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How do I go about proving this result?

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    Please make the body of the question self contained: it should make sense even in the absence of the title. Have you seen a book whose first sentence *is* its title?2012-04-25

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If the Gauss map is a diffeomorphism, then the derivative of the Gauss map is invertible everywhere on $M$, the original surface. The Gaussian curvature is the determinant of this derivative. So, is has the same sign everywhere. But the image of the Gauss map is the sphere, then the original surface is compact, because you are assuming the Gauss map a diffeomorphism.. But in this case, it has at least one point of positive curvature, then the curvature is positive everywhere.

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    Got it now. Thanks.2012-04-25