# Differential Geometry: Gaussian curvature question

• Jun 22nd 2010, 02:04 PM
jonathan122
Differential Geometry: Gaussian curvature question
Hi, I was wondering if anyone could help me - given the closed unit disc, with its boundary circle "glued" to the plane, I'm trying to prove that it is impossible to put a Riemannian metric on this space such that the Gaussian curvature is always non-negative, but I'm not really sure how to go about it. From the Gauss-Bonnet theorem, we have that the total curvature of the space is 2(pi), but that doesn't seem to help much...

(I think part of the problem is that I'm not really convinced that what I am trying to prove is true, even with the standard metric, but I've been assured that it is!)

Thanks very much for your help,

Jonathan.
• Jun 22nd 2010, 05:53 PM
xxp9
what is your space? glue the boundary of a unit disk to the whole plane will not give rise to a manifold.
• Jun 23rd 2010, 02:01 PM
jonathan122
Quote:

Originally Posted by xxp9
what is your space? glue the boundary of a unit disk to the whole plane will not give rise to a manifold.

Sorry! The space is a closed unit disc, but deformed so that it's boundary lies in a plane as an annulus.