Possible isometries of a positively curved $S^2\times S^2$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:29:57Z http://mathoverflow.net/feeds/question/90933 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90933/possible-isometries-of-a-positively-curved-s2-times-s2 Possible isometries of a positively curved $S^2\times S^2$ Renato G Bettiol 2012-03-11T21:04:38Z 2013-05-01T16:27:38Z <p>Just to put things in perspective, recall that the Hopf Conjecture asks whether $S^2\times S^2$ admits a metric of positive sectional curvature. By the work of <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jdg/1214443064" rel="nofollow">Hsiang-Kleiner</a>, it is known that, if $S^2\times S^2$ admits such a metric, then its isometry group cannot contain a circle, and is hence finite.</p> <blockquote> <p>Q: If $S^2\times S^2$ admits a metric with $sec>0$, what is known about its isometry group $G$?</p> </blockquote> <hr> <p>The only results I know of are:</p> <p>0) [Edit suggested by Misha] The diagonal antipodal action of $\mathbb Z_2$ on $S^2\times S^2$, i.e., $\pm 1\cdot(x,y)=(\pm x,\pm y)$, cannot be isometric if $S^2\times S^2$ is equipped with a metric of positive curvature. By Weinstein's Thm, an orientation-preserving isometry of an even-dimensional positively curved manifold has a fixed point (and the antipodal map does not). Equivalently, it would induce a positively curved metric on the $2$-fold orientable cover of $\mathbb R P^2\times \mathbb R P^2$, hence on $\mathbb R P^2\times \mathbb R P^2$, but this contradicts Synge's Thm.</p> <p>1) From <a href="http://wwwmath.uni-muenster.de/u/weckerm/sfb/about/publ/wilking3.ps" rel="nofollow">Wilking's thesis</a> (Prop 4.2), any simple subgroup of $G$ is either cyclic or isomorphic to a group in a finite list $F_1,\dots,F_k$ of simple groups. (This is actually true for any finitely generated subgroups of isometries of a manifold with $Ricâ‰¥0$).</p> <p>2) From <a href="http://arxiv.org/abs/math/0504504" rel="nofollow">Fang's paper</a> (Thm 1.2), $G$ cannot have a subgroup of sufficiently large odd order (but this lower bound is huge, since it is estimated with Gromov's universal constant for the total Betti number).</p> <hr> <p>Apart from these, are there other known restrictions on what $G$ can be like?</p> http://mathoverflow.net/questions/90933/possible-isometries-of-a-positively-curved-s2-times-s2/129325#129325 Answer by Luis Guijarro for Possible isometries of a positively curved $S^2\times S^2$ Luis Guijarro 2013-05-01T16:27:38Z 2013-05-01T16:27:38Z <p>You should look at Andrew Hick's thesis, </p> <p>Andrew Hicks, <em>Group actions and the topology of nonnegatively curved $4$-manifolds</em>, Illinois Journal of Mathematics. Volume 41, Issue 3 (1997), 421-437. </p> <p>A corollary to his Theorem 2 shows that for a positively curved metric on $S^2\times S^2$ with $\delta$ pinching, the size of the isometry group is bounded above by a constant related to the pinching. </p> <p>There are a few other interesting results about symmetries of nonnegatively curved 4-manifolds there. </p>