Curvature, integrability, and the six sphere

TL;DR

This paper investigates the relationship between almost-Hermitian and Riemannian geometries on manifolds, deriving an obstruction to integrability of orthogonal almost-complex structures under curvature constraints, with implications for the geometry of the 6-sphere.

Contribution

It introduces an obstruction equation linking curvature and integrability of almost-complex structures, providing new insights into the geometry of the 6-sphere.

Findings

Derived an obstruction equation for integrability

Provided a formula for the covariant derivative norm

Reinforced the non-Hermitian nature of the 6-sphere

Abstract

This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not hermitian. The partial proof is intrinsic in nature, and shows some level of promise when it comes to generalizing the non-complexity of the round $S^6$ result in new directions.