Every timelike geodesic in anti--de Sitter spacetime is a circle of the same radius

We refine and analytically prove an old proposition due to Calabi and Markus on the shape of timelike geodesics of anti--de Sitter space in the ambient flat space. We prove that each timelike geodesic forms in the ambient space a circle of the radius determined by $\Lambda$, lying on a Euclidean two--plane. Then we outline an alternative proof for $AdS_4$. We also make a comment on the shape of timelike geodesics in de Sitter space.