Theorems of Liuville types in theory mappings of the complete Riemannian manifolds

In the present paper we prove some vanishing theorems for projective and holomorphically projective diffeomorphisms of Riemannian and Kähler complete manifolds. We will use the well-known Liouville type theorems on harmonic, subharmonic and superharmonic functions on complete Riemannian manifolds for proofs of these theorems.