  We identify all metrics on a closed $n$-manifold with their Nash isometric
embeddings into a standard sphere of large, but fixed dimension, and use the
Palais' isotopic extension theorem to identify their deformations with the
isotopic deformations of their embeddings, the deformations of metrics in a
conformal class identified with their corresponding isotopic conformal
deformations. If $n\geq 3$, we characterize metrics of constant scalar
curvature in terms of properties of extrinsic quantities of their associated
embeddings, and prove that any metric on the manifold of constant positive
scalar curvature, which can be minimally embedded into this background sphere,
is a Yamabe metric in its conformal class. We then use Simons' gap theorem to
study the extrinsic quantities of almost complex Hermitian deformations, by
Yamabe metrics, of the standard minimal almost complex isometric embeddings of
$\mb{S}^6$, $\mb{S}^2 \times \mb{S}^4$, $\mb{S}^2\times\mb{S}^6$, and
$\mb{S}^6\times \mb{S}^6$, respectively, and prove that none of these manifolds
carry integrable almost complex structures.
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