# On what quantities is a (differential geometric) connection not allowed to depend?

If we parallel transport a vector $v$ along a closed curve $c_1$, the vector will end up as $v'=A_{c_1}v$, where $A_{c_1}$ is an element of the holonomy group. If we consider composition of curves curves $c_{21}=c_2\circ c_1$, then we get the transformation $A_{c_2}·A_{c_1}$.

Are there connections where $A_c$ is not $v$-independend? If not why, and where in the differential geometric framework are the roots for this "restriction"? I guess that if we consider the case where $A$ depends on the $v$ at hand, the destiction between the space and the tangent space might be a little blurry, but I can not really justfy that feeling. Are such ideas for connections "A(v)" investigated?

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