Twisted Sasaki metric on the unit tangent bundle and harmonicity

The paper deals with the twisted Sasaki metric on the unit tangent bundle of n-dimensional Riemannian manifold Mn . The main purpose of the research is to find deformations that preserve the existence harmonic left -invariant unit vector fields on 3 -dimensional unimodular Lie groups G with the left invariant metric and harmonic maps G -+ T1G in case of twisted Sasaki metric on the unit tangent bundle. The necessary and sufficient conditions for harmonicity of left -invariant unit vector field and map Mn -+ T1Mn are obtained. The necessary and sufficient conditions for harmonicity of left -invariant unit vector field and map M2 -+ T1M2 with respect to some orthonormal frame are obtained. Left -invariant harmonic unit vector fields and harmonic maps G -+ T1G , where G is a three-dimensional unimodular Lie group with left -invariant metric, using some orthonormal frame are described. Left -invariant harmonic unit vector fields which determine harmonic maps G -+ T1G , where G is a three-dimensional unimodular Lie group with left -invariant metric in the particular case of twisted Sasaki metric, namely the vertical rescaled metric are classified.