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Good afternoon,

there is an example of a Riemannian metric on S^2 \times S^2 of nonnegative sectional curvature that is not a product metric. I know there is one; however, I cannot find a specific reference. Any suggestions?

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The diffeomorphism group acts on Riemannian metrics... –  Tim Perutz Aug 10 '10 at 16:04
    
@Tim, you seem to be using a notion of the product metric that is not invariant under isometry. Formally, a product metric is the one that can be pulled by a self-diffeomorphism of $S^\times S^2$ to a metric of the form $(S^2, g_1)\times (S^2, g_2)$. –  Igor Belegradek Aug 10 '10 at 21:36
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3 Answers

up vote 5 down vote accepted

A discussion of nonnegatively curved metrics on $S^2\times S^2$ can be found in the survey by B. Wilking, see page 26 and last paragraph on page 25. In particular, there is a one parametric family of metrics of nonnegative curvature on $S^2\times S^2$ which are not moved by any diffeomorphism to a product metric.

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Good! - I did not expect to find it in that survey. –  Sergey O Aug 11 '10 at 14:43
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One has to be careful with perturbations of metrics of nonnegative curvature, because that may introduce negative curvature.

Here's another approach which gives you a nonnegatively curved metric.

Start with $S^3\times S^2$ with the product of round metrics. (Note that the round metric on S^3 is its biinvariant metric).

Consider the $S^1$ action on $S^3 \times S^2$ where it acts as the Hopf action on $S^3$ and simultaneously rotates the $S^2$ factor $2k$ times around for some integer $k$.

To make it explicit, thinking of $S^2$ as the unit sphere in the imaginary quaternions, the action can be described as $z*(p,q) = (zp, z^k q \overline{z}^k)$.

The action is clearly free and the quotient is diffeomorphic to $S^2 \times S^2$. Since the circle is acting isometrically, there is an induced submersion metric on $S^2\times S^2$. By the O'Neill formulas for a submersion, this metric has nonnegative curvature. When k = 0, one gets the usual product of round metrics, but when $k\neq 0$ the metric is, in general, not a product.

Edit I'm now no longer certain that for $k\neq 0$, the metric is not a product. I am confident that for $k\neq 0$, the metric is not a product of round metrics, but I don't see any reason they can't be a product of two nonnegatively curved metrics.

However, here is an example (sorry for doubling the length of my post!): Let $G = S^3\times S^3$. Let $g_0$ denote a biinvariant metric on $G$. Writing $\mathfrak{g}$ for the Lie algebra of $G$, set $\mathfrak{p}$ to be the Lie algebra of the diagonal $S^3$ and choose $\mathfrak{q}$ to be $g_0$-orthogonal to $\mathfrak{p}$.

Fix a positive real number $t$ and define a new inner product $g_1 = g_0|_{\mathfrak{q}} + \frac{t}{t+1}g_0|_{\mathfrak{p}}$ and left translate it around $G$ to give a left invariant, right $\Delta S^3$ invariant metric. Such a metric is called a Cheeger deformation of $g_0$ and it is known that $g_1$ has nonnegative sectional curvature.

Give $G\times G$ the product metric $g_0+g_1$ and consider the space $\Delta S^3 \backslash G\times G/ T^2$ where the $T^2$ acts on $G\times G$ as $(z,w)*(p,q,r,s) = (pz^{-1}, q, rw^{-1},sw^{-1})$.

(The map $G\times G\rightarrow G$ sending $(p,q,r,s)$ to $(r^{-1}p, s^{-1}q)$, or something like it if I've made a mistake, induces a diffeomorphism between $\Delta S^3\backslash G\times G/T^2$ and $G/T^2 = S^2\times S^2$, where the $G/T^2$ is referring to the action of $T^2$ on $G$ spelled out before the edit with k=1).

As above, there is an induced submersion metric of nonnegative sectional curvature by the O'Neill formulas. Finally, to prove that this is NOT a product metric, one observes that at generic points, there is a unique plane with 0 sectional curvature, while for a product metric, there should be infinitely many planes of 0 curvature.

The observation comes from

P.Müter, Krümmungserhöhende Deformationen mittels Gruppenaktionen, Ph.D. thesis, University of Münster, 1987.

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Can these metrics also be obtained by restricting the Fubini-Study metric on projective space to an embedding of the Hirzebruch surfaces $\mathbb{F}_{2k}$? –  damiano Aug 10 '10 at 16:02
    
I have no idea, sorry. Perhaps someone else can weigh in. –  Jason DeVito Aug 10 '10 at 16:03
    
As far as I understand, Sergey know some examples --- he needs a reference. –  Anton Petrunin Aug 10 '10 at 19:09
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@Anton, I interpreted "I know there is one" to mean "I know there SHOULD be an example, I just don't know of an explicit one". Since I don't know any references off the top of my head, I did the only thing I could. If you feel I should modify/delete the answer, just let me know. –  Jason DeVito Aug 10 '10 at 19:54
    
It actually takes some work to find a nonnegatively metric on $S^2\times S^2$ that is not isometric to a product metric, e.g. one probably has to show that there aren't too many planes of zero curvature. So I agree with Jason that Sergey probably felt there should be an example but did not know any. –  Igor Belegradek Aug 10 '10 at 21:23
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I strongly suspect that being a product metric is a closed condition, whereas having nonnegative sectional curvature is an open one, so a generic perturbation of the product metric should do the job.

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Having non-negative sectional curvature is not an open condition (just like being a non-negative real number isn't). Note also that the imbedded S^1xS^1 is flat. –  André Henriques Aug 10 '10 at 16:12
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Moreover, it's a longstanding conjecture of Hopf that $S^2\times S^2$ admits no Riemannian metric of strictly positive sectional curvature. –  Victor Protsak Aug 10 '10 at 16:45
    
Yes, sure. I had mistakenly taken strictly positive for nonegative in my answer, but as pointed out by Andre' and Victor this was a total nonsense when referred to $S^2\times S^2$. –  domenico fiorenza Aug 10 '10 at 17:34
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