A characterization of the dependence of the Riemannian metric on the curvature tensor by Young symmetrizers
The first method for the determination of a Riemannian metric from the components of the curvature tensor in normal coordinates was given by Herglotz. The so-called Herglotz relations are nonlinear differential equations and give (at least in the analytic case) the power series expansion of the metric. The related coefficients are determined by the symmetrized partial derivatives of the curvature tensor. Later on, P. Günther found a power series expansion of the metric with explicit coefficients which depend on covariant derivatives of the curvature tensor symmetrized in a certain way. In the paper under review, the author studies the difference between both kind of symmetrizations. For this reason the representation theory of the symmetric group \({\mathcal S}_r\) is used. The partial derivatives \(\partial_{i_1}\dots\partial_{i_r} R_{ijkl}\) induce group ring elements belonging to a direct sum \(J_{(r)}\oplus\widehat J_{(r)} \oplus \breve J_{(r)}\) of three minimal left ideals of the group ring \(\mathbb{C} [{\mathcal S}_{r+4}]\). Then the symmetrized partial derivatives \(\partial_{(i_1}\dots\partial_{i_r} R_{| a| i_{r+1} i_{r+2})b}\) correspond to a linear mapping of the direct sum into \(J_{ (r)} \cdot \varepsilon\), \(\varepsilon\in \mathbb{C} [{\mathcal S}_{r+4}]\) which maps \(\widehat J_{(r)} \oplus \breve J_{(r)}\) to 0. For the symmetrized covariant derivatives \(\nabla_{(i_1} \dots \nabla_{i_r)} R_{ijkl}\) and \(\nabla_{(i_1}\dots\nabla_{i_r} R_{| a| i_{r+1} i_{r+2})b}\) only the ideals \(J_{(r)}\) and \(J_{(r)} \cdot \varepsilon\) do appear. So, the inverse mapping \(J_{(r)} \cdot \varepsilon \to J_{(r)}\) gives a relation between both kinds of symmetrized covariant derivatives. This shows \({\mathcal R}= {\mathcal R}^s\), where \({\mathcal R}\) is the algebra of tensor polynomials generated by \(\nabla_{(i_1} \dots \nabla_{i_r)} R_{ijkl}\) and \({\mathcal R}^s\) is the algebra generated by \(\nabla_{(i_1} \dots \nabla_{i_r} R_{| k| i_{r+1} i_{r+2})l}\). To prove these results, the author uses purely algebraic methods like Young symmetrizers and the Littlewood-Richardson rule. Furthermore, the author constructs examples of metrics for which the group ring elements induced by \(\partial^{(r)}R\) belong to \(J_{(r)}\) only or admit a nontrivial part in the ideal \(\widehat J_{(r)} \oplus \breve J_{(r)}\).
partial derivatives of the curvature tensor
scientific article; zbMATH DE number 1129194