  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.
