On quarter-symmetric metric connections on pseudo-Riemannian manifolds Iulia Elena Hiric˘a and Liviu Nicolescu Abstract. The geometric significance of semi-symmetric connections was originally studied by K. Yano ([13]). The notion was extended to quarter symmetric connections by S. Golab ([3]). In the present paper the theory is extended and it is shown that the Golab algebra associated to a quarter symmetric metric connection is essential in order to characterize the geometry of a pseudo-Riemannian manifold. M.S.C. 2010: 53B20, 53B21, 50B20. Key words: quarter-symmetric metric connections, F -principal vector fields, Golab connections, associative deformation Golab algebras, Einstein spaces. Introduction Throughout this paper one considers M a connected paracompact, smooth man- ifold of dimension n. Let X (M ) be the Lie algebra of vector fields on M, T p M the vector space of tangent vectors in a point p ∈ M, T (r,s) (M ) the C ∞ (M )-module of tensor fields of type (r, s) on M, Λ p (M ) the C ∞ (M ) −module of p−forms on M . Let A be a (1, 2)−tensor field on M . The C ∞ (M ) −modul X (M ) becomes a C ∞ (M ) −algebra if we consider the multiplication rule given by X ◦ Y = A (X, Y ), ∀X, Y ∈X (M ) . This algebra is denoted by U (M,A) and it is called the algebra associated to A. If ∇ and ∇ ′ are two linear connections on M and A = ∇ ′ −∇, then U (M,A) is called the deformation algebra defined by the pair (∇, ∇ ′ ) ([10]). In the present paper we continue and develop the study of [4], generalizing the no- tion of quarter-symmetric metric connections along the line of symmetric connections on pseudo-Riemann manifolds. Interesting properties of semi-symmetric connections or quarter-symmetric connections can be obtained on manifolds endowed with special structures ([1], [6], [7]) and extensive literature with applications can be mentioned ([2], [12]). The aim of this work is to characterize the F -principal vector fields in the de- formation algebra of two linear connections. It is illustrated the close ties between * Balkan Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 56-65. c  Balkan Society of Geometers, Geometry Balkan Press 2011.