Differential $p$-forms and $q$-vector fields with constant coefficients

TL;DR

This paper characterizes differential $p$-forms and $q$-vector fields with constant coefficients on manifolds, explores their properties, introduces conformal variants, and discusses solving related PDE systems with applications.

Contribution

It provides a comprehensive characterization of constant coefficient forms and vector fields, introduces conformal variants, and analyzes associated PDE systems with computational methods.

Findings

Characterization of constant coefficient $p$-forms and $q$-vector fields.

Obstruction identified via Schouten-Nijenhuis bracket.

Reduction of PDE systems for solution computation.

Abstract

Differential $p$-forms and $q$-vector fields with constant coefficients are studied. Differential $p$-forms of degrees $p=1,2,n-1,n$ with constant coefficients on a smooth $n$-dimensional manifold $M$ are characterized. In the contravariant case, the obstruction for a $q$-vector field $V_q$ to have constant coefficients is proved to be the Schouten-Nijenhuis bracket of $V_q$ with itself. The $q$-vector fields with constant coefficients of degrees $q=1,2,n-1,n$ are also characterized. The notions of differential $p$-forms and $q$-vector fields with conformal constant coefficients are introduced. For arbitrary degrees $p$ and $q$, such differential $p$-forms and $q$-vector fields are seen to be the solutions to two second-order partial differential systems on $J^2(M,\mathbb{R}^n)$, which are reducible to two first-order partial differential systems by adding variables. Computational aspects in solving these systems are discussed and examples and applications are also given.