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For hypersurfaces in higher-dimensional Euclidean spaces, the principal curvatures may be defined in a directly analogous fashion. The principal curvatures are the eigenvalues of the matrix of the second fundamental form (,) in an orthonormal basis of the tangent space. The principal directions are the corresponding eigenvectors.
The two principal curvatures at p are the maximum and minimum possible values of the curvature of this plane curve at p, as the plane under consideration rotates around the normal line. The following summarizes the calculation of the above quantities relative to a Monge patch f(u, v) = (u, v, h(u, v)).
The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least. The theorem is named for Leonhard Euler who proved the theorem in . More precisely, let M be a surface in three-dimensional Euclidean space, and p a point on M.
The Gaussian curvature is the product of the two principal curvatures Κ = κ 1 κ 2. The sign of the Gaussian curvature can be used to characterise the surface. If both principal curvatures are of the same sign: κ 1 κ 2 > 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface ...
An example of negatively curved space is hyperbolic geometry (see also: non-positive curvature). A space or space-time with zero curvature is called flat. For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat spacetime. There are other examples of flat geometries in both settings, though.
For general Riemannian manifolds one has to add the curvature of ambient space; if N is a manifold embedded in a Riemannian manifold (M,g) then the curvature tensor R N of N with induced metric can be expressed using the second fundamental form and R M, the curvature tensor of M:
In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem. Later in the 1700s, the new French school led by Gaspard Monge began to make contributions to differential geometry.
Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in the direction ...