Note on the locally product and almost locally product structures

Abstract

This paper is concerned with a study of some of the properties of locally product and almost locally product structures on a differentiable manifold X _n of class C^k. Every locally product space has certain almost locally product structures which transform the local tangent space to X _n at an arbitrary point P in a set fashion: this is studied in Theorem (2.2). Theorem (2.3) considers the nature of transformations that exist between two co-ordinate systems at a point whenever an almost locally product structure has the same local representation in each of these co-ordinate systems. A necessary and sufficient condition for X _n to be a locally product manifold is obtained in terms of the pseudo-group of co-ordinate transformations on X _n and the subpseudo-groups [cf., Theoren (2.1)]. Section 3 is entirely devoted to the study of integrable almost locally product structures.