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A Dirac measure δ a supported at any point a has zero curvature. If μ is any measure whose support is contained within a Euclidean line L, then μ has zero curvature. For example, one-dimensional Lebesgue measure on any line (or line segment) has zero curvature. The Lebesgue measure defined on all of R 2 has infinite curvature.
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 ...
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "towards" the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. Mean Curvature may ...
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. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.
The mean curvature flow extremalizes surface area, and minimal surfaces are the critical points for the mean curvature flow; minima solve the isoperimetric problem. For manifolds embedded in a Kähler–Einstein manifold , if the surface is a Lagrangian submanifold , the mean curvature flow is of Lagrangian type, so the surface evolves within ...
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.
Minimal surface curvature planes. On a minimal surface, the curvature along the principal curvature planes are equal and opposite at every point. This makes the mean curvature zero. Mean curvature definition: A surface is minimal if and only if its mean curvature is equal to zero at all points.