It is shown that a geometric property of Riemannian manifolds, the
positivity of the curvature, is equivalent to an analytic property 
of the heat equation on differential forms, namely, the positivity 
preserving property of the heat semigroup generated by the 
Hodge-de Rham Laplacian.

To establish the correspondence one has to consider the heat equation no
longer on function spaces but rather on spaces of exterior differential 
forms where positivity is understood through a natural cone which 
generalises the one of positive functions.

The proof relies on the methods of noncommutative Dirichlet forms and one
application of the result is to establish the Maximum Principle for
solutions of the Dirichlet problem where boundary data are differential
forms.