A Characterization of the Dependence of the Riemannian Metric on the Curvature Tensor by Young Symmetrizers

Abstract

In differential geometry several differential equation systems are known which allow the determination of the Riemannian metric from the curvature tensor in normal coordinates. We consider two of such differential equation systems. The first system used by Gunther [8] yields a power series of the metric the coefficients of which depend on the covariant derivatives of the curvature tensor symmetrized in a certain manner. The second system, the so-called Herglotz relations [9], leads to a power series of the metric depending on symmetrized partial derivatives of the curvature tensor.
We determine a left ideal of the group ring $\mathbb{C}(S_{r+4})$ of the symmetric group $S_{r+4}$ which is associated with the partial derivatives ${\partial }^{(r)}R$ of the curvature tensor $R$ of order $r$ and construct a decomposition of this left ideal into three minimal left ideals using Young symmetrizers and the Littlewood-Richardson rule. Exactly one of these minimal left ideals characterizes the so-called essential part of ${\partial }^{(r)}R$ on which the metric really depends via the Herglotz relations. We give examples of metrics with and without a non-essential part of ${\partial }^{(r)}R$. Applying our results to the covariant derivatives of the curvature tensor we can show that the algebra of tensor polynomials $\mathcal{R}$ generated by ${▽}_{(i_1}...{▽}_{i_r)}{R}_{ijkl}$ and the algebra ${\mathcal{R}}^s$ generated by ${▽}_{(i_l}...{▽}_{i_r}{R}_{|k|i_{r+l^{s}r+2)}l}$ fulfil $\mathcal{R}={\mathcal{R}}^s$.