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General natural Einstein Kahler structures on tangent bundles
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| DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS |
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We study the conditions under which a Kahlerian structure (G. J), of general natural lift type on the tangent bundle T M of a Riemannian manifold (M, g), studied in [S. Druta, V. Oproiu, General natural Kahler 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 lambda 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). It follows that (T M. G. J) has constant holomorphic sectional curvature (Theorem 8). In the second case a certain second degree homogeneous equation in the proportionality factor lambda and its first order derivative lambda' must be fulfilled. After some quite long computations done by using the Mathematica package RICCI for doing tensor computations, we obtain an Einstein Kahler structure only on (TOM, G. J) c: (T M. G, J), where TOM denotes the subset of nonzero tangent vectors to M (Theorem 9). In the last case we obtain that the Kahlerian manifold (T M. G. J) cannot be an Einstein manifold. (C) 2008 Elsevier B.V. All rights reserved.
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Tangent bundle, Riemannian metric, General natural lift, Einstein structure
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