Questions tagged [jets]
The jets tag has no usage guidance.
30
questions
1
vote
0answers
85 views
Connection as a jet section [closed]
Let $\pi:E\longrightarrow M$ a smooth fibre bundle. A connection is a linear bundle homomorphism $\Phi:TE\longrightarrow TE$ such that $\Phi$ is a projection to the vertical bundle $VE\subset TE$.
I ...
2
votes
0answers
100 views
Morphism between jet spaces smooth
In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets:
Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets ...
16
votes
2answers
668 views
Functional approach vs jet approach to Lagrangian field theory
Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
4
votes
0answers
130 views
Holonomic sections $C^\infty(M)$-generate jet bundle
Given a vector bundle $E \to M$ with a corresponding $k$-th jet bundle $J^kE \to M$, denote by $j^k : \Gamma(E) \to \Gamma(J^kE)$ the $k$-th jet prolongation $(k \in \mathbb{N} \cup \{0\})$ and recall ...
5
votes
2answers
201 views
Equivalence of two definitions of jets of smooth functions
In the literature I have encountered two different definitions of jets of smooth functions, and I was wondering how one could identify these definitions.
One definition is the often encountered ...
1
vote
1answer
198 views
The bundle of symmetric affine connections as quotient of the second-order frame bundle
This post is not about finding an answer to a certain problem - because the answer already exists - but rather about finding the simplest possible answer.
The problem is: how to define the bundle $C(...
4
votes
1answer
150 views
Semi-holonomic jets in synthetic differential geometry
Anders Kock's two texts on synthetic differential geometry (SDG) are a great place to get geometric intuition, especially when it comes to jets. Unfortunately, he doesn't seem to cover semi-holonomic ...
8
votes
2answers
365 views
In which sense are Euler-Lagrange PDE's on fiber bundles quasi-linear?
In what follows, all manifolds are smooth, Hausdorff, paracompact, connected and oriented, and all maps between any two of them are assumed to be smooth. Let $\pi:E\rightarrow M$ be a fiber bundle ...
7
votes
1answer
357 views
A very basic question about projections in formal PDE theory
I am learning formal PDE theory for my research and I am currently struggling to have a basic understanding of the operations involved in completing a (say, linear) PDE system to an involutive one (...
3
votes
2answers
511 views
Integrability conditions for differential equations on $J^\infty$
Is there any result on the existence of solutions of differential equations of the form
$$
D_\alpha\Phi([u])=U_\alpha([u])\Phi([u]),
$$
where $[u]$ is an element of an infinite dimensional bundle $J^\...
16
votes
2answers
1k views
A variant of the Monge-Cayley-Salmon theorem?
Suppose one has a smooth non-degenerate curve $\gamma: [0,1] \to {\bf R}^n$ into Euclidean space (thus $\gamma'$ never vanishes), with the property that the velocity $\gamma'(t)$ and acceleration $\...
1
vote
0answers
88 views
Negatively curved jet pseudometrics from jet differentials: understanding a proposition by Green-Griffiths
In the paper by Green and Griffiths Two applications of algebraic geometry to entire holomorphic mappings (Proposition 2.5) it is proved that a jet pseudometric can be constructed on a projective ...
3
votes
1answer
130 views
Does the Banach algebra of jets have the approximation property?
To formulate my question I need the construction of the algebra $J^n_M(K)$ of jets of degree $n$ on a compact set $K$ of a smooth manifold $M$. I'll describe it for the simplest case of $M={\mathbb R}$...
2
votes
2answers
229 views
Constructing jet bundles from a cocycle of smooth transition functions
Suppose we are given an open cover $\mathcal{U}=(U_{i})_{i \in I}$ of a smooth manifold $M$, a cocycle of smooth transition functions $g_{ij}: U_{ij} \to G$ where $G$ is a Lie group, and a (not ...
7
votes
1answer
1k views
Short and elegant definition of the $C^1$ topology
A friend told me that the $\mathbf{C^1}$-topology on the set $C^\infty(M,N)$ of smooth functions between two smooth manifolds $M$ and $N$ can be defined as the coarsest topology making the map
$$
C^\...
16
votes
0answers
584 views
Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?
The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
2
votes
0answers
106 views
Are affine maps (wrt to a connection), which preserve a tensor field, given by a PDE?
Let $(M, \nabla)$ be a manifold together with a connection on $TM$ and let $T$ be a tensor field on $M$. Suppose the pseudogroup $\Gamma$ of locally defined smooth maps, that simultanously preserve ...
5
votes
0answers
163 views
How to visualize the dual objects of jets of functions?
I work with a smooth $f: M \to \Bbb C$ and I would like to have an object mimicking the concept of "$k$-th order differential" from multivariate calculus. For various reasons that are not important ...
7
votes
1answer
495 views
Jets in synthetic differential geometry
As I understand it from Kostecki's notes, the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$
where $$D_k(n) = \{(x_1, \ldots, ...
6
votes
1answer
209 views
When does 'Zariski tangent space derivative' vanishes everywhere imply that a section is constant?
Consider an abelian algebra, $R$, over the field $K$ with the properties that every residue field of $R$ is (canonically) isomorphic to $K$ (I'm not sure but I think this is necessary, otherwise we ...
4
votes
0answers
278 views
Differential ideals of Pfaffian forms on jet bundles (Integrability)
(I asked this question on math.stackexchange, but got no reaction in several weeks. So, my conclusion is, that it is harder to answer than I thought, and maybe admissible for the attribute 'research ...
2
votes
1answer
603 views
Induced Riemannian metric on Jet-Manifold
Suppose $(M,g)$ and $(N,g')$ are smooth Riemannian manifolds and $J^r(M,N)$ is the
smooth manifold of $r$-jets $j^r_xf$ of smooth maps $f:M\to N$.
Is there an 'induced' Riemannian metric $g''$ on $J^...
2
votes
0answers
285 views
Multivalued solution of PDE ${v_{xx}v_{yy}-v_{xy}^{2}}={(1+v_{x}^{2}+v_{y}^{2})^2}$
Let's start with a definition:
Definition: A scalar k-th order differential equation on a smooth manifold $M$,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\...
5
votes
2answers
640 views
Jets of Equivariant Vector Bundles
Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...
5
votes
1answer
2k views
1-jet bundle on vector bundle with metric connection
Background
I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to specify ...
2
votes
1answer
872 views
Tautological and normal bundles over flag manifolds and jet bundles
Hello! Recently, doing my research on jet bundles, I was led to consider the following construction.
Let $V$ be a real vector space of dimension $n$. Consider the flag manifold $G(V,k,l)$ and the two ...
-1
votes
2answers
355 views
Inverse Problem for jet equations
The following is a well known fact and due to the functorial properties of the jet functor:
Suppose you have two smooth manifolds $M$ and $N$ and maps $f:M \rightarrow N$ as well as
$g: M \rightarrow ...
3
votes
1answer
521 views
Jet spaces between non Hausdorff manifolds
I found it very hard to find literature about smooth manifolds that are not required to be Hausdorff. In particular I'm interested in their local properties:
1.) Are the $r$-th order jet bundles $J^r(...
0
votes
0answers
264 views
Jet spaces for maps with constraints
Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps:
Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. ...
5
votes
2answers
1k views
On the smooth structure of the spaces of $k$-jets
I was asking myself, if the following list of conditions is sufficient to determine the usual smooth structure on the spaces of $k$-jets.
the map $j^k f:M\ni x\to j_x^k f\in J^k(M,N)$ is smooth, for ...
