The phi-connections

Question

Let $\phi$ be an endomorphism of a fiber bundle. A $\phi$-connection is such that:
$$\nabla_X (fs)=Xf.\phi (s)+ f \nabla_X (s)$$
If $\phi \nabla=\nabla \phi$ then we can define the curvature:
$$R(X,Y)=\nabla_X \nabla_Y- \nabla_Y \nabla_X -\phi \nabla_{[X,Y]}$$
Has the $\phi$-connection a physical meaning?