**Formality** of a space is meant in the sense of Sullivan, e.g. a space $X$ is called formal, if it's commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is weakly equivalent (There is zig-zag of quasi-isomoprhisms.) to the rational cohomology algebra $(H^*(X,\mathbb{Q}),0)$ of $X$. Since I'm only interested in manifolds, this is equivalent to the case of weakly equivalence between the DeRham algebra of differential forms $(A_{DR}(X),d)$ and the real cohomology algebra of $X$.

A compact oriented manifold $X$ is called **geometric formal**, if it admits a formal metric, e.g. a Riemannian metric $g$, such that the wedge product of harmonic forms is harmonic again. The typical class of examples are symmetric spaces.

By using Hodge decomposition, it's easy to see, that geometric formality implies formality.

The converse is not true, geometric formality is much stronger als formality. A formal metric has topological consequences, which go beyond formality, one can for example show, that the betti numbers of a geometric formal manifold $X$ is bounded above by the one of a torus in the same dimension, e.g. $$b_i(X)\le b_i(T^{dim(X)})={n \choose i}.$$

This bound on the betti numbers is also known for **rationally elliptic** spaces $X$, that are spaces whose total dimension of their rational cohomology $H^*(X,\mathbb{Q})$ and their rational homotopy $\pi_*(X)\otimes\mathbb{Q}$ is finite. For compact manifolds, this is clearly equivalent having finite dimensional rational cohomology.

There are examples for rationally elliptic compact simply-connected manifolds, which are not geometric formal. (Some of them generalized symmetric spaces, see [On formality of generalised symmetric spaces][1] by D. Kotschick and S. Terzic)

**Are there examples of compact oriented manifolds, which are geometric formal, but not rationally elliptic? Are there even examples of simply connected ones?**


  [1]: http://arxiv.org/abs/math/0106135