What you are trying to express, is the following, imho. For the sake of clarity let us split $M$ into two manifolds, $M$, $N$. Consider the 1-jet bundle $\pi_{M\times N}:J^1(M,N)\to M\times N$, which is bundle isomorphic to $L(TM,TN)$. Given smooth $f:M\to N$, we get the 1-jet section $j^1f:M\to J^1(M,N)$ of $\pi_M: J^1(M,N)\to M$ which satisfies $\pi_N\circ j^1f = f:M\to N$.

Now your question is: Given a section $s:M\to J^1(M,N)$ of $\pi_M: J^1(M,N)\to M$, can you recognize when $s=j^1(\pi_N\circ s)$.

Answer: In fact you can. There is a bunch of canonical 1-forms (called contact forms or Lepage forms) on $J^1(M,N)$, namely those $\omega\in\Omega^1(J^1(M,N)$ with $\pi_M^*\omega=0$.

• We have $s=j^1(\pi_N\circ s)$ if and only if $s^*\omega = 0$ for each contact form. See Wikipedia.

Note that the gauge group $\operatorname{Gau}(M)$ acts from the right on $J^1(M,N)$, and $\operatorname{Gau}(N)$ acts from the left.

What you are trying to express, is the following, imho. For the sake of clarity let us split $M$ into two manifolds, $M$, $N$. Consider the 1-jet bundle $\pi_{M\times N}:J^1(M,N)\to M\times N$, which is bundle isomorphic to $L(TM,TN)$. Given smooth $f:M\to N$, we get the 1-jet section $j^1f:M\to J^1(M,N)$ of $\pi_M: J^1(M,N)\to M$ which satisfies $\pi_N\circ j^1f = f:M\to N$.

Now your question is: Given a section $s:M\to J^1(M,N)$ of $\pi_M: J^1(M,N)\to M$, can you recognize when $s=j^1(\pi_N\circ s)$.

Answer: In fact you can. There is a module (over $C^\infty(M)$) of canonical 1-forms (called contact forms or Lepage forms) on $J^1(M,N)$, (edited) locally generated by $dy^j - k^j_i\,dx^i$ in terms of coordinates $(x_i,y^j,k^j_i)$ on $J^1(M,N)$ induced by coordinates $(x^i)$ on $M$ and $(y^j)$ on $N$.

• We have $s=j^1(\pi_N\circ s)$ if and only if $s^*\omega = 0$ for each contact form. See Wikipedia.

Note that the gauge group $\operatorname{Gau}(M)$ acts from the right on $J^1(M,N)$, and $\operatorname{Gau}(N)$ acts from the left.