Questions tagged [jet-bundles]

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What exactly is $T^2f$?

Given a smooth map $f : M \to N$ between manifolds, the differential gives a map $Tf : TM \to TN$ between tangent bundles. Taking another differential gives a map $T^2f : T^2M \to T^2N$ between ...
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What does the "$k$-th tangent mapping" refer to?

I'm reading the section on jet bundles in the book Manifolds of Differentiable Mappings. The author defines a k-jet from $X$ to $Y$ as an equivalence class of pairs $$(f : X \to Y, x \in X),$$ where ...
Frank's user avatar
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Valid interpretation of Higher order frame bundles and their jet groups?

I've been trying to develop an intuition for higher order frame bundles to help me understand them and this is what I've come up with. Criticisms welcome, as I'm not sure it's valid? NOTE: Always I ...
R. Rankin's user avatar
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Condition for section of the frame bundle of $M$ to be the 1-jet of a local diffeomorphism $\phi: M \rightarrow \mathbb{R}^n$?

We're given an n-dimensional Riemannian manifold $M$ and its frame bundle $FM$. The tangent bundle can be locally regarded as an invertible map $\phi:M\rightarrow\mathbb{R}^{n}$. Let's confine $\phi$ ...
R. Rankin's user avatar
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Injective homomorphism of bundle of invertible 1-jets into the Frame bundle? Seeking representations of nonholonomic Jet Groups

I'm pretty new to this so here goes: Herein we will consider always the sub-bundle of invertible jets. Given an n-dimensional smooth manifold $M^n$ I know that that the bundle of 1-jets of the $C^{\...
R. Rankin's user avatar
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A particular Cauchy problem for first-order PDE

Is the following result true? Suppose $M$ is a smooth manifold, $W\subseteq J^1M\cong M\times T^\ast M$ is an open subset, $F\colon W\to \mathbb R$ is a smooth function, and $(x_0,u_0,p_0)\in W$. Let ...
Parco Macelli's user avatar
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Proposition 1.1.14 D.J Saunders on Bundle

everyone. I am studying D.J Saunders's book The Geometry of Jet Bundles. On proposition 1.1.14, a proof is given that the structure of the total space $E$ of a bundle $(E,\pi,M)$ depends on those of ...
Jeff 's user avatar
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Integral sections of higher order jet fields

Let us consider a bundle $(E,\pi, M)$ and let $k\in \mathbb N$. I am going to adopt the notations and conventions by Saunders. Preliminaries A first-order jet field on $\pi$ is a section of the bundle ...
Parco Macelli's user avatar
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Continuity of the jet prolongation

I am studying the article Immersion Theory for Homotopy theorists by Michael Weiss for my bachelor thesis. The main theorem states that the space of immersions and formal immersions between two smooth ...
vheerde's user avatar
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How does a vector field act on a differential form?

In the definition of the Euler-Lagrange operator (2.6, I. Anderson), the total differential vector fields $D_{i}, D_{ij}$ act on differential forms. How does that work? Feels like Lie derivatives to ...
Jian's user avatar
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Relationship (if any) between jet bundles and projectivized vector bundles

I am learning jet bundle theory for my research in physics. I have read 6 (of 7) chapters of "The Geometry of Jet Bundles" (D. J. Saunders), but I am confused about the relationships between ...
Meem's user avatar
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de Rham cohomology of the Grassmannian bundle

Let $X$ be a smooth $n+m$ dimensional manifold and $\pi:Y\rightarrow X$ the bundle whose fibre over any $x\in X$ is the Grassmann manifold of all $m$ dimensional subspaces of $T_xX$. This is ...
Bence Racskó's user avatar
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Linearization of a system of equation at the identity jet

Consider the following first order system $$X_y=X_u=0,\quad Y=y,\quad Y_y=1,\quad UX_x=u, \quad U_uX_x=1,$$ where $(x,y,u)$ are independent variables and $(X,Y,U)$ are dependent variables. If ‎ $$\...
Mostafa's user avatar
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The r-jet prolongation of a functional?

Suppose I have some fibered manifold $E$ with n-dimensional Riemannian base $M$ and sections $\theta:M\rightarrow E$. I have a functional of the form: $$I=\intop_{M}\mathcal{L}(\theta)d\mu(\theta) $$ ...
R. Rankin's user avatar
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Exterior Algebra with Jets instead of Forms?

For a smooth, $n$-dimensional manifold $M$ over $\mathbb{R}$, I would like to know if sections of the dual space of the order-$k$ jet bundle sit within a commutative differential graded algebra ...
richokicked800goals's user avatar
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Difficulty understanding the variational bicomplex

Let $E\rightarrow X$ be a smooth fiber bundle, with (infinite-order) jet bundle $J_\infty\rightarrow X$. I am reading about the variational bicomplex; as stated at that nLab link, it is a bicomplex ...
I.A.S. Tambe's user avatar
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Understanding the first variational formula

I am trying to understand the formula in section 1 of this nLab page. It says if $L$ is an $n$-form on the jet bundle $J\rightarrow X$ of a smooth bundle $E\rightarrow X$, then we can write $$dL = E-...
I.A.S. Tambe's user avatar
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What is the partial derivative of a section?

