# Convergence of Riemannian metrics

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?

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 $$\alpha$$.
• You also need some conditions on the chart $\varphi_\alpha$ so it doesn't do silly things like blowing up $g$ near the boundary of the chart, especially if $M$ is not compact. – user10354138 May 3 '19 at 11:17