Towards conformally flat isothermic metrics

TL;DR

This paper investigates conditions under which 4-binary metrics are conformally flat by analyzing Weyl tensor components, classifying metrics, and identifying those that are conformally flat within a specific subclass.

Contribution

It classifies 4-binary metrics based on Weyl tensor components and isolates conformally flat metrics within a particular class.

Findings

Classified 4-binary metrics into four classes based on Weyl tensor components.

Identified conditions for conformal flatness within the last class of metrics.

Abstract

According to [8] if the stationary Schroedinger equation on n-dim. Riemann space admits R-separation of variables (i.e. separation of variables with a factor R), then the underlying metric is necessarily isothermic. An important sub-class of isothermic metrics are the so called binary metrics. In this paper we study conditions for vanishing of components C_ijkl of Weyl tensor of arbitrary 4-binary metrics. In particular all 4-binary metrics for which C_ijij are the only non-vanishing components are classified into four classes. Finally, conformally flat metrics of the last class are isolated.