positive biorthogonal curvature
In light of the above construction, it is natural to inquire how restrictive the positive biorthogonal curvature condition is on 4-manifolds. As noted before, sec ⊥ > 0 automatically implies scal > 0, however it does not necessarily guarantee Ric > 0 (cf. Subsection 4.3). On the one hand, this means that sec ⊥ > 0 imposes rather restrictive topological conditions on 4-manifolds, e.g., vanishing of all the Seiberg-Witten invariants. On the other hand, such topological restrictions are by far not as strong as the ones implied by sec > 0, or even Ric > 0. For instance, sec ⊥ > 0 does not guarantee finiteness of the fundamental group, as illustrated by S 1 × S 3 with the standard product metric.
This suggests that more subtle Synge-type obstructions should also not detect sec ⊥ > 0. In fact, RP 2 × RP 2 admits metrics with sec ⊥ > 0, as discussed in Subsection 4.6. Another relevant example in this context is the nontrivial S 2bundle over S 2 , which is diffeomorphic to the connected sum CP 2 #CP 2 , where CP 2 denotes the manifold CP 2 with the opposite orientation from the one induced by its complex structure. We conclude by showing that this manifold also has sec ⊥ > 0, using arguments similar to the S 2 × S 2 case. It is important to observe that it is also currently unknown whether CP 2 #CP 2 admits a metric with sec > 0.
Proof. Similarly to the S 2 × S 2 case, CP 2 #CP 2 admits cohomogeneity one metrics with sec ≥ 0 invariant under the action of SU (2). In order to describe this initial metric, notice that the normal bundle of the usual embedding CP 1 ֒→ CP 2 can be identified with the vector bundle (S 3 × R 2 )/S 1 over CP 1 = S 3 /S 1 . Take two copies of the disk bundles given as tubular neighborhoods of the zero section of this vector bundle. Each one of them is the complement of a metric ball on CP 2 , that is deleted to carry out the connected sum. It is then easy to see that (5.1) CP 2 #CP 2 = (S 3 × S 2 )/S 1 , by gluing along the boundary these two disk bundles. Here, the S 1 -action on S 3 ×S 2 is a product action, on S 3 via the Hopf action and on S 2 by rotation. The standard product metric on S 3 × S 2 then induces a submersion metric 4 g 0 with nonnegative curvature on CP 2 #CP 2 . The cohomogeneity one action of SU(2) comes from the left-translation action of SU(2) = S 3 on the first factor of S 3 × S 2 , which induces an action on the quotient since it commutes with the above circle action. Both singular orbits of this cohomogeneity one action on (CP 2 #CP 2 , g 0 ) are 2-spheres, that correspond to the zero section of the disk bundles that were glued together. The zero curvature planes are images via the submersion of mixed planes on S 3 × S 2 that are spanned by vectors orthogonal to the circle action field, cf. Müter [6,Satz 4.29]. Thus, there is a circle's worth of zero curvature planes at every point, but any such planes tangent to a regular point must intersect. At singular points, there are zero curvature planes orthogonal to each other, but all of them are not tangent to the singular orbit. This scenario is totally analogous to the Cheeger deformed metrics on S 2 × S 2 , i.e., metrics obtained after the first step of our deformation (Proposition 2.3). More precisely, sec g0 ≥ 0, and sec ⊥ g0 > 0 on all regular points. Since the only points with zero biorthogonal curvature are along the singular orbits and all zero curvature planes are not tangent to these orbits, a first-order local conformal deformation using squared distance functions to the singular orbits, totally analogous to the one in Proposition 3.2, gives the desired metrics with sec ⊥ > 0 on CP 2 #CP 2 as a consequence of Proposition 3.3.
Remark 5.2. As shown above, in order to construct metrics with sec ⊥ > 0 on CP 2 #CP 2 , one can skip the first step in the construction for S 2 × S 2 . This is an important observation, because differently from S 2 × S 2 with the standard product metric, the Cheeger deformation of (CP 2 #CP 2 , g 0 ) with respect to the SU(2)-action does not destroy any zero curvature planes, see Müter [6,Satz 4.29].
Remark 5.3. Since there is a circle's worth of zero curvature planes at every point on (CP 2 #CP 2 , g 0 ), although the first-order local conformal deformation produces sec ⊥ > 0, it cannot be used to produce metrics with sec θ > 0 for every θ > 0. 4 We remark that this construction is very similar to the original gluing construction of nonnegatively curved metrics on the connected sum of two compact rank one symmetric spaces, which is due to Cheeger [1] and was later greatly generalized by Grove and Ziller [3]. The only subtle difference is that (CP 2 #CP 2 , g 0 ) has only one orbit that is a totally geodesic hypersurface (the boundary of the disk bundles glued together), while in the gluing construction the metric locally splits as a product near this hypersurface.