Surfaces of positive curvature in 3 space

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?