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 df = ∑n i=0 f θi, where each θi is a 1-form; the second generalization is represented as Fu = (log ρ)u + 2(I/ρ)e , Fv = −2Jρe−F , where I, J are given by ω = −Jdu+ Idv for a 1-form ω and ρ is given by Ω = ρdu∧dv for a nowhere zero 2-form Ω. Studying these systems, we will obtain generalizations and analogues of results we already have on over-determined systems on surfaces. Results on the second generalization yield corollaries for over-determined systems on surfaces in 3-dimensional non-flat space forms.