Authors: Steven Kenneth Kauffmann
Merriam-Webster's Collegiate Dictionary, Eleventh Edition, gives a technical definition of curvature, "the rate of change of the angle through which the tangent to a curve turns in moving along the curve and which for a circle is equal to the reciprocal of the radius". That precisely describes a curve's intuitive curvature, but the Riemann "curvature" tensor is zero for all curves! We work out the natural extension of intuitive curvature to hypersurfaces, based on the rates that their tangents develop components which are orthogonal to the local tangent hyperplane. Intuitive curvature is seen to have the form of a second-rank symmetric tensor which cannot be algebraically expressed in terms of the metric tensor and a finite number of its partial derivatives. The Riemann "curvature" tensor contrariwise is a fourth-rank tensor with both antisymmetric and symmetric properties that famously is algebraically expressed in terms of the metric tensor and its first and second partial derivatives. Thus use of the word "curvature" with regard to the Riemann tensor is misleading, and since it can't encompass intuitive curvature, Gauss-Riemann "geometry" oughtn't be termed differential geometry either. That "geometry" is no more than the class of the algebraic functions of the metric and any finite number of the metric's partial derivatives, which it is convenient to organize into generally covariant entities such as the Riemann tensor because those potentially play a role in generally-covariant metric-based field theories.
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