Trapped photons in Schwarzschild-Tangherlini spacetimes

We study the set of trapped photons outside the event horizon of Schwarzschild-Tangherlini black holes in dimension $d\ge 4$. We prove that the only trapped photons are those with constant Boyer-Lindquist coordinate radius. We then identify the set of trapped photons as a submanifold of the tangent bundle of the spacetime, and prove that this submanifold has topology $T^1 S^{n-1}\times $\mathbb{R}$^2$, as suggested in [7].