# Geometric Dirac structure

## Definition

A **geometric Dirac structure** on a Riemannian manifold is a Dirac structure where the connection on the manifold is the Levi-Civita connection. Explicitly, it is a quadruple where:

- is a Hermitian vector bundle on
- is a self-adjoint Clifford structure
- is the Levi-Civita connection on
- is a Hermitian connection on compatible with the Clifford multiplication