1. The norm defined by the Riemannian metric $g$ on the tangent and cotangent bundles naturally induces a norm on each tensor bundle. This follows from the fact that given vector spaces $V$ and $W$ with inner products, there is a naturally induced inner product on the vector space $V\otimes W$.
2. The metric at $t = 0$ is initially prescribed data and therefore has nothing to do with the Ricci flow itself. So your question is equivalen to: Given a Riemannian metric $g$, when do the covariant derivatives of order $k$ of Riemann curvature have pointwise bounded norm? A sufficient condition is that $g$ can be written in local co-ordinates as $g_ij\,dx^i\,dx^j$, where the function $g_{ij}$ are $C^{k+2}$ functions of the co-ordinates $x^1, \dots x^n$.
EDIT: My answer to #2 is incomplete, since we're working on a noncompact manifold. You also need uniform pointwise upper and lower bounds on the eigenvalues of $g_{ij}$, as well as its derivatives up to order $k+2$ with respect to local co-ordinates.
1. The norm defined by the Riemannian metric $g$ on the tangent and cotangent bundles naturally induces a norm on each tensor bundle. This follows from the fact that given vector spaces $V$ and $W$ with inner products, there is a naturally induced inner product on the vector space $V\otimes W$.
2. The metric at $t = 0$ is initially prescribed data and therefore has nothing to do with the Ricci flow itself. So your question is equivalen to: Given a Riemannian metric $g$, when do the covariant derivatives of order $k$ of Riemann curvature have pointwise bounded norm? A sufficient condition is that $g$ can be written in local co-ordinates as $g_ij\,dx^i\,dx^j$, where the function $g_{ij}$ are $C^k$ C^{k+2}$functions of the co-ordinates$x^1, \dots x^n$. 1 1. The norm defined by the Riemannian metric$g$on the tangent and cotangent bundles naturally induces a norm on each tensor bundle. This follows from the fact that given vector spaces$V$and$W$with inner products, there is a naturally induced inner product on the vector space$V\otimes W$. 2. The metric at$t = 0$is initially prescribed data and therefore has nothing to do with the Ricci flow itself. So your question is equivalen to: Given a Riemannian metric$g$, when do the covariant derivatives of order$k$of Riemann curvature have pointwise bounded norm? A sufficient condition is that$g$can be written in local co-ordinates as$g_ij\,dx^i\,dx^j$, where the function$g_{ij}$are$C^k$functions of the co-ordinates$x^1, \dots x^n\$.