# Dirac structure

## Definition

A Dirac structure on a Riemannian manifold is a quadruple where:

- is a Hermitian vector bundle on
- is a self-adjoint Clifford structure on
- is a metric connection on
- is a Hermitian connection on compatible with Clifford multiplication

A geometric Dirac structure is a Dirac structure where the metric connection is the Levi-Civita connection.