We develop the theory of weighted Ricci curvature in a weighted Lorentz–Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz–Finsler version of the Bonnet–Myers theorem based on a generalized Bishop inequality.

中文翻译：

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加权洛伦兹-芬斯勒流形的几何 I：奇点定理

我们在加权洛伦兹-芬斯勒框架中发展了加权 Ricci 曲率理论，并扩展了广义相对论的经典奇点定理。为了达到这个结果，我们将 Jacobi、Riccati 和 Raychaudhuri 方程推广到加权 Finsler 时空，并研究它们对沿因果测地线存在共轭点的影响。我们还展示了基于广义 Bishop 不等式的 Bonnet-Myers 定理的加权 Lorentz-Finsler 版本。