I guess the term "spacetime symmetry" of field indeed implies the symmetry has a strong connection with the Killing vector field. But I have no idea how to express it formally.

For Arnold's book mathematical methods of classical mechanics, the Noether's theorem is connected with a one-parameter diffeomorphism group.

A Lagrangian system $(M, L)$ admits the mapping $h$ if for any tangent vector $v\in TM$, $$L(h_* v) = L(v).$$

If the system $(M, L)$ admits the one-parameter group of diffeomorphisms $h^S: M \rightarrow M, s \in R$, then the lagrangian system of equations corresponding to $L$ has a first integral $I: TM \rightarrow R$

In local coordinates q on M the integral $I$ is written in the form

$$I(q,\dot q)=\frac{\partial L}{\partial \dot q}\frac{d h^s(q)}{ds}|_{s=0}$$

What I want is: let the one-parameter diffeomorphism group be the flow generated by the Killing vector field, then the first integral is the current I want.

But indeed there are many technical issues, like the generalization of the tangent bundle. So I wonder could someone help me to write it down or just let me know my thought is not correct.

Thank you for your answering.

I guess the term "spacetime symmetry" of field indeed implies the symmetry has a strong connection with the Killing vector field. But I have no idea how to express it formally.

For Arnold's book mathematical methods of classical mechanics, the Noether's theorem is connected with a one-parameter diffeomorphism group.

A Lagrangian system $(M, L)$ admits the mapping $h$ if for any tangent vector $v\in TM$, $$L(h_* v) = L(v).$$

If the system $(M, L)$ admits the one-parameter group of diffeomorphisms $h^S: M \rightarrow M, s \in R$, then the lagrangian system of equations corresponding to $L$ has a first integral $I: TM \rightarrow R$

In local coordinates $q$ on $M$ the integral $I$ is written in the form

$$I(q,\dot q)=\frac{\partial L}{\partial \dot q}\frac{d h^s(q)}{ds}|_{s=0}$$

What I want is: let the one-parameter diffeomorphism group be the flow generated by the Killing vector field, then the first integral is the current I want.

But indeed there are many technical issues, like the generalization of the tangent bundle. So I wonder could someone help me to write it down or just let me know my thought is not correct.