Volume-preserving maps, source-free systems and their local structures

Abstract

In this paper, we study local structures of volume-preserving maps and source-free vector fields, which are defined in the Euclidean n-space Rn with n ≥ 3. First, we prove that any volume-preserving map, defined in some neighbourhood of the origin, can be represented as a composition of n − 1 essentially two-dimensional area-preserving maps. This result can be viewed as an analogue of the following known fact (Feng and Shang 1995 Volume-preserving algorithms for source-free dynamical systems Numer. Math. 71 451–63): any source-free vector field on Rn can be represented as a sum of n − 1 essentially two-dimensional Hamiltonian vector fields. Then, we present a local representation of source-free vector fields under volume-preserving coordinate changes. Finally, we construct a Lie algebra of skew-symmetric tensor potentials of second order associated with source-free vector fields. The Lie algebra turns out to be isomorphic to the Lie algebra of source-free vector fields.