I made an attempt to understand the jet bundle after reading up on fiber bundles but I got stuck at the first definition, how is the partial derivative of a section defined? The total space contains ...
Emil's user avatar
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Calculating KJets of SO(2)

I am currently working through definitions and examples of KJets and one-parameter point groups. The text I am working with describes kjets as vector fields defined by: \begin{equation} (j_k T_\...
Ragon's user avatar
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11 votes
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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 ...
Sophie's user avatar
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1-jets of $\mathscr{C}^1$ curves in $\mathbb{R}P^n$

For a $\mathscr{C}^{1}$ curve $c:(-\epsilon,\epsilon) \to \mathbb{R}^{n+1} \setminus \{0\}$, let $\overline{c} = \pi \circ c$, where $\pi: \mathbb{R}^{n+1} \setminus \{0\} \to \mathbb{R}P^n$ is the ...
Paul Joh's user avatar
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What is the tensor power series of a vector?

In this Wikipedia article https://en.m.wikipedia.org/wiki/Jet_(mathematics) there’s Taylor’s theorem for functions $f: \mathbb{R}^m \to \mathbb{R}^n$ where $f(x)=\sum_{n\in \mathbb{N}} D^n f(x_0) (x-...
Lave Cave's user avatar
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Easy computation by Ravi Vakil (jet bundles)

I landed on some short notes by Ravi Vakil from the 90s, the Beginner's Guide to Jet Bundles from the Point of View of Algebraic Geometry. The notes are very clear but on the very first page there is ...
math_ews's user avatar
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1 answer
66 views

Jets on affine spaces $\mathbb R^n$

I do not understand here in $0.1$ how follows $$J^r(\mathbb R^n,\mathbb R^m)=\mathbb R^m \times \mathbb R^n \times J_0^r(\mathbb R^m,\mathbb R^n)_0$$ from $$j_0^r t_{f(x)}\circ j_0^r (t_{-f(x)}\circ ...
user122424's user avatar
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4 votes
1 answer
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Contangent space as a jet space, (inconsistency ?), Renteln

In Renteln's, Manifolds, Tensors and Forms, p. 81, The cotangent space as a jet space$^*$, we have the following definitions Let $f:M \to \mathbb R$ be a smooth function, $p \in M$, and $\{x^i\}$ ...
Physor's user avatar
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The variety of $\mathbb{C}[t]_{< d}$-points on a variety

Let $X \subseteq \mathbb{C}^n$ be an affine variety defined by $f_i(x_1, \ldots, x_n)=0, 1 \le i \le m$. I am interested in the points $(g_1, \ldots, g_n) \in \mathbb{C}[t]^n$ where $\deg(g_i) < d$ ...
Kevin's user avatar
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1 answer
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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 ...
I. Roperval's user avatar
2 votes
0 answers
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calculate the kernel of a germ map

I have a question, more about how to calculate the kernel of a certain map: Let the ring $\varepsilon_n=\lbrace f\colon (\mathbb{R}^{n},0)\longrightarrow \mathbb{R}: f \quad\textit{is map germ}\...
logarithm's user avatar
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1 answer
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A Formula of Jet Bundles in Gelfand's Book

In Gelfand's book Discriminants, resultants, and multidimensional determinants he gives a formula: $$J_{1}(L)\cong J_{1}(\mathcal{O}_{X})\otimes L$$ Here $L$ is a line bundle on some complex variety. ...
Chenxi Yin's user avatar
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0 answers
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Proof that equality of derivative of section is coordinate independent

Lemma 4.1.1 in D.J Saunders 's book (The geometry of jet bundles) says: Let $(E,\pi,M)$ be a bundle, and let $p$ $\in M$. Suppose that $\phi$ and $\psi$ are section that satisfy $\phi(p)=\psi(p)$. Let ...
amilton moreira's user avatar
1 vote
1 answer
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Does the natural exact sequence of the holomorphic jet bundle spilt?

Let $X$ be a complex manifold. Let us consider the jet bundle of the trivial line bundle on $X$. We denote it as $J_{1}(\mathbb{C})$. We have the short exact sequence: $$0\rightarrow\Omega_{X}\...
Chenxi Yin's user avatar
1 vote
1 answer
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Proving a general formula for "tangent vectors" (actually $k$-th partial derivatives) when changing coordinates

I want to prove that the order of contact of two functions $f, g: M \to N$ between manifolds is well defined. Definition. Let $f, g : M^{n} \to N^{m}$ be smooth functions. We say $f$ has order of ...
Matheus Andrade's user avatar
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0 answers
93 views

Ring structure on a space of functions between vector spaces?

