Connections on tangent bundles and double tangent bundles

Question

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.

Answer 1

This is wrong even locally. You want: $$T^2f\big(x,y;\xi,\Gamma^1_x(y,\xi)\big) = \big(f(x),df(x)y;df(x)\xi,d^2f(x)(y,\xi) + df(x)\Gamma^1_x(y,\xi)\big)$$ to be of the form $$ \big(f(x),df(x)y;df(x)\xi,\Gamma^2_{f(x)}(df(x)y,df(x)\xi)\big).$$ But if $y\ne 0$ with $df(x)y=0$ but $d^2f(x)(y,\xi)\ne0$, then the lower formula is 0 in the last component whereas the upper one is $\ne0$, for any choice of of $\Gamma^1$ and $\Gamma^2$.

If $f$ is an immersion then you can have it (pull back a Riemannian metric from $N$ to $M$ and consider the Levi-Civita connections.