This is a screenshot of page 5 of the article called "Matsuki correspondence for sheaves"
Here $f$ is a smooth function defined on a Riemannian manifold $X$ and $\nabla f$ is the gradient vector field of $f$.
I was thinking that saying that "the vector field $\nabla f$ is tangent to $G$-orbits" is the same as saying that "the trajectory of $x$ by the gradient flow $\nabla f$ lies in the $G$-orbit $G.x$" but reading this passage in page 5, I realize that it is not, anyone explain the difference between the two statements, please ?
In general, if $M$ is a manifold and $N$ is a submanifold of $M$. Let $\xi$ is a vector field on $M$. What is the difference between saying that $\xi$ is tangent to $N$ and saying that at each point $x$ the flow of $\xi$ lies in $N$ ?