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What are examples of non-compact complete Riemannian manifolds with everywhere positive curvature?

Can you give examples of 2-dimensional surfaces in $\mathbb{R}^3$ with this property?


Note that by Bonnet-Myers theorem, if the curvature is bounded from below by a positive number, then the manifold is compact, so the curvature must decrease to 0.

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    $\begingroup$ Think about bowls $\endgroup$
    – Deane
    Feb 15, 2021 at 19:18

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Think about a paraboloid like $z=x^2+y^2$. The idea is that the curvature is positive but tends towards zero as we take $x^2+y^2=r^2$ to infinity. This is a good calculation to work out as an example.

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  • $\begingroup$ Is this true also for the two sheeted hyperboloid? Is there some condition on a curve that will make its revolution surface of positive curvature? $\endgroup$
    – Ur Ya'ar
    Feb 16, 2021 at 17:57
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    $\begingroup$ I never really remember these formulas by heart, but you can definitely find them for instance here: math.stackexchange.com/questions/529499/…) The moral of the story is that there are explicit formulas for Gaussian Curvatures for surfaces of revolution, and so these will allow you to understand explicitly the Gaussian curvature of (for instance) the paraboloid in terms of $r^2$. $\endgroup$ Feb 16, 2021 at 20:04

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