The paper considers stationary critical points of the heat flow in sphere SN and in hyperbolic space ℍN, and proves several results corresponding to those in Euclidean space SN which have been proved by
Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary
critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy
problems for the heat equation, it is shown that the solution u has a stationary critical point if and only
if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x0, is fixed
and initial-boundary value problems are considered for the heat equation on bounded domains containing
x0. It is shown that for any initial data satisfying the balance law with respect to x0 (or being
centrosymmetric with respect to x0) the corresponding solution always has x0 as a stationary critical point,
if and only if the domain is a geodesic ball centred at x0 (or is centrosymmetric with respect to x0,
respectively).