  We study the spatial critical points of the solutions $u=u(x,t)$ of the
fractional heat equation. For the Cauchy problem, we show that the origin $0$
satisfies $\nabla_x u(0,t) = 0$ for $t>0$ if and only if the initial data
satisfy a balance law of the form $\int_{\mathbb{S}^{N-1}} \omega u_0(r\omega)
\, \mathrm{d} \omega=0$ for a. e. $r \ge 0$. Moreover, for the Dirichlet
initial-boundary value problem, we prove two symmetry results: $\Omega$ is a
ball centered at the origin if and only if $\nabla_x u(0,t) = 0$ for $t>0$
provided that the initial data satisfies the above-mentioned balance law;
$\Omega$ is centrosymmetric if and only if $\nabla_x u(0,t) = 0$ for $t>0$
provided that the initial data is centrosymmetric. These results extend some
theorems obtained by Magnanini and Sakaguchi in 1997-1999 for the (local) heat
equation to the fractional context. These extensions are nontrivial because of
the nonlocal nature of the fractional Laplacian. Among others, the proof of the
characterization of a ball in the Dirichlet initial-boundary value problem for
the fractional heat flow not only works for the classical heat flow but also
gives a new insight into the problem.
