Necessary and Sufficient Conditions for the Exitsence of Metric in Three-Dimensional Affine Manifololds

鄭國順(Kuo-Shung Cheng) ； 倪維斗(Wei-Tou Ni)

《Chinese Journal of Physics》 17卷4期 (1979/12) Pp. 167-172

並列摘要

Explicit necessary and sufficient conditions for the existence of metric in three-dimensional affine manifolds are found in this paper. These conditions can be grouped into two kinds: (i) those involving the covariant derivatives of the Riemannian tensor R^α βγδ, and (ii) those involving R^α βγδ only. The first group consists of eighteen equations of the form (R^2 131R^3 112－R^3 181R^2 112)(R^1 131R^3 112－R^3 131R^1 112);λ=(R^2 131R^3 112－R^3 131R^2 112);λ‧(R^1 131R^3 112－R^3 131R^1 112) and their permutations. The second group contains three third-degree conditions, three fourth-degree conditions and thirteen sixth-degree conditions. In case the above necessary and sufficient conditionsare satisfied, the solutions for metric are obtained.