This can be viewed as a sequel to my previous question on double tangent bundle. Where I learned that the double tangent bundle $TTM$ is not **natural** diffeomorphic to $\oplus^3 TM$.

Recently, I also learned that the double tangent bundle $TTM$ is however **non-canonical** diffeomorphic to $\oplus^3 TM$, and the way to split $TTM$ to $\oplus^3 TM$ corresponds to a linear connection on $TM\to M$. This is studied in the frame work of DVB double vector bundle ($TTM$ is a DVB), see for example arXiv:0810.0066v2.

Now I am wondering the following question, it may seem evil. Given a smooth map of manifolds: $f: M\to N$. Can we find splittings of $TTM\cong \oplus^3 TM$ and $TTN\cong\oplus^3 TN$ such that the splittings respect $f$. Or in terms of connections, can we find connections $\omega_1$ on $M$ and $\omega_2$ on $N$ such that $f$ preserves the connections (I do not know how to formulate the intuition, it should not be $f^*\omega_2=\omega_1$ since $TM$ is not pull back bundle of $TN$). My intuition is that there may be some topological obstruction.