Conditions for a symmetric connection to be a metric connection

• Published in: Journal of mathematical physics 1992-11, Vol.33 (11), p.3716-3722
• Main Author: Edgar, S. Brian
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Geometry, differential geometry, and topology; Mathematical methods in physics; Physics
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.529867

A recent result−in ‘general circumstances’ for a four‐dimensional space–time−giving algebraic conditions on a curvature tensor (of a symmetric connection) so that the connection be metric, is shown to be a special case of a more general result; both of these results are shown formally to be of a generic nature. In this new result the conditions are imposed on a tensor of a more general character than the curvature tensor. In addition it is shown that once the symmetric connection is known to be metric, the metric is uniquely defined (up to a constant conformal factor). For those special curvature tensors which are excluded from the original result, supplementary conditions are suggested, which, alongside the original conditions are sufficient to ensure that most of these excluded curvature tensors are also Riemann tensors.