  We identify the cone of smooth metrics $\mc{M}(M)$ on a manifold $M^n$ with
the space of smooth isometric embeddings $f_g: (M,g) \rightarrow
(\mb{S}^{\tn},\tg)$ into a standard sphere of large dimension $\tn=\tn(n)$, and
their isotopic deformations, and the space $\mc{C}(M)$ of conformal classes
with the embeddings of metrics in the same conformal class, isotopic to each
other by conformal deformations. Uniform limits of $C^1$ isometric embeddings
of a metric, or isometric embeddings of metrics on $M$ with a different smooth
structure, and their deformations, are carried by $(\mb{S}^{\tn},\tg)$ also,
but when the latter exist, they do not embed into a smooth flow of any
$f_g(M)\hookrightarrow \mb{S}^{\tn}$. We characterize the metrics of constant
scalar curvature by properties of their isometric embeddings, and use it to
prove a homotopy lifting property of the bundle $\mc{M}(M)
\stackrel{\pi}{\rightarrow } \mc{C}(M)$ by Yamabe metrics. When $M$ carries an
almost complex structure $J_0$, this result extends to a homotopy lifting
property of the bundle $\mc{M}^{J_0}(M) \stackrel{\pi}{\rightarrow }
\mc{C}^{J_0}(M)$ of metrics, compatible with some $J$ in the same orientation
class as $J_0$, and their classes, the lift now by almost Hermitian Yamabe
metrics. We use these results and the gap theorem of Simons to study the
existence and integrability properties of almost complex structures on spheres,
and products. We find the sigma invariants of $Sp(2)$ and the $M^7_k$ spheres
of Milnor, and with one exception, the almost Hermitian sigma invariant of any
product of spheres with almost complex structures. We organize all manifolds
with Yamabe, or almost Hermitian Yamabe metrics on them, in a Pascal like
triangle, set according to their dimensions and symmetries, and the values of
their associated conformal invariants.
