$\mathbb{S}^6$ (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

TL;DR

The paper investigates the geometric properties of certain high-dimensional spheres and products of spheres, demonstrating that some cannot admit integrable almost complex structures by analyzing their embeddings and scalar curvature characteristics.

Contribution

It characterizes metrics of constant scalar curvature via extrinsic embedding properties and proves the non-existence of integrable almost complex structures on specific sphere products.

Findings

Metrics of constant positive scalar curvature can be realized as Yamabe metrics.

Certain sphere products cannot support integrable almost complex structures.

Embedding techniques reveal geometric obstructions to complex structures.

Abstract

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. :