Questions tagged [jets]

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How is the $k$-times iterative frame bundle $FF\cdots FM$ associated to the higher order frame bundle $F^k M$?

$\DeclareMathOperator\Gl{Gl}$As I understand it a natural bundle is one for which a diffeomorphism on the base space lifts to an automorphism on the total space of the bundle. It is my understanding ...
R. Rankin's user avatar
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Torsion free connection $\implies$ Jet coordinates $=$ Taylor expansion coefficients?

Suppose we have some smooth n-dimensional manifold $M$ endowed with basis 1-forms $\theta^a$ with $a=1\cdots n$. Then $\theta^a$ are sections of the coframe bundle $F^* M$. In local coordinates ($x^a$ ...
R. Rankin's user avatar
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3 votes
1 answer
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Precise definition of a linear total differential operator

In the works of A. M. Vinogradov on calculus on the infinite jet space, differential equations and "diffieties", a central notion is that of a $\mathcal C$-differential operator. If $\pi:Y\...
Bence Racskó's user avatar
3 votes
0 answers
53 views

Jet at a singular point or a submanifold

Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$. We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ ...
Peter Kravchuk's user avatar
1 vote
0 answers
32 views

Nonholonomic version of bijection between $r$th order connection on $TM$ and principal connection on $r$th order frame bundle $P^r M$?

Given a smooth manifold $M$, Kolář - On the torsion of linear higher order connections showed that there is an equivalence between a linear, $r$-th order connection the tangent bundle $TM$ of $M$, and ...
R. Rankin's user avatar
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2 answers
251 views

A linear representation of the group of jets at 0 under composition

Let $G$ be the set of sequences $(f'(0), f''(0), f'''(0), \ldots)$ of derivatives at zero of functions $f : \mathbb{R} \to \mathbb{R}$ with $f(0) = 0$ and $f'(0) \ne 0 $. The set is a group under ...
Carlos Tomei's user avatar
3 votes
0 answers
61 views

Integral sections of higher-order jet fields

I posted this topic on StackExchange, but it may suit this forum better. Consider a bundle $(E,\pi, M)$ and let $k\in \mathbb N$. I am going to adopt the notations and conventions by Saunders. ...
Parco Macelli's user avatar
2 votes
0 answers
61 views

How far can one get by counting spaces of solutions this way?

I am quite used to "counting"/computing finite dimensions. For example, one would expect a hypersurface in $\mathbb{C}^3$ to have dimension $3 - 1 = 2$. But it is often the case that the ...
Malkoun's user avatar
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3 votes
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131 views

How is differentiation defined on the Green-Griffiths jet bundles?

In their original paper Green and Griffiths remark that there is a differentiation operation on their jet bundles: $$ (-)' : \mathcal{J}_{k,m} \to \mathcal{J}_{k+1,m+1} $$ Which they define on p.47 ...
Ben C's user avatar
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$N$th-order approximation of point stabilizing diffeomorphisms by $N$th-order jet group?

NOTE: migrated from math SE. I was wondering if ever higher jet groups of frames on a (possibly pseudo) Riemannian manifold $M$ approximate the point stabilizing subgroup of diffeomorphisms on $M$ as ...
R. Rankin's user avatar
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65 views

Internal symmetries of partial differential relation via the nonholonomic jet bundle

On a smooth n-dimensional Riemannian manifold $M$, suppose I have the kth order partial differential relation (PDR) written in the form: $$\mathcal{R}=\mathcal{R}\left(x^{i},u^{i},u_{j}^{i},u_{jk}^{i}\...
R. Rankin's user avatar
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7 votes
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187 views

What is the relationship between higher-order derivations (in the sense of Hasse-Schmidt) and differential operators?

Let $A$ and $B$ be $R$-algebras. A Hasse-Schmidt $m$-derivation $D : A \to B$ is a tuple $(D_0, D_1, \dots, D_m)$ of $R$-linear maps $A \to B$ satisfying the generalized Leibniz law, $$ D_k(xy) = \...
Ben C's user avatar
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4 votes
0 answers
178 views

Physical intuition for curvature on higher order frame bundles?

