Let $G$ be a compact Lie group and let $T$ be a maximal torus of $G$. Consider the flag manifold $M=G/T$. Let's fix a regular element $r$ of $\mathfrak{t}^*$ and identify $M$ with the coadjoint orbit $G.r$. Let $f:M \rightarrow \mathbb{R}$ be a smooth function.
I've come across the following statement "Let's denote $\nabla f$ the gradient vector field of $f$." This was without specifying any metric on $M$, why is that ? Is it because there is a canonical metric on $M$ (if so, how it is defined), or is it because $\nabla f$ independent of the choice of a metric on $M$ ?
The definition of gradient vector field I'm using is the following
Definition: Let (M,g) be a Riemannian manifold. The gradient of a smooth function $f:M\to \mathbb{R}$ with respect to the metric $g$ is the unique smooth vector field $\text{grad}_g\, f$ on $M$ such that for all smooth vector fields $Y$ on $M$, $$g(\text{grad}_g\, f, Y) = df(Y).$$
The Killing metric is a natural choice for any compact semi-simple group. It is bi-invariant therefore induces a natural $G$ invariant metric on any homogenous space $G/K$. But other choices are useful. Indeed, any $K$ invariant quadratic form on the Lie algebra of $G$ induces a invariant metric on $G/K$. In the case of $G/T$ you obtain a family of Kähler $G$-invariant metrics.