The authors consider a geometric problem concerning a second order linear partial differential equation [see N. Teodorescu, Bull. Math. Soc. Roum. Sci. 41, No. 2, 101-109 (1939; Zbl 0025.17902); ibid. 42, No. 1, 79-89 (1941; Zbl 0025.40801); ibid. 43, 59-68 (1941); ibid. 44, 71-84 (1942)]. The results can be described as follows. Let \(ds^ 2=a_{ij}dx^ idx^ j\) be a (pseudo) Riemannian metric in a domain \(\Omega_ n\subset R^ n\). The authors look for a (pseudo) Riemannian metric \(d\bar s^ 2=\bar a_{ij}dx^ idx^ j\) in \(\Omega_ n\) such that \({\bar \nabla}_ ka_{ij}=\rho_ ka_{ij}\) where \({\bar \nabla}\) is the Levi-Civita connection of \(d\bar s^ 2\). They obtain \(d\bar s^ 2\) explicitly in the case where \(ds^ 2\) is Riemannian, reducible or irreducible. If \(ds^ 2\) is pseudo-Riemannian then it is reducible and the authors obtain the corresponding expressions of \(d\bar s^ 2\).