The flat torus cannot be smoothly embedded in $\mathbb{R}^3$ isometrically. Since the torus is compact, any embedded torus lies in the interior of some sphere centered at the origin. Decrease the radius of the sphere until it just touches the torus for the first time. At the point of contact, the curvature of the torus agrees with that of the sphere, and is hence positive.
The flat torus cannot be embedded in $\mathbb{R}^3$ isometrically. Since the torus is compact, any embedded torus lies in the interior of some sphere centered at the origin. Decrease the radius of the sphere until it just touches the torus for the first time. At the point of contact, the curvature of the torus agrees with that of the sphere, and is hence positive.