Norms of higher derivatives of mappings between Riemannian manifolds

2011-01-12T11:23:54Z

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 $D \Phi^t$.)

Answer by Deane Yang for Norms of higher derivatives of mappings between Riemannian manifolds

2011-01-12T16:17:37Z

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.

Norms of higher derivatives of mappings between Riemannian manifolds - MathOverflow

Norms of higher derivatives of mappings between Riemannian manifolds

Answer by Deane Yang for Norms of higher derivatives of mappings between Riemannian manifolds