Compatible and almost compatible metrics (Q2710711)

The author calls two pseudo-Riemannian contravariant metrics \(g_1\) and \(g_2\) almost compatible if for any linear combination \(g = \lambda_1 g_1 + \lambda_2 g_2\) with constant coefficients the corresponding Levi-Civita connection and the Riemannian curvature tensor are connected by the same linear combination. The metrics are called compatible, if this condition is fullfilled for metrics \(g\) with det \(g \neq 0\). The compatibility is related to the vanishing of a certain Nijenhuis tensor. No proofs are given.