Projectively recurrent semi-definite Riemannian manifolds.
Let \(M\) be a \(n\) (\(\geq 3\))-dimensional semi-definite Riemannian manifold with semi-definite Riemann metric \(g\), Riemannian connection \(\nabla\), Riemannian curvature tensor \(R\) and Ricci tensor \(S\). Locally, the projective curvature tensor \(W\) with respect to an orthogonal frame, is given by NEWLINE\[NEWLINEW_{ijkl} = R_{ijkl} - \frac{1}{n-1}\epsilon_i (S_{jk}g_{il} - S_{jl}g_{ik}).NEWLINE\]NEWLINE The manifold \(M\) is called projectively recurrent, if there exists a 1-form \(\alpha\) such that \(\nabla W = \alpha \otimes W\). If \(\nabla W = 0\), the manifold is called projectively symmetric. The author shows, that a projectively symmetric Riemannian manifold of dimension \(n \geq 3\) is locally symmetric and further that a projectively recurrent manifold is of constant curvature or locally symmetric.
scientific article; zbMATH DE number 1657763