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Let $(E,\nabla)$ be a vector bundle with connection over $M$. I define a proper value function of $\nabla$ as a function $\lambda \in {\cal C}^{\infty}(M)$ such that:

$$\nabla_X (s)=X(\lambda).s$$

for any vector field $X$ over $M$; $s$ is a fixed section of $E$, a proper vector of the connection.

Can we decompose the space of sections of $E$ over proper vectors of the connection?

If $\nabla (s)=(d\lambda) s$, then we have:

$$R_{\nabla} (s)=0$$

where $R_{\nabla}$ is the curvature of the connection, so that it is maybe impossible that we had:

$$\nabla (s)= (d\lambda)s$$

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