Norm of Riemannian curvature tensor

Question

For the background, I have taken a course on Differential Geometry of curves and surfaces, and am currently trying to self-learn Ricci flows from a book written by Chow and Knopf. In this book, I am encountering an expression $|\text{Rm}|$. I have read what the definition of the Riemannian curvature tensor $\text{Rm}$ on a Riemannian manifold is, but I cannot seem to find a proper definition for $|\text{Rm}|$, which I am guessing is some kind of norm for $\text{Rm}$. Can someone please tell me how it is defined? In particular, if we are on a surface, does $|Rm|$ have any relations with the Gaussian curvature?

Answer 1

This is just the Frobenius/$\ell^2$ norm extended to general tensors - take the tensor square and then contract over corresponding indices: $$|\mathrm{Rm}| = \sqrt{R_{ijkl} R^{ijkl}}.$$

On a surface the sum has four non-zero terms, each of which is $\kappa^2$; so you get $|\mathrm{Rm}| = 2|\kappa|$.