Let (E,M) be a vector bundle, ϕ an endomorphisms and X a vector fields of M. I call ∇X⊗ϕ a double connection if:
∇:E→TM∗⊗End(E)∗⊗E
∇X⊗ϕ(s+s′)=∇X⊗ϕ(s)+∇X⊗ϕ(s′)
∇X⊗ϕ(f.s)=X(f).ϕ(s)+f.∇X⊗ϕ(s)
With s,s′ two sections of E, and f a smooth function of M.
What is the space of double connections?