Curvature, integrability, and the six 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 -sphere 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 result in new directions.
Cite
@article{arxiv.2108.02586,
title = {Curvature, integrability, and the six sphere},
author = {Gabriella Clemente},
journal= {arXiv preprint arXiv:2108.02586},
year = {2023}
}