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Start with a smooth closed surface of positive Gauss curvature in 3 space. The differential of the Gauss mapping induces a new metric on the surface by letting the inner product of two tangent vectors be the inner product of their images in the tangent space of the 2 sphere.

For the sphere and nearly spherical surfaces the new metric will still have positive Gauss curvature.

  • What is an example where the Gauss curvature is not everywhere positive?

  • what is an example where the new metric can not be embedded in 3 space - if any?

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Note that you are using the pullback metric from the sphere. Thus the new curvature will be identically one.

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  • $\begingroup$ Thanks. I did not know the Nirenberg result. I was aware that positive Gauss curvature is required for the induced metric but its seems possible that the induced metric may have regions of negative or zero curvature. $\endgroup$ – lavinia Dec 16 '13 at 12:14
  • $\begingroup$ @lavinia: After some thought the answer should be way simpler. I have edited the answer. $\endgroup$ – user99914 Dec 18 '13 at 7:49

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