2

I know that a compact surface $S \subset \mathbb{R}^3$ of positive Gaussian curvature is homeomorphic to a sphere from Gauss-Bonnet's theorem. This led me to ask if there is a characterization for compact and orientable hypersurfaces with positive sectional curvature with possibly some additional hypothesis. I think maybe need some additional hypothesis because if there was a characterization without more hypothesis, then the Hopf's conjecture would be solved.

Thanks in advance!

George
  • 3,817
  • 2
  • 15
  • 36
  • What do you mean by "hypersurface"? – Kajelad Aug 25 '21 at 00:03
  • A Riemannian manifold immersed in another Riemannian manifold such that the codimension is $1$. – George Aug 25 '21 at 00:46
  • Isometrically immersed, or just immersed? Regardless, in dimension $\ge 3$ there are lots of nonhomeomorphic compact orientable manifolds of positive sectional curvature, so you won't be able to get anything as nice as the 2-dimensional result. – Kajelad Aug 25 '21 at 01:01
  • Isometrically immersed. Do you have a reference for the case when dimension $\geq 3$? – George Aug 25 '21 at 01:12
  • You might want to look into Lens spaces and spherical 3-manifolds, which inherit orientations and metrics of constant positive sectional curvature from $S^3$. Similar examples exist in higher dimensions, and all of these can appear as immersed hypersurfaces of suitable ambient manifolds. – Kajelad Aug 25 '21 at 05:08

0 Answers0