We establish a one-to-one correspondence between static spacetimes and Riemannian manifolds that maps causal geodesics to geodesics, as suggested by L. C. Epstein. We then explore constant curvature spacetimes - such as the de Sitter and the anti-de Sitter universes - and find that they map to constant curvature Riemannian manifolds, namely the Euclidean space, the sphere and the hyperbolic space. By imposing the conditions required to map to the sphere, we obtain the spherically symmetric metrics for which there is radial oscillatory motion with a period independent of the amplitude. We then consider the case of a perfect fluid and an Einstein cluster and determine the pressure and density profiles required to find this type of motion. Finally, we give examples of surfaces corresponding to certain types of motion for metrics that do not exhibit constant curvature, such as the Schwarzschild, Schwarzschild de Sitter and Schwarzschild anti-de Sitter solutions, and even for a simplified model of a wormhole.

中文翻译：

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黎曼流形对偶静态时空

正如 LC Epstein 所建议的，我们在静态时空和黎曼流形之间建立了一对一的对应关系，将因果测地线映射到测地线。然后我们探索恒定曲率时空——例如德西特宇宙和反德西特宇宙——并发现它们映射到恒定曲率黎曼流形，即欧几里得空间、球面和双曲空间。通过施加映射到球体所需的条件，我们获得了具有与振幅无关的周期的径向振荡运动的球对称度量。然后我们考虑完美流体和爱因斯坦星团的情况，并确定发现这种运动所需的压力和密度分布。最后，