Surface can deviate from flatness {curvature, tensor}.
local
Curvature is at point and is local property.
intrinsic
Curvature is intrinsic to surface and does not depend on outside reference points.
curve
For curves, curvature at point is curvature-radius reciprocal.
surface
For surface points, curvature R is product of reciprocals of maximum curvature radius R1 and minimum curvature radius R2: R = (1 / R1) * (1 / R2). Rotating plane around normal to surface can find both.
surface: triangle
Triangles are three geodesics. Curvature is triangle-angle sum minus pi radians, all divided by triangle area: r = (angle sum - pi) / area.
surface: sphere
Surface curvature is area by surface projection onto a sphere, divided by surface area: r = (projection area) / area.
surface: normals
Curvature is solid angle in radians by normals to surface over surface region, divided by surface area.
surface: volume
Volume can also find curvature.
surface: hexagon
To measure curvature around point, use regular hexagon around the point and measure angles.
bending
If surface bends without stretching, curvature stays constant, because one radius increases and other radius decreases proportionally.
space-time: curvature tensor
In space-time, geodesic deviations, measured along each dimension, are metric second derivative and are second-order differential forms. The result is fourth-order tensor whose matrix coefficients have one covariant and three contravariant indices. For four-dimensional space-time, tensor has 20 independent terms. Metric coefficients are potentials.
Tensor is skew-symmetric and cyclic symmetric and is commutative, with diagonal terms equal zero.
space-time: Weyl tensor
Tensors {Weyl tensor} can measure gravity-field tidal distortion.
space-time: Riemann curvature tensor
At points, total space curvature depends on Ricci tensors and Weyl tensors. Curved space can be Euclidean space {Riemannian manifold} locally. Curved space-time can be Minkowski space {Lorentzian manifold} {pseudo-Riemannian manifold} {semi-Riemannian manifold} locally.
space-time: distance curvature
Second-order tensor {distance curvature} can derive from vector curvature, by contracting matrix coefficients.
space-time: scalar curvature
Tensor trace is scalar invariant. First-order tensors {scalar curvature} can derive from distance curvatures, by contracting to put metric-coefficient second derivatives into linear forms. Scalar curvature is the only such invariant. Physical world has four dimensions, because only such manifold results in curvature invariance.
Mathematical Sciences>Calculus>Vector>Tensor>Kinds
3-Calculus-Vector-Tensor-Kinds
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Date Modified: 2022.0224