General natural Einstein Kähler structures on tangent bundles

We study the conditions under which a Kählerian structure (G,J)(G,J), of general natural lift type on the tangent bundle TM   of a Riemannian manifold (M,g)(M,g), studied in [S. Druţă, V. Oproiu, General natural Kähler structures of constant holomorphic sectional curvature on tangent bundles, An. St. Univ. “Al.I. Cuza” Mat. 53 (2007) 149–166], is Einstein. We found three cases. In the first case the first proportionality factor λ is expressed as a rational function of the first two essential parameters involved in the definition of J and the value of the constant sectional curvature c   of the base manifold (M,g)(M,g). It follows that (TM,G,J)(TM,G,J) has constant holomorphic sectional curvature (Theorem 8). In the second case a certain second degree homogeneous equation in the proportionality factor λ   and its first order derivative λ′λ′ must be fulfilled. After some quite long computations done by using the Mathematica package RICCI for doing tensor computations, we obtain an Einstein Kähler structure only on (T0M,G,J)⊂(TM,G,J)(T0M,G,J)⊂(TM,G,J), where T0MT0M denotes the subset of nonzero tangent vectors to M   (Theorem 9). In the last case we obtain that the Kählerian manifold (TM,G,J)(TM,G,J) cannot be an Einstein manifold.