ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N.42–2019 (51–58) 51 Semi-symmetry properties of the tangent bundle with a pseudo-Riemannian metric Aydin Gezer * Ataturk University Faculty of Science Department of Mathematics 25240, Erzurum Turkey agezer@atauni.edu.tr Cagri Karaman Ataturk University Faculty of Science Department of Mathematics 25240, Erzurum Turkey cagri karamannn@hotmail.com Abstract. In this note, we consider the tangent bundle TM equipped with a pseudo- Riemannian metric  g over a Riemannian manifold (M,g). We investigate semi-symmetry properties of the tangent bundle TM with respect to the Levi-Civita connection  ∇ and a metric connecton  ∇ with torsion. Keywords: metric connection, pseudo-Riemannian metric, Ricci semi-symmetry, semi-symmetry, tangent bundle. 1. Introduction A Riemannian manifold (M,g) is said to be locally symmetric if the Riemannian curvature tensor R is parallel with respect to the Levi-Civita connection ∇, i.e., ∇R = 0. A natural generalization of the notion of local symmetry is semi- symmetry. A semi-symmetric space is a (pseudo-) Riemannian manifold (M,g) such that its curvature tensor R satisfies the condition R(X, Y ) · R =0 for all vector fields X and Y on M , where R(X, Y ) is a linear operator acting as a derivation on the curvature tensor R. This class of spaces was first studied by E. Cartan. Nevertheless, N. S. Sinjukov first used the name ”semi-symmetric spaces” for manifolds satisfying the above curvature condition [5]. Later, Z. I. Szabo gave the full local and global classification of semi-symmetric spaces *. Corresponsing author