Morphisms of the heat equation (Q1381939)

The author extends the Fuglede-Ishihara characterization of harmonic morphisms to maps \(f:M\times \mathbb{R}^+\to N\times \mathbb{R}^+\) that pull back local solutions of the heat equation on \(N\) to those of \(M\), as well as to maps \(f:M\times M\times \mathbb{R}^+\to N\times N\times \mathbb{R}^+\) that pull back the heat kernel of \(N\) to that of \(M\).