2
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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 $\alpha$.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – user10354138 May 3 '19 at 11:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.