  Let $\{B_\beta (x), x \in \mathbb{S}^N\}$ be a fractional Brownian motion on
the $N$-dimensional unit sphere $\mathbb{S}^N$ with Hurst index $\beta$. We
study the excursion probability $\mathbb{P}\{\sup_{x\in T} B_\beta(x) > u \}$
and obtain the asymptotics as $u\to \infty$, where $T$ can be the entire sphere
$\mathbb{S}^N$ or a geodesic disc on $\mathbb{S}^N$.
