While reading "Hopper, Andrews - The Ricci Flow in Riemannian Geometry" I came across Shi's global derivative estimates, which posed two problems for me:

1. For a manifold (M,g) with curvature tensor $Rm$: how exactly are $\left| Rm\right|$ and $\left| \nabla^k Rm\right|$ at a point $p\in M$ defined? Is there some kind of standard tensor norm you can use here?
2. Since Shi's result [for a ricci flow solution $(M,g(t))\_{ t\in [0,T]}$] only gives us bounded curvature derivatives $\left| \nabla ^k Rm (t)\right|$ for times $t>0$ , I want to know under which circumstances the derivatives $\left| \nabla ^k Rm (0)\right|$ are also bounded. (Is e.g. the compactness of $(M,g(0))$ sufficient for this bound?)