AI Summary & Key Takeaways
AI InsightHere, $a_{ml}={\lambda }_{m|l}+{\lambda }_m{\lambda }_{l\ }$, $\ b_{ml}={\text{\^{A} $\mu$ }}_{m|l}+{\lambda }_m{\text{\^{A} $\mu$% }}_l$ and $c_{ml}={({\lambda }_m{\gamma }_l+\gamma }_{m|l})$ are non-zero covariant tensors field of second order, respectively.
📌 3 Key Takeaways
Abstract
In this work, we introduce and investigate two new classes of Finsler spaces: generalized $W^{\ }_{|h}${}-recurrent spaces and generalized $W^{\ }_{|h}${}-birecurrent spaces, which extend classical notions of recurrence and birecurrence to a broader geometric setting. These spaces are defined via specific covariant derivative conditions imposed on Weyl's projective curvature tensor $W^i_{jkh}${}, involving scalar functions ${\lambda }_m$,$\ {\mu }_m$, and ${\gamma }_m${}. We establish several structural identities characterizing the recurrence properties of these tensors and examine higher-order covariant derivatives. Additionally, we explore the decomposition of the curvature tensor $W^i_{jkh}$ in terms of lower-rank tensors, considering cases where the tensor is expressed as a product of a covariant vector and a mixed-type tensor. The recurrence and birecurrence properties of such decompositions are studied under various tensorial conditions. Our analysis shows that under appropriate assumptions on the covariant vector and tensor fields, the decomposed tensors inherit generalized birecurrence properties, and remain non-vanishing, emphasizing the consistency of the new structure with existing geometric frameworks. Here, $a_{ml}={\lambda }_{m|l}+{\lambda }_m{\lambda }_{l\ }$, $\ b_{ml}={\text{\^{A} $\mu$ }}_{m|l}+{\lambda }_m{\text{\^{A} $\mu$% }}_l$ and $c_{ml}={({\lambda }_m{\gamma }_l+\gamma }_{m|l})$ are non-zero covariant tensors field of second order, respectively.