Monogenic Functions in Conformal Geometry

TL;DR

This paper explores monogenic functions within conformal geometry, focusing on their definitions, properties, and the conformally invariant extension of the Dirac equation on Riemannian spin manifolds.

Contribution

It provides a clear exposition of monogenic functions and introduces the conformally invariant extension of the Dirac equation on Riemannian spin manifolds.

Findings

Identification of two natural extensions of the Dirac equation

Selection of the conformally invariant extension

Clarification of monogenic functions in conformal geometry

Abstract

Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition.