The aim of this paper is to provide the geometrical structure of a gravitational field that includes the addition of dark matter in the framework of a Riemannian and a Riemann–Sasaki spacetime. By means of the classical Riemannian geometric methods we arrive at modified geodesic equations, tidal forces, and Einstein and Raychaudhuri equations to account for extra dark gravity. We further examine an application of this approach in cosmology. Moreover, a possible extension of this model on the tangent bundle is studied in order to examine the behavior of dark matter in a unified geometric model of gravity with more degrees of freedom. Particular emphasis shall be laid on the problem of the geodesic motion under the influence of dark matter.

中文翻译：

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黎曼和佐佐木时空的暗引力场

本文的目的是提供一种引力场的几何结构，其中包括在黎曼和黎曼-萨萨基时空框架内添加暗物质。通过经典的黎曼几何方法，我们得出了修正的测地线方程，潮汐力以及爱因斯坦和雷绍杜里方程，以解决额外的暗重力。我们进一步研究了这种方法在宇宙学中的应用。此外，研究了该模型在切线束上的可能扩展，以便在具有更大自由度的统一重力几何模型中检查暗物质的行为。应特别强调暗物质影响下的测地运动问题。