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 $(E,c,\nabla ^{E},\nabla ^{M})$ where:

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