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## Main Question or Discussion Point

Hi I know that the sphere has positive curvature everywhere, and the torus has positive an negative curvature. Is there a space homeomorphic to the sphere such that this space has negative curvature everywhere?

And, is there a space homeomorphic to the torus such that the curvature is always positive (negative)?

Thanks

And, is there a space homeomorphic to the torus such that the curvature is always positive (negative)?

Thanks