Euclidean hypersurface normals, shape operators and curvature

For $n\ge 2$, a smooth embedded hypersurface $S\subset {\mathbb{R}}^n$ has the embedded-submanifold meaning of Embedded submanifolds and slice charts. If $X:U\subset {\mathbb{R}}^{n-1}\to S$ is a smooth local parametrization of rank $n-1$, define $T_{X(y)}S=\mathrm{im}DX(y)$ with its Euclidean inner product. A smooth local unit normal is a smooth map $\nu :V\to {\mathbb{R}}^n$ on a relatively open subset $V\subseteq S$ with $\mathrm{\mid }\nu \mathrm{\mid }=1$ and $\nu \perp TS$. Define the Euclidean shape operator by $S_{\nu }v=-d{\nu }_p(v)$ on $T_{p}S$, and the extrinsic Gaussian (Gauss–Kronecker) curvature by $K_{\nu }(p)=\det S_{\nu }(p)$. Here $d{\nu }_p(DX(y)u)=D(\nu \circ X)(y)u$. Smooth functions and compact supports on $S$ use its subspace topology and these local parametrizations. Nonvanishing curvature means $K_{\nu }\mathrm{\ne }0$ for either choice of local unit normal at each point. The graph and localization lemma Smooth Euclidean hypersurface graphs and compact localization ↗ proves that these definitions are independent of parametrization, that the derivative takes values in $T_{p}S$, and that changing the unit normal only changes the sign of the shape operator. This is the Euclidean specialization of the usual Weingarten definition; the equivalence is proved there, without requiring the later Riemannian theory.