$\DeclareMathOperator\SO{SO}$A priori: I apologize if this isn't up to Mathoverflow standards, I've had very little luck getting questions on this subject answered elsewhere. I'm looking for a physics ...
R. Rankin's user avatar
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Codimension of cusp singularities in the space of 2-jets

In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem. For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\...
Overflowian's user avatar
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Notation for jet bundles of mixed order?

This question is motivated by the consideration of linear control systems in Brunovsky normal form. The idea is that you have $m$ smooth functions with unconstrained dynamics, and the control input ...
Jeanne Clelland's user avatar
2 votes
0 answers
160 views

The variety of $\mathbb{C}[t]_{< d}$-points on a variety

(This was posted to https://math.stackexchange.com/q/4244260/799193 where it did not receive an answer.) Let $X \subseteq \mathbb{C}^n$ be an affine variety defined by $f_i(x_1, \ldots, x_n)=0, 1 \le ...
Kevin's user avatar
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2 votes
1 answer
145 views

Splitting of higher order jet sequence

Let $X$ be a smooth variety. Because $\mathcal{O}_X$ admits a canonical connection $\mathrm{d} : \mathcal{O}_X \to \Omega_X$ the sequence, $$ 0 \to \Omega_X \to J^1(\mathcal{O}_X) \to \mathcal{O}_X \...
Ben C's user avatar
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7 votes
1 answer
357 views

What would be a good introductory reference for learning jet-bundle theory?

I am interested in learning the theory of Jet bundles, and am aware of the standard reference "The geometry of jet bundles" by D. J. Saunders. However this appears to be a detailed book, ...
user90041's user avatar
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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 ...
alexpglez98's user avatar
2 votes
0 answers
177 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 ...
user267839's user avatar
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22 votes
2 answers
2k 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 ...
Bence Racskó's user avatar
7 votes
0 answers
303 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 ...
Lukas Miaskiwskyi's user avatar
5 votes
2 answers
257 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 ...
user337331's user avatar
1 vote
1 answer
223 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(...
Giovanni Moreno's user avatar
4 votes
1 answer
228 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 ...
ಠ_ಠ's user avatar
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8 votes
2 answers
595 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 ...
Pedro Lauridsen Ribeiro's user avatar
7 votes
1 answer
598 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 (...
Pedro Lauridsen Ribeiro's user avatar
3 votes
2 answers
719 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^\...
Javier's user avatar
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17 votes
2 answers
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 $\...
Terry Tao's user avatar
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1 vote
0 answers
97 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 ...
Francesco Genovese's user avatar
3 votes
1 answer
144 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 $M={\mathbb R}$...
Sergei Akbarov's user avatar
2 votes
2 answers
323 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 ...
ಠ_ಠ's user avatar
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7 votes
1 answer
3k 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^\...
xxpauly's user avatar
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18 votes
0 answers
1k 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\...
Pedro Lauridsen Ribeiro's user avatar
2 votes
0 answers
120 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 ...
hase_olaf's user avatar
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5 votes
0 answers
253 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 ...
Alex M.'s user avatar
  • 5,197
7 votes
1 answer
608 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, ...
ಠ_ಠ's user avatar
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6 votes
1 answer
336 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 ...
James Hanson's user avatar
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3 votes
0 answers
302 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 ...
cknoll's user avatar
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3 votes
1 answer
789 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^...
Nevermind's user avatar
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3 votes
0 answers
309 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 $\...
user avatar
5 votes
2 answers
719 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$. ...
Jean Delinez's user avatar
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5 votes
1 answer
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 ...
Tobias Ohrmann's user avatar
2 votes
1 answer
1k 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 ...
Giovanni Moreno's user avatar
-1 votes
2 answers
373 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 ...
Nevermind's user avatar
  • 614
3 votes
1 answer
568 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(...
Mirco's user avatar
  • 137
0 votes
0 answers
270 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 $M$. ...
Mirco's user avatar
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5 votes
2 answers
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 ...
agt's user avatar
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