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Assume that $M$ is a 3-dimensional complete open Riemannian manifold homeomorphic to a open solid torus. If $M$ has no conjugate point, then it has a constant sectional curvature.

Is it right? Thank you in advance.

(This question is arise when I consider the following : Conjugate points in ${\rm Sl}(2,\mathbb{R})$)

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closed as off-topic by Hee Kwon Lee, Chris Godsil, Stefan Kohl, Jan-Christoph Schlage-Puchta, Neil Hoffman Mar 23 '17 at 16:12

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    $\begingroup$ The product of a unit disk in the hyperbolic plane, by the circle gices you a solid torus but it does not have constant sectional curvature. Anyway this should be asked at MSE. $\endgroup$ – Mikhail Katz Mar 22 '17 at 9:29