Conformal properties of spheres

TL;DR

This work develops a unifying extrinsic view of Yamabe theory by identifying ${\mathcal M}(M)$ with Nash embeddings into a fixed sphere ${\mathbb S}^{\tilde n}$ and studying the induced extrinsic functionals; it proves a homotopy lifting property for Yamabe metrics along conformal paths and extends it to almost Hermitian Yamabe metrics when an almost complex structure exists. It then applies these tools to spheres and products to derive obstructions to almost complex structures, characterize constant scalar curvature via embedding data, and compute sigma and almost Hermitian sigma invariants for ${\rm Sp}(2)$, Milnor's exotic ${M^7_k}$, and Calabi–Eckmann products, culminating in a conformal Pascal triangle organizing these invariants. The results reveal a deep interplay between intrinsic conformal geometry and extrinsic embedding data, providing a framework to compare conformal invariants across manifolds with different smooth structures and to understand when extremal metrics can be realized within almost Hermitian or related geometric structures.

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 $\mc{M}(M) \stackrelπ{\rightarrow } \mc{C}(M)$ by Yamabe metrics, and when $M$ carries an almost complex structure $J_0$, extend it to a homotopy lifting property of the bundle $\mc{M}^{J_0}(M) \stackrelπ{\rightarrow } \mc{C}^{J_0}(M)$ 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.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 10
• Theorem 11