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 RN 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).
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