A Characterization of Whitney forms

TL;DR

This paper characterizes Whitney forms on simplices, showing they are uniquely determined by their affine coefficients and their integrals over faces, linking cochains to differential forms.

Contribution

It provides a new characterization of Whitney forms, establishing their uniqueness based on affine properties and face integrals, enhancing understanding of their geometric and algebraic structure.

Findings

Whitney forms are uniquely determined by their face integrals.

The form $Wc$ has affine coefficients and pulls back to constant forms on faces.

The characterization links cochains directly to differential forms with specific properties.

Abstract

We give a characterization of Whitney forms on an $n$-simplex $\sigma$ and prove that for every real valued simplicial $k$-cochain $c$ on $\sigma$, the form $Wc$ is the unique differential $k$-form $\varphi$ on $\sigma$ with affine coefficients that pulls back to a constant form of degree $k$ on every $k$-face $\tau$ of $\sigma$ and satisfies $\int_{\tau} \varphi = <c,\tau>$.