Orthogonal almost complex structures on the Riemannian products of even-dimensional round spheres

TL;DR

This paper investigates the conditions under which orthogonal almost complex structures are integrable on Riemannian products of even-dimensional spheres, addressing a specific question about complex structures on such products.

Contribution

It provides a partial answer to Calabi's question about the existence of complex structures on products of round 2- and 4-spheres, focusing on integrability conditions.

Findings

Identifies conditions for integrability of orthogonal almost complex structures

Provides partial classification of complex structures on sphere products

Addresses a specific open problem posed by Calabi

Abstract

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.