Definition:Gradient Operator/Riemannian Manifold

Definition 1
Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.

Let $f \in \map {\CC^\infty} M$ be a smooth mapping on $M$.

The gradient of $f$ is defined as:

where $\d_{\d R}$ is de Rham differential.