Publ. Math. Debrecen 89/4 (2016), 499–511 DOI: 10.5486/PMD.2016.7545 H-projectively Euclidean K¨ahler tangent bundles of natural diagonal type By CORNELIA-LIVIA BEJAN and SIMONA-LUIZA DRUT¸ ˘ A-ROMANIUC Dedicated to Professor Lajos Tam´assy Abstract. We obtain the characterization of the natural diagonal K¨ ahler mani- folds (TM,G,J ) which have constant holomorphic sectional curvature, or equivalently, which are H-projectively Euclidean. Moreover, we classify the natural diagonal K¨ ahler manifolds (TM,G,J ) which are horizontally H-projectively flat (resp. vertically H- projectively flat). 1. Introduction The holomorphically planar curves were introduced in 1954 by Otsuki and Tashiro [19] to generalize in some extent, in the K¨ahlerian context, the notion of geodesics from the Riemannian case. In this sense, the projective transfor- mations, i.e. the transformations preserving the geodesics (see [2], [8], [9], [26]), have as a K¨ahlerian correspondent the holomorphically projective transforma- tions, i.e. the transformations preserving the holomorphically planar (H-planar) curves (see [19]). A well-known result is that a K¨ahlerian space holomorphically projective to an Euclidean space (called also H-projectively Euclidean space) has constant holomorphic sectional curvature (see [27]). Mathematics Subject Classification: 53C55, 53C15, 53C05. Key words and phrases: tangent bundle, natural diagonal metric, K¨ahler manifold, holomor- phically projective transformation, H-projective curvature tensor field, holomorphic sectional curvature.