To the question of interrelation between the metric and the curvature tensor in an \(n\)-dimensional Riemannian space (Q1288107)

Let \(D_n= \{x\mid x=(x_1, x_2, \dots, x_n) \in {\mathbb R}^n\), \(0< x_i <1\), \(i= 1, \dots, n \}, n\geq 2\), be a domain in \(\mathbb R^n\) and let \(x_1, x_2,\dots ,x_n\) be a semigeodesic coordinate system in \(D_n\). The authors study uniqueness of a Riemannian metric \(g_{ij}\) in \(D_n\) satisfying the Cauchy conditions \[ g_{km}\big| _{x_1=0}=g^0_{km},\quad \frac{\partial g_{km}}{\partial x_1}\bigg| _{x_1=0}=g^1_{km} \] with prescribed functions \(g^0_{km}\) and \(g^1_{km}\) and having prescribed values for some components of the curvature tensor \(R^i_{qks}\) in \(D_n\). The authors obtain two sets of ``important'' components of the curvature tensor (those are components which allow us to prove the uniqueness theorem) and give an example of one ``not important'' set of components. Previously, some similar results were obtained in the authors' article [Dokl. Math. 54, 863-864 (1996; Zbl 0895.53038)].