Metrics compatible with a symmetric connection in dimension three
The paper is dealing with the problem of characterizing symmetric connections \(\nabla\) that are Levi-Civita connections of a (pseudo-Riemannian) metric \(g\), mainly in the case of 3-dimensional manifolds. After reexamining the classical prescriptive solution of the problem, the main result provides necessary and sufficient conditions for the existence of a metric \(g\) compatible with a given \(\nabla\). They are expressed in terms of the curvature tensor \(R\), the Ricci tensor \(K\), commutators of \(R\) and \(\nabla R\), commutators of \(R\) and \(\nabla^2 R\), the endomorphism \[ \nabla_V \nabla_W R(X,Y)-\nabla_{\nabla_WV} R(X,Y), \] where \(X\), \(Y\), \(V\), \(W\) are arbitrary vector fields of the manifold under consideration. In particular, the second derivatives of \(R\) characterize the exceptional metrics. An explicit metric satisfying these conditions is also given. It should be noted that the problem has two main cases according to the rank of \(K\) (1 or 2).
scientific article; zbMATH DE number 894314