Mechanics of Continuous Media in (Ln, g)-spaces. I. Introduction and mathematical tools S. Manoff Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Department of Theoretical Physics, Blvd. Tzarigradsko Chaussee 72 1784 Sofia - Bulgaria e-mail address:
[email protected] Abstract Basic notions and mathematical tools in continuum media mechanics are recalled. The notion of exponent of a covariant differential operator is intro- duced and on its basis the geometrical interpretation of the curvature and the torsion in (Ln,g)-spaces is considered. The Hodge (star) operator is general- ized for (Ln,g)-spaces. The kinematic characteristics of a flow are outline in brief. PACS numbers: 11.10.-z; 11.10.Ef; 7.10.+g; 47.75.+f; 47.90.+a; 83.10.Bb 1 Introduction 1.1 Differential geometry and space-time geometry arXiv:gr-qc/0203016v1 5 Mar 2002 In the last years, the evolution of the relations between differential geometry and space-time geometry has made some important steps toward applications of more comprehensive differential-geometric structures in the models of space-time than these used in (pseudo) Riemannian spaces without torsion (Vn-spaces) [1], [2]. 1. Recently, it has been proved that every differentiable manifold with one affine connection and metrics [(Ln,g)-space] [3], [4] could be used as a model for a space- time. In it, the equivalence principle (related to the vanishing of the components of an affine connection at a point or on a curve in a manifold) holds [5] ÷ [11], [12]. Even if the manifold has two different (not only by sign) connections for tangent and co-tangent vector fields [(Ln,g)-space] [13], [14] the principle of equivalence is fulfilled at least for one of the two types of vector fields [15].