The spinorial connections

Question

Let $(M,g)$ be a spin manifold with spinors bundle $\Sigma$, a spinorial connection is an operator:

$$\nabla : \Sigma \rightarrow \Sigma$$

$$\nabla(f \psi)= (df)^* .\psi + f \nabla \psi$$

with $\psi$, a spinor and $f$ a smooth function.

What are the properties of the spinorial connections?

Answer 1

We have:

$$\nabla ={\cal D}+A$$

Where $\cal D$ is the Dirac operator and $A$ is an endomorphism.

Comments

Is this supposed to be an answer to your question "What are the properties of the spinorial connections?" ?

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It is a response for the case where the fiber bundle is trivial, otherwise we have non trivial connections.