Timeline for Norms of higher derivatives of mappings between Riemannian manifolds
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2011 at 17:02 | comment | added | Deane Yang | Assuming that what you're trying to do is reasonable, I'm pretty sure one of these approaches will work. I am pretty sure in the first approach, if you use the right norms, then the Christoffel symbol terms do no harm. And in the second approach, it is also a matter of using the right topology matched to your particular situation. Is it possible for you to find someone nearby who is experienced in these matters to discuss this? | |
| Jan 14, 2011 at 11:29 | comment | added | Jaap Eldering | If I'd have $M = N = \mathbb{R}$ flat manifolds and $f(x) = x$, but a Nash embedding of $N$ into $\mathbb{R}^n$ that is strongly curved, then the representation of $D^k f$ would pick up these curvature terms, which can globally be unbounded for embeddings of noncompact manifolds. I don't immediately see how to correct for this. | |
| Jan 14, 2011 at 11:19 | comment | added | Jaap Eldering | Indeed I agree about the dependence of $D^k f$ on $j < k$ terms, but the weighted degree of all $j_i$ terms should be at most $k$, as in en.wikipedia.org/wiki/Faà_di_Bruno's_formula. I think that here I get one term $D f$ too many due to the pushforward along $f$ in the induced connection. This would prevent me from getting the right (inductively obtained) exponential growth behaviour $D^k \Phi^t \le C_k e^{k \rho t}$ when $\D \Phi^t \le C e^{\rho t}$. | |
| Jan 13, 2011 at 17:40 | comment | added | Deane Yang | I don't quite understand why Nash would not work for you, but I don't know the details. If you are not varying the Riemannian metric on $N$, then the isometric embedding is fixed. So all it does is add lots of constants to your estimates. The curvature in the normal directions should not need to be controlled, since they never change in what you do. | |
| Jan 13, 2011 at 17:39 | comment | added | Deane Yang | The norm of $Df$ is well-defined, since it does not use the induced connection at all. The norm of $D^2f$ does, but that shouldn't be a problem. In general, the norm of $D^kf$ does depend on $D^jf$ for $j < k$, but that shouldn't be a serious issue. | |
| Jan 13, 2011 at 10:35 | comment | added | Jaap Eldering | Thanks for these suggestions! I'm trying to understand whether they are useful for my problem. For the pullback construction, I understand that one should interpret $D f$ as a vector bundle map $T M \to f^\star(T N)$. But then the induced connection on $f^\star(T N)$ involves the pushforward along $f$, right? This would seem to introduce additional terms that I think I don't want. Unfortunately, I think Nash doesn't help me. I'm working on noncompact spaces and require uniform estimates (by bounded geometry assumptions). A Nash embedding doesn't control curvature in the normal directions. | |
| Jan 12, 2011 at 16:17 | history | answered | Deane Yang | CC BY-SA 2.5 |