2D Riemann-Christoffel curvature tensor via a 3D space using a specialized permutation scheme

Abstract

When a space in which Christoffel symbols of the second kind are symmetrical in lower indices exists, it makes for a supplement to the standard procedure when a 2D surface is normally induced from the geometry of the surrounding 3D space in which the surface is embedded. There it appears appropriate to use a scheme for straightforward permutation of indices of Gkij, when such a space would make this transformation possible, so as to obtain the components of the 2D Riemann-Christoffel tensor (here expressed in geodetic coordinates for an ellipsoid of revolution, of use in geophysics). By applying my scheme I find the corresponding indices in 2D and 3D supplement-spaces, and I compute components of the Riemann-Christoffel tensor. By operating over the elements of the projections alone, the all-known value of 1/MN for the Gaussian curvature on an ellipsoid of revolution is obtained. To further validate my scheme, I show that in such a 3D space the tangent vector to a PHI-curve for LAM=const1 would be parallel to a tangent vector to a PHI-curve for LAM=const2 on the surface of an ellipsoid of revolution. Surfaces parameterized by Gauss surface normal coordinates, such as the Earth, now can have the Riemann-Christopher curvature tensor computed in a straightforward fashion for the topographic surface hel(PHIel, LAMel) of the Earth, given Christoffel symbols for such a representation in terms of orthonormal functions on the ellipsoid of revolution.