  We construct and study an $H$-space multiplication on $\mathcal R^+(M)$ for
manifolds $M$ which are nullcobordant in their own tangential $2$-type. This is
applied to give a rigidity criterion for the action of the diffeomorphism group
on $\mathcal R^+(M)$ via pullback. We also compare this to other known
multiplicative structures on $\mathcal R^+(M)$.
