No abstract

Abstract. This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints.Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialise these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of "positive space" and its rational powers. Positive spaces are "semi-vector spaces" on which the group of positive real numbers acts freely and transitively through the scalar multiplication. Their tensor multiplication with vector spaces yields "scaled spaces" that are suitable to describe spaces with physical dimensions mathematically. We also deal with scales regarded as fields over a given background (e.g., spacetime).

We define an almost-cosymplectic-contact structure which generalizes cosymplectic and contact structures of an odd dimensional manifold. Analogously, we define an almost-coPoisson-Jacobi structure which generalizes a Jacobi structure. Moreover, we study relations between these structures and analyse the associated algebras of functions.As examples of the above structures, we present geometrical dynamical structures of the phase space of a general relativistic particle, regarded as the 1st jet space of motions in a spacetime. We describe geometric conditions by which a metric and a connection of the phase space yield cosymplectic and dual coPoisson structures, in case of a spacetime with absolute time (a Galilei spacetime), or almost-cosymplectic-contact and dual almost-coPoisson-Jacobi structures, in case of a spacetime without absolute time (an Einstein spacetime).2000 Mathematics Subject Classification. 53B15, 53B30, 53B50, 53D10, 58A10, 58A32 .

We apply the graph complex method of [7] to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an 'ideal' basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi-Ricci identities without the correction terms.Plan of the paper. In Sections 1 and 2 we recall classical reduction theorems and the Bianchi-Ricci identities. The main results of this paper, Theorems A-F, are formulated in Section 3. Sections 4, 5 and 6 contain necessary notions and results of the graph complex theory and related homological algebra. Section 7 provides proofs of the statements of Section 3.