Let $\pi\colon TM\to M$ be the tangent bundle of a differentiable manifold, let $E=TM\backslash 0$ be the slit tangent bundle, and let $V_eE$ be the kernel of $\pi_*$ at $e\in E$. The set $VE=\cup_{e\in E} V_eE$ is the vertical bundle. A non-linear connection is a splitting $TE=VE\oplus HE$, where $HE$ is the horizontal bundle. Every vector $v\in TM$ can be lifted to a vector $V\in HE$, such that $\pi_*(V)=v$. Similarly one can lift closed curves of $M$ starting at $p$ to (non-)closed curves on $E$. For every closed curve $\gamma$ this gives a map $\Gamma:T_pM\to T_pM$. If the connection is linear it is well known that this holonomy is a linear map. Is the converse true?

In coordinates, if $\{x^i\}$ are coordinates on $M$ then on TM we have coordinates $\{x^i,y^i\}$ and the basis of $HE$ obtained lifting the holonomic basis is $$ \frac{\partial}{\partial x^i}-N_i^j(x,y) \frac{\partial}{\partial y^j}. $$ The coefficients $N^i_j$ are said to be the coefficients of the non-linear connection, so I am asking if they are linear in $y$ under the assumpton of linear holonomy.