A coordinate-free reduction for flows on the volume manifold

Abstract

In this paper, instead of previous methods such as the Lie group method dependent of coordinate transformation, a coordinate-free and pure geometric reduction procedure for flows preserving volume form on any n-dimensional volume manifold is given by the general differential form theory. That is, a volume form-preserving flow with a r-parameter Abelian volume-preserving symmetry group on any n-dimensional volume manifold can be reduced into a volume-preserving flow on the corresponding ( n-r)-dimensional manifold by the differential geometric method.