Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I have two questions
What can we say about the integral curve of $Z$ on the product manifold? I think it is a diagonal product of the two integral curves. Note that the product of the two curves is a $2-$ dimensional surface.
I do not understand the following expressions: (1) $Y(f)=0$ since $Y$ is defined on a manifold and $f$ is defined on another one (2) $X(f)Y$ as a vector field on $N$ since $X(f)$ is a function on $M$ where $Y$ is a vector field on $N$.
Thank you.