Almost Complex Structure on $S^{2n}$

TL;DR

This paper proves that there is no complex structure near the space of orthogonal almost complex structures on spheres of dimension greater than two, using the first Chern class of a related vector bundle.

Contribution

It demonstrates the non-existence of complex structures in a neighborhood of orthogonal almost complex structures on high-dimensional spheres, advancing understanding of complex structures on manifolds.

Findings

No complex structure exists near the space of orthogonal almost complex structures on $S^{2n}$ for n>1.

The proof involves analyzing the first Chern class of the tangent bundle.

The result constrains possible complex structures on spheres.

Abstract

We show that there is no complex structure in a neighborhood of the space of orthogonal almost complex structures on the sphere $S^{2n}, \ n>1$. The method is to study the first Chern class of vetcor bundle $T^{(1,0)}S^{2n}$.