Metric connections in projective differential geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.Comment: 10 page

On compact holomorphically pseudosymmetric K\"ahlerian manifolds

For compact K\"ahlerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric K\"ahlerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric K\"ahlerian manifolds are discovered by W. Jelonek. In his examples, the structure functions change their signs on the manifold