标题： FLRW宇宙中标量场的艾森哈特提升 显示英文标题

标题： Eisenhart Lift for Scalar Fields in the FLRW Universe

摘要： 黎曼类型的 Eisenhart 提升描述了一个粒子的运动，将其视为高维黎曼流形中的测地线，该流形具有一个额外的坐标。最近，通过引入一个额外的矢量场，它已被推广到标量场系统。我们将这种方法应用于 Friedmann-Lemaître-Robertson-Walker 宇宙中的标量场系统，并分类了系统的对称性。特别是，对于由指数函数组合的平方构成的标量场势，其中特定指数为$e^{\pm\frac{\sqrt{6}}{4}\phi}$，我们找到了非平凡的（共形）Killing 矢量场和 Killing 张量场。此外，对于势能表示为一般指数的指数组合的指数形式的势能，我们发现了非平凡的共形 Killing 矢量场。通过沿共形 Killing 矢量场引入坐标，我们可以完全求解运动方程。我们还对多个标量场系统的对称性进行了分类。 显示英文摘要

摘要： The Eisenhart lift of Riemannian type describes the motion of a particle as a geodesic in a higher-dimensional Riemannian manifold with one additional coordinate. It has recently been generalized to a scalar field system by introducing one additional vector field. We apply this approach to a scalar field system in the Friedmann-Lemaitre-Robertson-Walker universe and classify the symmetries of the system. In particular, for a scalar field potential consisting of the square of a combination of exponential functions with specific index $e^{\pm\frac{\sqrt{6}}{4}\phi}$, we find nontrivial (conformal) Killing vector fields and Killing tensor fields. Moreover, for a potential written as an exponentiation of a combination of exponential potentials with general index, we find nontrivial conformal Killing vector fields. By introducing the coordinate along the conformal Killing vector field, we can solve the equations of motion completely. We also classify the symmetries of multiple scalar field system.