Convergence of Riemannian metrics

Question

Let $M$ be a smooth manifold.Let $(g_k)_k$ be a sequence of Riemannian metrics. Let $g$ be another Riemannian metric. What does it mean that $g_k$ converges to $g$ in $C^2$ norm?

Answer 1

This means, there is a (locally finite) open cover $\{U_\alpha\}_{\alpha\in I}$ of $M$, and a coordinate system $\varphi_\alpha: U_\alpha\to {\mathbb R}^n$ for each $\alpha\in I$, so that $(\varphi_\alpha^{-1})^*g_k$ (this is just $g_k$ written in this coordinate system, say as an $n\times n$ matrix valued function on the domain $\varphi_\alpha (U_\alpha)\subset {\mathbb R}^n$) converges to $ (\varphi_\alpha^{-1})^*g$, in $C^2$ for every $\alpha$.