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?