The rational homotopy type of a manifold $X$ (or any topological space) discards all torsion from the homotopy (or homology) groups of $X$.
Its dominating role for partial differential equations stems from its close relation to differential forms which was discovered in the pioneering work
of Sullivan and Quillen in the early 70's. In fact, the rational homotpy type of a simply connected compact manifold is completely
determined by a suitable differentially graded algebra (DGA) of differential forms. This minimal model however is not natural.

On a Riemannian manifold, the metric determines such an DGA which partly remedies the non-functoriality of the minimal model.
We will construct this metric minimal model and discuss some applictions in the calculus of variations.
In part this is a generalization of investigations of geometrically formal manifolds by Kotschick and Terzic.
The metric minimal model of such spaces is just the algebra of harmonic forms.