$(M,g)$ is a Riemannian manifold, $(g_{t})_{t >0}$ is a family of Riemannian metrics on $M$. What does ''$g_{t} \xrightarrow[t \to 0]{} g$ in the $\mathcal{C}^r$ sense for any $r \ge 0$'' means?
More precisely, I know the convergence in the Hölder sense for functions, using the Hölder norms, but I don't know about Hölder norms for metrics.