Consider a closed manifold $M $ embedded in $\mathbb{R}^3$. Assume $M$ is diffeomorphic to a torus. Then I want to know is it possible to have $K(p)=0$ for every point $p\in M$? Here $K$ is Gaussian curvature.
First what I know, is diffeomorphic does not guarantee the Gaussian curvature of $M$ and Gaussian curvature of the torus are the same. How one can prove or disprove such a statement?
A smooth isometric embedding ($C^\infty$) of the flat torus in $\mathbb{R}^3$ is not possible, but according to the Nash-embedding-theorem it is possible to have a $C^1$ map. See this Math overflow question .