Generalized Latitude and Longitude in a General Riemannian Space, with a Specialization for Hotine's (Omega, Phi, Nu) Coordinate System
Abstract
The present study seeks to find out whether Hotine's coordinate system (omega, phi, N) could be amissible in a class of general (curved) Riemannian spaced. The adopted approach postulates generalized quantities omega and phi to be coordinates and then examines, via commutators, in what kind of space this may hold true; the third coordinate remains N, a differentiable scalar function of position, which is assumed to be admissible in any space. The commutators lead to six conditions for six independent components of the covariant Riemann-Christoffel tensor. In the case of the original Hotine's coordinates omega and phi, it is shown that these components are identically zero, and, therefore, that the space must be flat. This implies that all orders of covariant derivatives of A sub r, B sub r, C sub r must be zero, where the orthonormal triad A, B, C represents Hotine's base vectors. Besides disproving the admissibility of the (omega, phi, N) coordinate system in any but the flat space, the analysis cross-validates a number of equations from Zund, 1990, applicable to the flat space, and presents several new relations that do not have equivalents in (Hotine, 1969) or Zund, 1990.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1991
- Accession Number
- ADA235584
Entities
People
- Georges Blaha