On some manifolds with positive sigma invariants and their realizing conformal classes

We prove that the metric of the Riemannian product $(\mb{S}^k(r_1)\times \mb{S}^{n-k}(r_2), g^n_k)$, $r_1^2+r_2^2=1$, is a Yamabe metric in its conformal class if, and only if, either $g^n_k$ is Einstein, or the linear isometric embedding of this manifold into the standard $n+1$ dimensional sphere is minimal. We combine this result with Simons' gap theorem to show that, for $2\leq k\leq n-2$, the conformal class of the product metric with minimal embedding, which is at the upper end of Simons' gap theorem, realizes the sigma invariant of $\mb{S}^k\times \mb{S}^{n-k}$, and that this is the only class that achieves such a value. Similarly, we use coherent minimal isometric embeddings of suitably scaled standard Einstein metrics $g$ on $\mb{P}^n(\mb{R})$, $\mb{P}^n(\mb{C})$, and $\mb{P}^n(\mb{H})$ into unit spheres, and determine the sigma invariant of these projective spaces, prove that in each case the conformal class $[g]$ realizes it, and that this realizing class is unique.