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

Source 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. Let k M be the [Formula: see text] order extended complex manifold of the manifold [Formula: see text]. A tensor field [Formula: see text] on k M 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 manifold k M. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure [Formula: see text] on the extended complex manifold k M 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 on k M are investigated. Finally, for more detailed explanation and better understanding a tensor field [Formula: see text] of type [Formula: see text] is introduced on k M, proving that it is a metallic structure on k M. A study of a golden structure, which is a type of metallic structure, is also carried out.