Abstract
It is well known that the tensor field J of type (1, 1) on the manifold M is an almost complex structure if J(2) = -I, I is an identity tensor field and the manifold M is called the complex manifold. Let M-k be the k order extended complex manifold of the manifold M. A tensor field J(k) on M-k is called extended almost complex structure if (J(k))(2) = -I. This paper aims to study the higher order complete and vertical lifts of the extended almost complex structures on an extended complex manifold M-k. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure J(k) on the extended complex manifold M-k are proved. Also, a tensor field (J) over tilde (k) of type (1, 1) 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 M-k are investigated. Finally, for more detailed explanation and better understanding a tensor field J(k)* of type (1, 1) is introduced on M-k, proving that it is a metallic structure on M-k. A study of a golden structure, which is a type of metallic structure, is also carried out.