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

TL;DR

This paper introduces a specialized permutation scheme to compute the 2D Riemann-Christoffel curvature tensor from 3D space, simplifying calculations on surfaces like the Earth's ellipsoid using geodetic coordinates.

Contribution

The paper presents a novel permutation scheme for deriving 2D curvature tensors directly from 3D space, facilitating curvature computations on geophysical surfaces.

Findings

Successfully computes the Gaussian curvature of an ellipsoid of revolution.

Validates the scheme by showing tangent vectors to specific curves are parallel.

Enables straightforward curvature calculations for Earth's topographic surfaces.

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.