# Questions tagged [jets]

The jets tag has no usage guidance.

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### 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 ...

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### 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 ...

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### 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(...

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### 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 ...

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### 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 ...

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### 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 (...

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307 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^\...

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### 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 $\...

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### 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 ...

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### 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 $M={\mathbb R}$...

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185 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 ...

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### 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^\...

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### 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\...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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530 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^...

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276 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 $\...

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### 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$. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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(...

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256 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 $M$. ...

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### 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 ...