Conformal properties of spheres

Abstract

We identify the smooth metrics $\mc{M}(M)$ on a manifold $M^n$ with the smooth isometric embeddings $f_g: (M,g) \rightarrow (\mb{S}^{\tn}, \tg)$ into a standard sphere of large dimension $\tn=\tn(n)$, and their Palais isotopic deformations, and the space $\mc{C}(M)$ of conformal classes with the space of classes of metrics whose embeddings are isotopic to each other by conformal deformations. Isometric embeddings of a metric on the manifold with a different smooth structure, and their deformations, are carried by the same background also, but when they exist, they do not embed into a smooth flow of any $f_g(M)\hookrightarrow \mb{S}^{\tn}$. We characterize metrics of constant scalar curvature by properties of extrinsic quantities of their embeddings, and prove a homotopy lifting property of the bundle $\mc{M}(M) \stackrel{\pi}{\rightarrow } \mc{C}(M)$ by Yamabe metrics, and when $M$ carries an almost complex structure $J_0$, extend it 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 almost complex structures in the same orientation class as $J_0$, and their conformal 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 $Sp(2)$, the $M^7_k$ spheres of Milnor, or any other, and except for $\mb{P}^1(\mb{C})\times \mb{P}^1(\mb{C})$, the almost Hermitian sigma invariant of product of spheres carrying almost complex structures, and organize manifolds with these invariants into a Pascal like triangle set according to the symmetries of the metrics, and the values of their associated conformal invariants.