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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 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.
In these coordinates the first and second fundamental forms are written in a way that makes clear the Gaussian curvature is −1 for any solution of the sine-Gordon equations. Then any solution to the sine-Gordon equation can be used to specify a first and second fundamental form which satisfy the Gauss–Codazzi equations.
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.
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
The Gaussian curvature of a surface is given by = =, where L, M, and N are the coefficients of the second fundamental form. Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that K is in fact an intrinsic invariant of the 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 ...