Two generalizations of an over-determined system on a surface

Abstract

We introduce and study two generalizations of an over-determined system on a surface. The first generalization is an over-determined system of polynomial type, which is in the form of [Formula: see text], where each [Formula: see text] is a one-form; the second generalization is represented as [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are given by [Formula: see text] for a one-form [Formula: see text] and [Formula: see text] is given by [Formula: see text] for a nowhere zero two-form [Formula: see text]. Studying these systems, we will obtain generalizations and analogs of results we already have on over-determined systems on surfaces. In addition, we will obtain new kinds of results on the systems. Over-determined systems of the first generalization appear on admissible hypersurfaces in [Formula: see text] ([Formula: see text]). Results on the second generalization yield corollaries for over-determined systems on surfaces in three-dimensional non-flat space forms.