Questions tagged [jet-bundles]
81 questions
101
votes
0 answers
Why is a PDE a submanifold (and not just a subset)?
I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold.
Let $\pi: E \to M$ be a smooth locally trivial fibre bundle.
In Gromovs words a partial differential relation of order $k$ is a subset of the $k$th…
hase_olaf
- 1,618
15
votes
1 answer
What does prolongation mean in differential geometry?
What is the meaning of the term "prolongation" in differential geometry? Differential geometers often talk about "prolonging" a system of differential equations, or jet prolongation of bundle sections, but I don't really understand what mental…
ಠ_ಠ
- 11,310
11
votes
1 answer
Is there a more straightforward way to define the jet bundle?
Everywhere I have looked, the jet bundle is defined as the fiber bundle of equivalence classes for the partial derivatives of functions from one manifold to another. However, it is easy to see that the 1-jet bundle for real functions on a manifold…
Sophie
- 288
11
votes
1 answer
Is a germ equivalent to an infinite jet?
Not all smooth functions are analytic, as it is well known, so they in general cannot be represented as a power series.
If we restrict our attention to analytic functions, then a specification of the values of all derivatives of a function at a…
Bence Racskó
- 7,919
8
votes
1 answer
High-order derivatives are independent of the chart
I am studing Ehresmann's jet bundles on manifolds and I came up with a (maybe silly) question.
In order to make it easy I skip the details of the definition and go directly to the part that I don't quite understand.
Given an $n$-dimensional manifold…
I. Roperval
- 438
8
votes
2 answers
Introductory material for jets and jet bundles
A student of mine would like to learn more about jets and jet bundles, and more in general about how to treat derivatives and differential equations in an invariant way.
She's also interested in the reformulation of some of the basic concepts of…
geodude
- 8,357
8
votes
1 answer
Compact-open and Whitney $C^\infty$-topologies agree on $C^\infty(M, N)$ for compact $M$.
Let $M$ and $N$ be smooth manifolds. There are different topologies we can equip the space $C^\infty (M, N)$ of smooth mappings between them with. Two of them are the compact-open topology and the Whitney $C^k$-topologies for $k \in \mathbb{N} \cup…
Carlos Esparza
- 3,747
8
votes
1 answer
What is the algebraic structure of higher-order jet spaces?
Firstly excuse any sloppiness here -- I'm not a mathematician by training so I've had a difficult time formalizing my question and tracking down relevant material.
Consider a point in a smooth manifold, $p \in M$. Linear approximations to functions…
8
votes
1 answer
What is the origin of the terms 'jet' and 'prolongation' in differential geometry?
I am just curious what is the reason for the terms 'jet' and 'prolongation' in differential geometry? Is there some mental imagery that these names are supposed to evoke? Or are they so-named because of some particular example that was later…
ಠ_ಠ
- 11,310
7
votes
1 answer
Does every section of $J^r L$ come from some section $s\in H^0(C,L)$, with $L$ line bundle on a compact Riemann surface?
I am working with jet bundles on compact Riemann surfaces. So if we have a line bundle $L$ on a compact Riemann surface $C$ we can associate to it the $r$-th jet bundle $J^rL$ on $C$, which is a bundle of rank $r+1$. If we have a section $s\in…
Brenin
- 14,592
6
votes
1 answer
What, precisely, is the space of jets of a vector bundle on a scheme?
Apologies in advance if this question is too basic. Let $X$ be a scheme. One defines the $n$-th jet space of $X$ to be the scheme representing the functor
$$
A\mapsto X(A[t]/t^{n+1}) .
$$
It is well-known that this functor is represented by a…
Daniel Miller
- 2,395
6
votes
2 answers
$k$-jet transitivity of diffeomorphism group
Given a connected smooth manifold $M$ and an invertible jet $\xi \in {\rm inv} J^k_p(M,M)_q$, what are the required conditions for the existence of a diffeomorphism $\phi \in {\rm Diff}(M)$ such that $j^k_p \phi = \xi$? What about if we want $\phi$…
Anthony Carapetis
- 36,533
5
votes
1 answer
Why a conformal map is determined by its 2-jet at a point?
I have heard somewhere that a conformal map between Riemannian manifolds is determined by its second jet at a single point. (assuming the source manifold is connected).
Where can I find a reference for this fact?
Asaf Shachar
- 25,967
4
votes
1 answer
Reference Request concerning Jet Bundles..
can anyone recommend me a nice reference concerning jet bundles? I've been looking for one for a long time but I couldn't find it...Thanks..
PtF
- 9,895
4
votes
0 answers
When is a PDE a subbundle of a jet bundle (as opposed to a fibered submanifold or just a closed embedded submanifold)?
Nowadays it is common to see PDEs defined as either closed embedded submanifolds of a jet bundle (of some appropriate order) or else, if some further conditions are satisfied, as a fibered submanifold. In some earlier literature, though (e.g.,…
Jim
- 71