On the curvatures of Einstein spaces (Q1891982)

Let \((M,g)\) be a pseudo-Riemannian manifold of dimension \(n\), \(n \geq 3\), and let \(V\) be a \(k\)-dimensional non-degenerate subspace of \(T_p M\) with \(2 \leq k \leq n\). For a frame \(\{e_1, \dots, e_k\}\) of \(V\), the scalar curvature function \(S(V)\) of \(V\) is defined as \[ S(V) = \sum^k_{i, j = 1} K(e_i, e_j), \] where \(K(e_i, e_j) = g(R_{e_i e_j} e_i, e_j)/(g(e_i, e_i) g(e_j, e_j) - g(e_i, e_j)^2)\). The scalar curvature function \(S(V)\) is independent of the choice of the frame of \(V\) and is a generalization of the scalar curvature \(S\) of \((M,g)\). The authors give some characterizations of Einstein spaces and spaces of constant curvature in terms of this scalar curvature function. Among others, they prove that a pseudo-Riemannian manifold \((M^n, g)\) is an Einstein space if and only if for each point \(p \in M\), \(S(V) = c(p)\), a constant, for every \((n - 1)\)-dimensional non-degenerate subspace \(V\) of \(T_p M\). Finally, the authors apply their results to Lorentz manifolds.