Complete curvature

total curvature

The complete curvature at a point on a surface $\Phi$ in the Euclidean space $\mathbf R^3$ is the scalar quantity $K$ equal to the product of the principal (normal) curvatures $k_1$ and $k_2$ calculated at the point on the surface: $K=k_1k_2$; it is usually called the Gaussian curvature of the surface (cf. Principal curvature). The notion of Gaussian curvature may be extended to a hypersurface in the Euclidean space $\mathbf R^{n+1}$, $n>2$. In that case it is the quantity $K=k_1\dots k_n$, where $k_i$ is the principal curvature at a point on the hypersurface in the principal direction $i$.

The Gaussian curvature at a point on a two-dimensional surface in a three-dimensional Riemannian space is equal to the difference between the interior curvature (the sectional curvature of the two-dimensional surface) and the exterior curvature (the sectional curvature of the ambient space in the direction of the tangent plane to the surface at that point).

The total curvature of a region $D$ on a surface $\Phi$ in the Euclidean space $\mathbf R^3$ is the quantity $\int\int_DK\,d\sigma$, where $K$ is the Gaussian curvature of the surface at a point and $d\sigma$ is the area element of the surface. Similarly, one defines the total curvature of a region in a certain Riemannian manifold, where by $K$ one understands the sectional curvature of the manifold calculated at the points on the manifold in the directions of the tangential planes, while the integration is taken over the area (measure) of the region in the manifold.

Comments

The phrase "complete curvature" is not used in the Western literature. The Gaussian curvature for a hypersurface in $\mathbf R^{n+1}$ is commonly called the Lipschitz–Killing curvature. The latter has also been defined in higher codimensions for a prescribed normal direction. The Gaussian curvature is also called Gauss–Kronecker curvature.

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1969) pp. Chapt. 7; Chapt. 5