An interrelation between geometries of a third-order tangent bundle and the Whitney sum

An affine connection on an $n$-dimensional differentiable manifold $M_n$ gives rise to a diffeomorhism $\sigma$ of the third order tangent bundle $T^3M_n$ into the Whitney sum $TM_n\oplus TM_n\oplus TM_n$. This diffeomorphism carries differential geometric objects from $T^3M_n$ to $TM_n\oplus TM_n\oplus TM_n$. For an arbitrary base $M$ we find the tensor of affine deformation between complete lifts of connections into $T^3M_n$ and into $TM_n\oplus TM_n\oplus TM_n$. In case the connection on the base is torsion-free we demonstrate that this tensor can be expressed in terms of the curvature tensor of the connection given on the base and covariant derivatives of this tensor. Moreover, $\sigma$ carries the connection of complete lift on $T^3M_n$ into the connection of complete lift in $TM_n\oplus TM_n\oplus TM_n$ if and only if the base is flat.