# Exponential of a connection

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Let $E$ be a vector bundle over $M$ and $\nabla$, a connection, I can define $e^{\nabla_X}$, it verifies:

$$e^{\nabla_X}(f.s)= (e^X f).e^{\nabla_X}(s)$$

where $s$ is a section, $f$ a smooth function over $M$ and $X$ a vector field over $M$. I define:

$$\tilde R_{\nabla}(X,Y)= e^{t\nabla_X}e^{t\nabla_Y}e^{-t\nabla_X}e^{-t\nabla_Y}e^{-t^2 \nabla_{[X,Y]}}$$

It verifies:

$$\tilde R_{\nabla}(X,Y)=1+t^2 R_{\nabla}(X,Y)+o(t^2)$$

where $R_{\nabla}$ is the curvature of $\nabla$. Is it bound to the holonomy of the connection $\nabla$?

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