Space of split curvature

Abstract We investigate(pseudo-)Riemannian spaces whose curvature tensor possesses a specific structure called split curvature. First, we provide a direct and simplified proof that every space of split curvature is semisymmetric. This result refines previous approaches by eliminating unnecessary assumptions and clarifying inaccuracies in earlier classifications. We then propose a natural classification of these spaces according to the rank of the Ricci tensor, distinguishing three geometric types. The results of Sinyukov and Mikeš on geodesic mappings of semisymmetric spaces and on equidistant spaces are employed here. Furthermore, we analyze geodesic mappings of spaces of split curvature and demonstrate the existence of spaces that admit nontrivial geodesic mappings while having non-constant scalar curvature. This disproves a previously published theorem by Kiosak asserting that the scalar curvature must be constant in this context. By constructing explicit examples, we provide a more precise and transparent description of the geometric structure of spaces of split curvature, including their classification, mapping properties, and curvature characteristics. Special attention is given to the role of concircular vector fields in equidistant spaces, which yields new insights into their structure and mapping behavior.