Proposed theorems for lifts of the extended almost complex structures on the complex manifold

TL;DR

This paper investigates higher-order lifts of extended almost complex structures on complex manifolds, proving theorems about their Nijenhuis tensor, introducing a related tensor field, and exploring Lie derivatives and almost analytic vectors.

Contribution

It presents new theorems on the Nijenhuis tensor for extended almost complex structures and introduces a new tensor field demonstrating extended almost complex structure properties.

Findings

Proved theorems on the Nijenhuis tensor of extended structures

Introduced a tensor field J~k as an extended almost complex structure

Studied Lie derivatives and almost analytic vectors in this context

Abstract

The present paper aims to study the higher-order complete and vertical lifts of the extended almost complex structures on an extended complex manifold kM. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure Jk on the extended complex manifold kM are proved. Also, a tensor field J~k of type (1,1) is introduced and shows that it is an extended almost complex structure. Finally, the Lie derivative concerning higher-order lifts is studied and basic results on the almost analytic complex vector concerning an extended almost complex structure on kM are investigated.