Generalized Conharmonically Recurrent Pseudo-Riemannian Manifolds and Their Applications to f (R)-Gravity

This paper studies generalized conharmonically recurrent pseudo-Riemannian manifolds and their geometric and physical implications, with particular attention to applications in fR-gravity. Explicit relations are established among the conharmonic, Riemann, Ricci, Weyl conformal, and Schouten curvature tensors. It is shown that, for n≠2, in both conharmonically symmetric and conformally flat manifolds, the associated recurrence 1-forms are explicitly determined by the scalar curvature. In four-dimensional perfect fluid spacetimes, the imposed curvature conditions yield strong kinematical restrictions: the fluid flow is geodesic, shear-free, irrotational, and nonexpanding, and the four-velocity field is covariantly constant. As an application, conharmonically flat and conharmonically symmetric spacetimes satisfying the fR-gravity field equations are examined. The resulting geometric conditions impose strong constraints on the energy-momentum tensor and admit only highly restricted matter configurations, including stiff-matter and dark-energy–type models.