Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M^m\subseteq {\bar{M}}^{m+1}$ be a hypersurface of positive dimension $m\ge 1$, equipped with a supplied smooth unit normal field $\nu$. At each $p\in M$, the shape operator $S_{\nu ,p}:T_{p}M\to T_{p}M$ is self-adjoint by Weingarten equation and adjointness of the shape operator. Its real eigenvalues, counted with algebraic multiplicity and regarded as an unordered multiset

$\{{\kappa }_1(p),\dots ,{\kappa }_m(p)\},$

are the principal curvatures at $p$; the corresponding eigenspaces are the principal directions. The real spectral theorem supplies an orthonormal eigenbasis at each fixed point, but no smooth ordering of the eigenvalues and no global eigenframe is asserted.

The extrinsic Gaussian curvature (also called Gauss–Kronecker curvature in higher dimension) and the scalar mean curvature with respect to $\nu$ are

$K_{\mathrm{e}\mathrm{x}\mathrm{t}}:=\det S_{\nu }={\prod }_{i=1}^m{\kappa }_i,H_{\nu }:=\frac{1}{m}\mathrm{tr}{S}_{\nu }=\frac{1}{m}{\sum }_{i=1}^m{\kappa }_i\mathrm{.}$

Trace and determinant are basis independent, so these functions do not depend on an ordering or eigenbasis. They are smooth even where individual ordered eigenvalue functions need not be smooth, because the matrix coefficients of $S_{\nu }$ are smooth and trace and determinant are polynomial in those coefficients.

Normal-linearity in Shape operator gives $S_{-\nu }=-S_{\nu }$. Therefore reversing the normal negates every principal curvature and $H_{\nu }$, while

$K_{\mathrm{e}\mathrm{x}\mathrm{t}}(-\nu )=\det (-S_{\nu })=(-1{)}^m{K}_{\mathrm{e}\mathrm{x}\mathrm{t}}(\nu )\mathrm{.}$

The assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited exactly through the smooth normal-bundle and shape-operator construction. Applying the finite-dimensional spectral theorem at a supplied point makes no additional family choice. The definitions apply to an empty hypersurface of fixed positive dimension, in dimension one, and at boundary points. Dimension zero is excluded explicitly because the averaging factor $1\mathrm{/}m$ is undefined, and degenerate metrics are outside the Riemannian hypothesis.