Geometric calculus on pseudo-Riemannian manifolds

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.