Geometric calculus on pseudo-Riemannian manifolds

TL;DR

This paper introduces a direct, elementary approach to geometric calculus on pseudo-Riemannian manifolds, avoiding embeddings and aligning with general relativity pedagogy, to develop differential calculus for various fields.

Contribution

It presents a novel axiomatic method for geometric calculus on pseudo-Riemannian manifolds that does not rely on embedding into tensor or vector spaces.

Findings

Develops a full differential calculus for vector, multivector, and tensor fields.

Provides a pedagogical framework aligned with general relativity.

Avoids complex embeddings, simplifying the understanding of geometric calculus.

Abstract

This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra or within a vector manifold framework, here we define geometric calculus directly, by elementary methods. In particular we use an axiomatic approach that directly parallels textbook introductions to general relativity and pseudo-Riemannian geometry, so that no structure outside the metrical Clifford bundle of the manifold need be introduced. On this basis we develop the full theory of differential calculus for vector, multivector, and tensor fields.