On Riemannian manifolds admitting infinitesimal projective transformations (Q1093928)

A transformation \(\phi\) : \(M\to M\) is called projective, if it preserves geodesics of M. Let \(\phi_ t\) be a one-parameter group of projective transformations, generated by a vector field X, called an infinitesimal projective transformation. X is called an infinitesimal affine transformation, if \(L_ X \Gamma^ i_{jk}\equiv 0\), where \(L_ X\) denotes the Lie derivative and \(\Gamma^ i_{jk}\) are the Christoffel symbols. Let \(\nabla\) be the covariant derivative and \(K_{ij}\) its Ricci tensor. The main result of the article is Theorem 1. Let M be a connected complete Riemannian manifold with constant positive scalar curvature and satisfying the condition \(\nabla_ k K_{ij}=\nabla_ j K_{ki}\). If M admits a non-affine infinitesimal projective transformation then M is a space of positive constant curvature. This theorem is a generalization of Nagano's theorem: ``Let M be a manifold as in Theorem 1 with parallel Ricci tensor. If M admits a non- affine infinitesimal projective transformation then M is a space of positive constant curvature.''