Einstein and Maximally Symmetric Space Condition for 4D Metric with Torus Symmetry

摘要

In this paper we present Joyce construction's of selfdual metric in four dimensions and made a comparison on Einstein and maximally symmetric space condition. By calculating both condition using Levi-Civita connection in holonomic (coordinate) bases, we find that both condition satisfies the zero proportionality constant, that is, Ricci flat for Einstein condition and zero Gaussian curvature for maximally symmetric space. From these results, we conclude that Joyce's metric satisfies the flat space condition.