In this Wikipedia article about jets, in the section about rigorous definitions, for the algebro-geometric definition, they take the vector space $C^\infty_p(\mathbb R^n,\mathbb R^m)$ of germs of ...
Vercassivelaunos's user avatar
3 votes
1 answer
94 views

Open subsets of Jet Bundles

I am currently reading through 'Stable mappings and their singularities' by Golubitsky and Guillemin. There they define $$ J^k(X, Y)_{x, y} $$ to be the set of equivalence classes of smooth maps $f: X\...
user-j's user avatar
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Total derivative as generalized vector field

I've seen that it is possible to think the total derivative as a generalised vector field between $d_\alpha:J^1B\rightarrow TB$ Where $J^1B$ is the first order jet bundle. But how an element of $J^1B$ ...
Alabarda 980's user avatar
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1 answer
188 views

Is the restriction map continuous?

Consider $C^{\infty}(U, \mathbb{R}^{k})$ and $C^{\infty}(V, \mathbb{R}^{k})$, where $V\subset U $, with $U$ and $V$ open subsets of $\mathbb{R}^n$. Is it true that the restriction \begin{align} R:C^{...
mathUser32's user avatar
3 votes
0 answers
111 views

Question Regarding Jet Bundle

I am reading an appendix on jet bundles, and I am confused on the following question. The note I am reading (Singularity of Mappings by Mond and Nuno-Ballesteros, Appendix A) says the following: ...
Winnie_XP's user avatar
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8 votes
2 answers
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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 ...
geodude's user avatar
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1 vote
1 answer
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Formula for sum of horizontal and vertical differentials

I am working through the book The Geometry of Jet Bundles by D.J. Saunders and came across the formula $$d_h+d_v = \pi_{k+1,k} ^\star \circ d,$$ which I am unable to prove. Here $d$ denotes the ...
David's user avatar
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7 votes
1 answer
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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 ...
Carlos Esparza's user avatar
2 votes
0 answers
187 views

Canonical subbundle of cotangent bundle of a jet manifold

In the first chapter of Griffiths' Exterior Differential Systems and the Calculus of Variations, he discusses a bit of the language of jet manifolds. Namely, for a smooth manifold $M$, he considers ...
Ivo Terek's user avatar
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3 votes
1 answer
142 views

Interior product of a exterior derivative in jet bundles

Let $(E,\pi,M)$ be a bundle which coordinates is given by $(x^i,u^\alpha)$ and $J^1\pi$ the first jet associated with this bundle which coordinates is given by $(x^i,u^\alpha,u^\alpha_i)$ Given a ...
amilton moreira's user avatar
2 votes
0 answers
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Jets of functions from the line to a Banach (Hilbert) space

Good morning, for research-related reasons I'm very often using the concept of jets of smooth (i.e. $C^\infty$) mappings $\varphi:I\to B$, where $I\subset \mathbb{R}$ is an open interval containing, ...
Gil Sanders's user avatar
1 vote
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Notation D.J Saunders 's book (The geometry of jet bundles)

in D.J Saunders 's book (The geometry of jet bundles) on page 25 he defines the vertical bundle by : Given the bundle $(E,\pi,M)$ the subset $$\{\xi \in TE:\pi_*(\xi)=0 \in TM\}$$ is called the set ...
amilton moreira's user avatar
3 votes
1 answer
202 views

Expression in D.J Saunders 's book (The geometry of jet bundles)

Lemma 4.1.1 in D.J Saunders 's book (The geometry of jet bundles) says: Let $(E,\pi,M)$ be a bundle, and let $p$ $\in M$. Suppose that $\phi$ and $\psi$ are section that satisfy $\phi(p)=\psi(p)$. ...
amilton moreira's user avatar
11 votes
1 answer
483 views

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 ...
Bence Racskó's user avatar
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Worked example of computation/identification of jet group $G_{k}^{n}$ for Lorenz system

Start with, for example, the Lorenz system $$\begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma (y - x), \\[6pt] \frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\[6pt] \frac{\...
graveolensa's user avatar
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Poincaré-Cartan form and Legendre transformation from the variational principle

I am learning classical field theory at the moment. Right now, I am trying to understand where the Legendre transform, or the multi-symplectic structure emerges. To make the setting clear, say we ...
jpdm's user avatar
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0 answers
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Jet prolongation of a distribution on a manifold

I'm trying to work with the first jet prolongation of a $k$-distribution on a manifold $M$ of dimension $n$. My intuition is to consider the Grassmann bundle $X=Gr_k(TM)\to M$ and look at the first ...
ChesterX's user avatar
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What does it really mean for a Lagrangian to be independent of the base coordinates?

In this question I will be considering what is called in physics a "classical Lagrangian field theory" from a geometric point of view. One is given an $n$ dimensional (smooth, real) "spacetime ...
Bence Racskó's user avatar