Hypersurface is not curved, normal vector

Question

Let $S$ be a hypersurface in $\mathbb{R}^n$. Is there a simple way to say that $S$ is flat by describing the normal vectors on $S$?

Like $S$ is flat if the normal vectors on $S$ are all identical..

Answer 1

It depends on whether you mean "flat" in the intrinsic or extrinsic sense. If you mean it in the extrinsic sense (i.e., $S$ is contained in an affine hyperplane), then the necessary and sufficient condition is that all of the unit normals are identical (assuming $S$ is connected). But if you mean it in the intrinsic sense of being locally isometric to $\mathbb R^{n-1}$ with its Euclidean metric, then the necessary and sufficient condition is that the curvature tensor is identically zero. The curvature tensor can be computed from the second fundamental form (via the Gauss equation), which in turn can be computed from the derivatives of the unit normal vector (via the Weingarten equation). So yes, you can determine flatness from the normal vector, but it's not straightforward.