Duality Mappings and Metric Extensor

TL;DR

This paper introduces duality mappings and metric extensor, revealing their fundamental identities and equivalence, and constructs metric products with new inversion formulas, providing deeper insights into geometric algebra structures.

Contribution

It presents the concepts of duality mappings and metric extensor, establishing their identities and equivalence, and develops metric products with novel inversion formulas.

Findings

Fundamental identities of duality mappings are established.

The equivalence between metric tensor and metric extensor is demonstrated.

A surprising inversion formula for the metric extensor is derived.

Abstract

We introduce the key concepts of duality mappings and metric extensor. The fundamental identities involving the duality mappings are presented, and we disclose the logical equivalence between the so-called metric tensor and the metric extensor. By making use of the duality mappings and the metric extensor, we construct the so-called metric products, i.e., scalar product and contracted products of both multivectors and multiforms. The so-known identities involving the metric products are obtained. We find the fundamental formulas involving the metric extensor and, specially, we try its surprising inversion formula. This proposal unveils, once and for all, an unsuspected meaning of the metric products.