Bull. Korean Math. Soc. 55 (2018), No. 3, pp. 871–879 https://doi.org/10.4134/BKMS.b170344 pISSN: 1015-8634 / eISSN: 2234-3016 ON NEARLY PARAK ¨ AHLER MANIFOLDS Aydin Gezer and Sibel Turanli Abstract. The purpose of the present paper is to study on nearly para- K¨ahlermanifolds. Firstly, to investigate some properties of the Ricci tensor and the Ricci* tensor of nearly paraK¨ahler manifolds. Secondly, to define a special metric connection with torsion on nearly paraK¨ahler manifolds and present its some properties. 1. Introduction An almost product structure on a 2k-dimensional smooth manifold M is a (1, 1)-tensor field P squaring to the identity. In this case, the pair (M,P ) is called an almost product manifold. An almost paracomplex manifold is an almost product manifold (M,P ) such that the two eigenbundles T + M and T − M associated with the two eigenvalues ±1 of P have the same rank. The Nijenhuis tensor N of an almost paracomplex structure P is given by N P (X, Y )=[PX,PY ] − P [PX,Y ] − P [X,PY ]+[X, Y ]. It is well known that an almost paracomplex structure is integrable if and only if the corresponding Nijenhuis tensor N vanishes. An integrable almost para- complex structure is a paracomplex structure. For a survey on paracomplex geometry we refer to [1]. An almost paraHermitian manifold consists of a smooth manifold M en- dowed with an almost paracomplex structure P and a pseudo-Riemannian metric g compatible in the sense that (1.1) g(PX,Y )= −g(X,PY ) or equivalently g(PX,PY )= −g(X, Y ). Note that the metric g is neutral, i.e., it has signature (k,k) and the eigen- bundles T ± M are totally isotropic with respect to g. The condition (1.1) also implies that g is hybrid with respect to P . The 2-covariant skew-symmetric tensor field F defined by F (X, Y )= g(PX,Y ) is the fundamental 2-form of the almost paraHermitian manifold (M,g,P ). Recall the defining conditions of some of the classes: −N P = 0, paraHermitian manifolds, Received April 14, 2017; Accepted September 14, 2017. 2010 Mathematics Subject Classification. Primary 53C55; Secondary 53C05. Key words and phrases. metric connection, nearly paraK¨ahler manifold, Ricci tensor. c 2018 Korean Mathematical Society 871