Lichnerowicz-Witten differential, symmetries and locally conformal symplectic structures

Abstract

A locally conformal symplectic structure (for short l.c.s.) on a smooth manifold M is a generalisation of a symplectic structure. In this paper at first the theory of locally conformal symplectic structures is reviewed, and a description of the Lichnerowicz-Witten (LW) deformed differential operator is given. Using the exterior algebra of the LW differential operator, Hamiltonian vector fields associated to such l.c.s. structures are introduced. Several useful identities of the deformed exterior calculus are derived. The theory of symmetries of such locally conformal symplectic structures is developed. We show examples of the applications of our formalism, in particular, we present nonholonomic oscillator equation which admits a locally conformal symplectic structure. We study canonoid transformations of a locally Hamiltonian vector field on a locally conformal symplectic manifold. In particular, we present a generalized geometric theory of canonoid transformation in the l.c.s. structure setting.