On the existence of turning points in D-dimensionsal Schwarzschild-de Sitter and anti-de Sitter spacetimes

We investigate the motion of a test particle in a d-dimensional, spherically symmetric and static space-time supported by a mass $M$ plus a $\mathrm{\Lambda }$-term. The motion is strongly dependent on the sign of $\mathrm{\Lambda }$. In Schwarzschild-de Sitter (SdS) space-time ($\mathrm{\Lambda }>0$), besides the physical singularity at $r=0$ there are cases with two horizons and two turning points, one horizon and one turning point and the complete absence of horizon and turning points. For Schwarzschild-Anti de Sitter (SAdS) space-time ($\mathrm{\Lambda }<0$) the horizon coordinate is associated to a unique turning point.