  We study Serrin's overdetermined boundary value problem \begin{equation*}
  -\Delta_{S^N}\, u=1 \quad \text{ in $\Omega$},\qquad u=0, \; \partial_\eta
u=\textrm{const} \quad \text{on $\partial \Omega$} \end{equation*} in
subdomains $\Omega$ of the round unit sphere $S^N \subset \mathbb{R}^{N+1}$,
where $\Delta_{S^N}$ denotes the Laplace-Beltrami operator on $S^N$. A
subdomain $\Omega$ of $S^N$ is called a Serrin domain if it admits a solution
of this overdetermined problem. In our main result, we construct Serrin domains
in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular
neighborhoods of the equator. Our result provides the first example of Serrin
domains in $S^{N}$ which are not bounded by geodesic spheres.
