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We identify all metrics on a closed $n$-manifold with their Nash isometricembeddings into a standard sphere of large, but fixed, dimension, and usePalais' isotopic extension theorem conveniently to analyze their deformationscorresponding to variations of the metrics within a conformal class. If $n\geq3$, we characterize metrics of constant scalar curvature in terms of propertiesof extrinsic quantities of their associated embeddings, and prove that anymetric of constant positive scalar curvature that can be minimally embeddedinto this background sphere is a Yamabe metric in its conformal class. We thenuse Simons' gap theorem to study the extrinsic quantities of almost complexHermitian deformations by Yamabe metrics of the standard minimal almost complexisometric embeddings of $\mathbb{S}^6$, $\mathbb{S}^2 \times \mathbb{S}^4$,$\mathbb{S}^2\times\mathbb{S}^6$, and $\mathbb{S}^6\times \mathbb{S}^6$,respectively, and deduce that none of these manifolds carry integrable almostcomplex structures.

$\mathbb{S}^6$ (or any of $\mathbb{S}^2 \times \mathbb{S}^4$,  $\mathbb{S}^2\times\mathbb{S}^6$, or $\mathbb{S}^6\times \mathbb{S}^6$,  respectively) is not diffeomorphic to a complex manifold