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

Source abstract

This paper studies generalized conharmonically recurrent pseudo‐Riemannian manifolds and their geometric and physical implications, with particular attention to applications in f ( R )‐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 f ( R )‐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.