I am wondering if the flow of a Killing vector field preserves the submanifolds?
In fact, on a Riemannian manifold $(M,h)$ I have a Killing vector field $W$. Let $\varphi:I\times M\to M$ be its flow. We know that $\varphi_t:=\varphi(t,.):M\to M$ is an isometry.
I can say that given a submanifold $A\subset M$, $\varphi_t(A)\subset A$ and therefore, $d\varphi_t:T_pA\to T_pA$?
Or, in othe wrods, $\forall u\in T_pA$, $d\varphi_t(u)\in T_pA$?
I will appreciate any comments or answers in advance!
Consider $\mathbb{R}^2$ with the Eucledean metrix $\phi_t(x)=x+tu$ is the flow of the vector field $X(x)=u$ where $u\in\mathbb{R}^2$ is not zero. Take the circle of radius $1$ center at $0$, it is not preserved by $\phi_t$ and $X$ is tangent to $C$ at $(0,1)$ if $u=(1,0)$.