New subclasses of J- pseudo harmonic morphisms F of a (semi-) Rumanian manifold ( M, g) into a metric (para-) f-manifold (N,h,J ) are introduced, namely; nearly, quasi, semi homothetic harmonic maps. On the way, some characterizations of its tension field of F are given. Also liftings of J-pseudo harmonic morphisms F to the tangent bundles TM and TN, with various type of lifted metric (para-) f-structures are considered. Finally, some supporting examples are provided.
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