Morphisms of the Heat Equation

Abstract

In this article we consider a certain class of mappings between Riemannian manifolds of the type M × R^{+ *}, which preserve two objects associated to the heat equation. First we show that maps which pull back local solutions of the heat equation to local solutions of the heat equation, called ’heat equation morphisms‘ have to be the product of a homothetic submersion and an affine map of R^{+ *}. Then using the above characterisation, we study maps which pull back the heat kernel to the heat kernel. We call such maps ‘heat kernel morphisms’. We show in Theorem 3 that, in the case of compact manifolds, a map Φ : M × M × R^{+ *} → N × N × R^{+ *} of the form Φ (x,y,t) = (φ (x), φ (y),h(t)), with φ surjective, is a heat kernel morphism if and only if φ is a homothetic covering of N(φ)-sheets and constant dilation λ such that N(φ) = λ^m (m = dim M) and h(t) = λ²t. In particular, if φ is bijective then it must be an isometry and h(t) = t. A similar problem was considered from a different point of view in [6].