Suppose M is a Riemannian manifold and the function $f: \mathcal{M} \rightarrow \mathbb{R}$ be a scalar function and $\mathrm{grad}f(x) \in T_x M$ is a gradient computed at the point $x \in \mathcal{M}$.
Here, I would like to calculate the norm of the gradient again, that is,
$\mathrm{grad}(\|\mathrm{grad}f(x)\|_x)$
If $\mathcal{M} = \mathbb{R}^d$ (Euclidean space), then this gradient is easily computed as $\displaystyle{\frac{1}{\|\nabla f(x)\|}} H\nabla f(x)$.
For a general Riemannian manifold, is this the same?