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For example, a sphere of radius r has Gaussian curvature 1 / r 2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
A sphere of radius R has constant Gaussian curvature which is equal to 1/R 2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling. Conversely, the surface of a sphere cannot be unfolded onto a flat plane without distorting the distances.
The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a ...
The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically. Its mean curvature is not zero, though; hence extrinsically it is different from a plane.
This immersion cannot be because a small oscillating sphere would provide a large lower bound for the principal curvatures, and therefore for the Gauss curvature of the immersed sphere, but on the other hand if the immersion is this has to be equal to 1 everywhere, the Gauss curvature of the standard , by Gauss' Theorema Egregium.
Chapters 15 through 17 concern the Gauss curvature flow, a different way of generalizing the curve-shortening flow to higher dimensions using Gaussian curvature in place of mean curvature. Although Gaussian curvature is intrinsic, unlike mean curvature, the Gauss curvature flow is extrinsic, because it involves the motion of an embedded surface ...
The Minkowski problem, despite its clear geometric origin, is found to have its appearance in many places. The problem of radiolocation is easily reduced to the Minkowski problem in Euclidean 3-space: restoration of convex shape over the given Gauss surface curvature. The inverse problem of the short-wave diffraction is reduced to the Minkowski ...
The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...