A differential approach to Maxwell-Cremona liftings

TL;DR

This paper generalizes Maxwell-Cremona liftings to higher dimensions using differential forms, enabling the study of self-stressed frameworks in multidimensional spaces and extending classical concepts to complex structures.

Contribution

It introduces differential liftings for frameworks in higher dimensions, extending classical planar liftings to graphs and complexes in $d$-space.

Findings

Differential liftings are defined using homotopy group-based differential forms.

The approach extends classical liftings to higher-dimensional frameworks.

A representation of liftings as functions on Grassmannians is provided.

Abstract

In 1864, J. C. Maxwell introduced a link between self-stressed frameworks in the plane and piecewise linear liftings to 3-space. This connection has found numerous applications in areas such as discrete geometry, control theory and structural engineering. While there are some generalisations of this theory to liftings of $d$-complexes in $d$-space, extensions for liftings of frameworks in $d$-space for $d\geq 3$ have been missing. In this paper, we introduce and study differential liftings on general graphs using differential forms associated with the elements of the homotopy groups of the complements to the frameworks. Such liftings play the role of integrands for the classical notion of liftings for planar frameworks. We show that these differential liftings have a natural extension to self-stressed frameworks in higher dimensions. As a result we generalise the notion of classical liftings to both graphs and multidimensional $k$-complexes in $d$-space ($k=2,\ldots, d$). Finally we discuss a natural representation of generalised liftings as real-valued functions on Grassmannians.