  In this study, firstly, the k-th order extension of complex product manifold
is consid- ered. Then the higher order vertical, complete lifts of geometric
structures on product manifold to its extended spaces are given. Also higher
order lifts of tensor field of type (1,1) are presented. And then extended
contact manifolds are defined. Finally higher order vertical and complete lifts
of time dependent complex Hamiltonian equations on contact manifold to its
extensions are introduced. In conclusion, geometric meaning of Hamiltonian
mechanical systems is discussed.
