J. geom. 00 (2001) 000 – 000 0047–2468/01/000000 – 00 $ 1.50 + 0.00/0 © Birkh¨ auser Verlag, Basel, 2001 On almost paracontact Riemannian manifolds of type (n, n) Mancho Manev and Maria Staikova Abstract. In this paper we give a classification with eleven basic classes of almost paracontact Riemannian mani- folds of type (n, n) with respect to the covariant derivative of the (1, 1)-tensor of the almost paracontact structure. Mathematics Subject Classification (2000): 53C15, 53C25. Key words: Almost paracontact, Riemannian manifolds. 1. Introduction In 1976 I. Sato [1] introduced the concepts of almost paracontact manifolds and of almost paracontact Riemannian manifolds as analogues of almost contact manifolds and of almost contact Riemannian manifolds. After that S. Sasaki [2] defined the notion of an almost paracontact Riemannian manifold of type (p,q) and arbitrary dimension, where p and q are the numbers of the multiplicity of the structural eigenvalues 1 and −1, respectively. In addition, there is a simple eigenvalue 0. In this paper we consider almost paracontact Riemannian manifolds of type (n, n), i.e. p = q = n. We put this fixation in view of reasons of later investigations relevant to 2n-dimensional Riemannian almost product manifolds (M 2n ,P,g) with structural group O(n) × O(n), which are classified in [3]. In this reason the manifolds in our consideration could be construct by natural way as a direct product of (M 2n ,P,g) and a real line or as a hypersurface of (M 2n ,P,g). The method used in the present paper is analogous to the methods of classification in [4] and partly to those in [5] for the almost contact metric manifolds and for the almost contact manifolds with B -metric, respectively. 2. Preliminaries A (2n + 1)-dimensional real differentiable manifold M is said to have an almost paracontact structure (φ,ξ,η) of type (n, n), if it admits a (1, 1)-tensor φ, a vector field ξ and a 1-form η satisfying the following conditions: η(ξ) = 1, φ 2 = I 2n+1 − η ⊗ ξ, trφ = 0. (1) 1