I'm studying a paper by Demir N. Kupeli, On submanifolds in spacetimes, and during a proof of a proposition, the author say:
"Extend $X\in T_p S$ by making it invariant under the flow generated by K, ..."
I would like to know what's means making it invariant under the flow.
I found in web a definition, but I have some doubts about. Actually, I don't understand the definition.
Let be $K$ a vector field tangent to $S$ and $u\in T_p S$ a irrotational vector. $K$ is called invariant under the flow if
$$[K,u]=\nabla_uK-\nabla_Ku=0.$$
How can I prove that the vector field $[K,u]$ exist?
Why is necessary $u$ be irrotational vector?
Someone has any reference about this thing?
Thanks!
@Irddo a vector field $Z$ is said to be invariant under the flow $\varphi_t$ generated by a vector field $X$ if
$(\varphi_t)_*(Z) = Z$
for all values of $t$ for which the flow exists, where $(\varphi_t)_*$ is the push-forward map induced by the diffeomorphism $\varphi_t$.
As a reference for such things, well, basically any book on differential geometry probably contains the answer (at least if it is advanced enough to talk about differentiable manifolds). A sort of easier to read book which contains the answer would be Boothby's wonderful book "an introduction to differentiable manifolds and Riemannian geometry". Or you can try Lee's book "an introduction to smooth manifolds", which is at a more advanced level than Boothby's previously mentioned book. Anyway, there are tons of books which contain the answer, and these are just two of them.