What does "flat hypersurface" mean?

Question

If $S$ is a flat hypersurface with boundary in $\mathbb{R}^n$, what does it mean?

Is it just a simple open domain (found in most PDE contexts)?

Answer 1

It is an $n-1$-dimensional submanifold with boundary, whose Gaussian curvature is $0.$ An example is a standard Mobius strip, or the same rectangle of paper glued into a cylinder (this, for $n=3.$)

Answer 2

The flat hypersurface in $E_{n}$ is a hypersurface with at least one of the principle curvatures(curvatures in principle directions) is zero. For example the cylinder is flat one since one of the principle curvatures is zero while the sphere has two none zero principle curvatures.