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

Abstract

It is well known that the tensor field [Formula: see text] of type [Formula: see text] on the manifold [Formula: see text] is an almost complex structure if [Formula: see text] is an identity tensor field and the manifold [Formula: see text] is called the complex manifold. LetkM be the [Formula: see text] order extended complex manifold of the manifold [Formula: see text]. A tensor field [Formula: see text] onkM is called extended almost complex structure if [Formula: see text]. This paper aims to study the higher order complete and vertical lifts of the extended almost complex structures on an extended complex manifoldkM. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure [Formula: see text] on the extended complex manifoldkM are proved. Also, a tensor field [Formula: see text] of type [Formula: see text] is introduced and shows that it is an extended almost complex structure. Furthermore, the Lie derivative concerning higher-order lifts is studied and basic results on the almost analytic complex vector concerning an extended almost complex structure onkM are investigated. Finally, for more detailed explanation and better understanding a tensor field [Formula: see text] of type [Formula: see text] is introduced onkM, proving that it is a metallic structure onkM. A study of a golden structure, which is a type of metallic structure, is also carried out.