Bulletin of the Iranian Mathematical Society Vol. 38 No. 4 (2012), pp 955-972. PARA-K ¨ AHLER TANGENT BUNDLES OF CONSTANT PARA-HOLOMORPHIC SECTIONAL CURVATURE S. L. DRUT ¸ ˘ A-ROMANIUC Communicated by Jost-Hinrich Eschenburg Abstract. We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagonal (almost) para-K¨ ahlerian structures on the tangent bundle, and we study the conditions under which they have constant para-holomorphic sectional curvature. 1. Introduction The natural fiber bundles over manifolds, and in particular the tan- gent and cotangent bundles endowed with various structures of natural lift type, were studied in papers such as [1, 10, 11] [16]-[18], [23]-[32], [35]. Roughly speaking, a natural operator is a fibred manifold mapping, which is invariant with respect to the group of local diffeomorphisms of the base manifold. The results from [16]-[18] allowed the extension of the Sasaki metric, which is very rigid in certain senses, to the metrics of natural lift type, leading to interesting geometric structures and to MSC(2010): Primary: 53C05; Secondary: 53C15, 53C55. Keywords: Natural lifts, almost product structures, para-Hermitian structures, para-K¨ahler structures, para-holomorphic sectional curvature. Received: 29 Septeber 2010, Accepted: 11 June 2011. c  2012 Iranian Mathematical Society. 955