# Endo-connections

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Let $(E,A,M)$ be a vector bundle over a manifold $M$, and $A$ is a commutativ subalgebra of endomorphisms of $E$, $\tilde \nabla$ is a connection of $A$. An endo-connection $\nabla$ over $(E,A,M)$ is an operator $\nabla$ such that:

$$\nabla : E \rightarrow TM \otimes A \otimes E$$

$$\nabla_{X\otimes a}(s)=a.\nabla_{X \otimes 1}(s)$$

$$\nabla_{X\otimes a } (a'.s)=a.\tilde \nabla_X a'.s+ a'. \nabla_{X\otimes a} (s)$$

With $X$ a vector, and $a,a'$ sections of $A$ and $s$ a section of $E$.

What is the space of endo-connections?

asked Nov 27, 2021
edited Nov 27, 2021

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