Geometric formality of rationally elliptic manifolds in small dimensions

We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rational cohomology $S^2\times {\mathbb C}P^2$. An infinite family of six-dimensional simply connected biquotients whose second Betti number is three, different from Totaro's biquotients, is considered and it is proved that none of biquotient from this family is geometrically formal.