Let $M, N$ be Riemannian manifolds and $f: M \to N$ be a smooth map (I'm actually only considering diffeomorphisms (flows) $\Phi^t: M \to M$, but just for the sake of generality).
The first derivative of $f$ can be understood as its tangent map $T f: T M \to T N$. Higher derivatives can abstractly be viewed as maps between higher order tangent bundles.
I want to make estimates on the (operator norm) size of these higher derivatives. In the higher order tangent spaces (see also the recent question http://mathoverflow.net/questions/2019/) I'd have to use induced metrics, which I don't readily know how to work with, and besides, I think these would include the base, lower order derivatives as well.
I would prefer to keep things defined on the tangent/tensor bundle, in a similar way as taking covariant derivatives for vector fields, but I don't know how to do this for for maps $f: M \to N$.
So my question roughly is: are there natural/practical representations of norms of higher order derivatives of maps between manifolds?
One thing I did come up with is representing $f$ in normal coordinates, as these are the most canonical charts and then use the norms in the tangent spaces at the argument and image points $x$ and $y = f(x)$.
(The basis for this question is that I want to obtain a Gronwall-like growth estimate for the higher derivatives of a flow $\Phi^t$ in terms of the exponential growth of its tangent flow $D \Phi^t$.)