Feynman path integrals on nonlinear phase space

Definition of Feynman continual integral in Hamiltonian form on cotangential fibering of the Riemann space $M$ is given. Representation of the solution of parabolic type equation on $M$ in the form of the continual integral is established. It is shown that at the Feynman quantization (when operators are put into correspondence to functionals by means of continual integral) function of the functional of the form $\int\limits_0^1 Hd\sigma$ corresponds to the function of the operator $\hat H$. Extension of this result to the case of functions. of noncommuting operators is given.