Geometric Dirac structure

Definition

A geometric Dirac structure on a Riemannian manifold $(M,g)$ is a Dirac structure where the connection on the manifold is the Levi-Civita connection. Explicitly, it is a quadruple $(E,c,{\mathrm{\nabla }}^E,{\mathrm{\nabla }}^M)$ where:

• $E$ is a Hermitian vector bundle on $M$
• $c:T^{*}(M)\to End(E)$ is a self-adjoint Clifford structure
• ${\mathrm{\nabla }}^M$ is the Levi-Civita connection on $M$
• ${\mathrm{\nabla }}^E$ is a Hermitian connection on $E$ compatible with the Clifford multiplication