A Riemannian metric on a differentiable manifold $M$ is a collection $g$ of inner products

that is differentiable in the sense that for any two differentiable vector fields $X$ and $Y$, the function

is differentiable

In this context, given a chart $\varphi:U\longrightarrow M$ inducing

the (local) functions

are differentiable, and we may express