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

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