2
$\begingroup$

As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature.

But Hopf's conjecture is unsolved, i.e., the conjecture: $S^2\times S^2$ has a positive curvature.

So I think that the quasi-positive curvature-condition can be weakened by conditions : non-negative curvature and non-flatness.

Here my concrete question is : Is there a simply connected compact simple Lie group which does not have a metric with a positive curvature and has a metric with a non-negative curvature ?

[Definitions]

Here we say that a Riemannian manifold $M$ has positive (resp. non-negative) curvature if all sectional curvatures are positive (resp. non-negative) at all points of $M$ And a manifold with quasi-positive curvature is a manifold with a non-negative curvature and a point at which all tangent 2-planes have positive curvature.

$\endgroup$
  • 5
    $\begingroup$ Could you please clarify the question? (I suppose that the last sentence of your post is in doubt and you would like to know if it's a true statement?) $\endgroup$ – SashaKolpakov Oct 10 '13 at 4:23
  • $\begingroup$ Okay I will clarify my question. $\endgroup$ – Hee Kwon Lee Oct 10 '13 at 8:17
  • $\begingroup$ what you call "to be positively curved" should be termed "to admit a positively curved Riemannian metric" $\endgroup$ – YCor Oct 10 '13 at 8:35
  • $\begingroup$ That's right. And I eddited. $\endgroup$ – Hee Kwon Lee Oct 10 '13 at 8:42
  • 4
    $\begingroup$ What you are asking about is a well-known open problem. See the survey by Ziller: math.upenn.edu/~wziller/papers/SurveyMexico.pdf for most recent summary of obstructions and examples. Also the biinvariant metric on any simply-connected compact Lie group has nonnegative curvature (this is an easy exercise in Do Carmo's "Riemannian geometry''). $\endgroup$ – Igor Belegradek Oct 10 '13 at 11:32
1
$\begingroup$

For high rank Lie group, this is in fact the generalized Hopf conjecture which is still open.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.