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 In how many ways can an iterated tangent bundle (T^k)M be viewed as a fibre bundle over (T^(k-1))M?) 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 $D \Phi^t$.)
Best Answer
You could indeed pull back each tensor bundle of $N$ via the map $f$ to $M$ and use the naturally induced metric and connection to define norms of higher covariant derivatives of $f$. You can do this for every order greater than or equal to $1$. Depending on your needs, this might work fine.
If you need something that will work under weaker a priori assumptions, then you might need to embed $N$ isometrically into a higher dimensional Euclidean space (via Nash's theorem) and treat $f$ as a vector-valued function. This allows you to work with maps $f$ that are not necessarily continuous and define its weak derivatives. This is often used in the study of harmonic maps.