Questions tagged [jet-bundles]

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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 ...
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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_\...
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1 answer
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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 ...
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1 answer
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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 ...
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  • 345
2 votes
0 answers
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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-...
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  • 551
4 votes
1 answer
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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 ...
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1 answer
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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 ...
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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 $\{x^i\}$ ...
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1 vote
1 answer
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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 $\deg(g_i) < d$ ...
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  • 111
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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 ...
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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}\...
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2 votes
1 answer
126 views

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. ...
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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 ...
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1 answer
124 views

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}\...
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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 ...
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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 ...
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3 votes
1 answer
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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\...
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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 $J^1B$ ...
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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^{...
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3 votes
0 answers
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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: ...
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8 votes
2 answers
436 views

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 ...
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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 ...
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5 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 ...
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2 votes
0 answers
142 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 ...
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3 votes
1 answer
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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 ...
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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, ...
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1 vote
0 answers
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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 ...
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3 votes
1 answer
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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 $\phi(p)=\psi(p)$. ...
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10 votes
1 answer
304 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 ...
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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{\...
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1 vote
0 answers
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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 ...
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1 vote
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 ...
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3 votes
0 answers
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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 ...
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3 votes
1 answer
73 views

Differential operators on a smooth mainifold

Let $M$ be a smooth manifold. If I'm not wrong, the set of differential operators on $M$ is defined as $\mathcal{D}_M $ can be defined by using vector fields. I.e. for each $D \in \mathcal{D}$ we ...
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2 votes
1 answer
243 views

Structure of the Infinite Jet Spaces

I'm curious about possible equivalent formulations of the infinite jet bundle. It seems like all the jet bundles could be equivalently constructed by modding out by the equality of partial derivatives ...
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3 votes
1 answer
222 views

Reference for Taylor's Theorem $\mathbb R^n \to \mathbb R^m$

I am looking for a reference/ book recommendation that in detail exhibits the theory behind the general Taylor theorem using jets for functions $f\colon \mathbb R^n \to \mathbb R^m$ \begin{align} f(...
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4 votes
0 answers
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Degrees of freedom of a metric up to coordinate changes (precise formulation)

Let $M$ be a smooth $n$-dimensional manifold. I have heard that a Riemannian metric on $M$, depends locally on $ n(n+1) / 2 - n = n(n-1) /2$ "independent" functions up to coordinate changes. I can ...
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  • 23.5k
4 votes
0 answers
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Space of principal connections is affine modelled on $\Lambda^1(M;\mathfrak{g})$?

I'm working within the jet-formulation espoused by Saunders in "The Geometry of Jet Bundles" and am struggling to prove the stated result. I would like to stay in this context and understand the ...
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4 votes
0 answers
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sheaf of principal parts and sheaf of jets

Let $(M,\mathcal{O}_M)$ be a complex manifold. Let $\Delta\colon M\to M\times M$ be the diagonal map and $\mathcal{I}$ the kernel of $\Delta^{-1}\mathcal{O}_{M\times M}\to \mathcal{O}_M$. In this ...
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5 votes
1 answer
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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 ...
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6 votes
1 answer
584 views

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 ...
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12 votes
1 answer
1k views

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,...
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5 votes
2 answers
247 views

$k$-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 $...
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1 vote
1 answer
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High-dimensional version of Borel's Lemma as a corollary of Whitney's extension theorem

A version of Borel's Lemma states that, given a sequence $(f_0, f_1, f_2, \ldots)$ of functions in $C^\infty(\mathbb{R}^{n-1})$, there always exists a function $F$ in $C^\infty(\mathbb{R}^{n})$ such ...
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4 votes
1 answer
264 views

Tangent space of Jets pace

I would like to understand what the tangent space of a jet space is. For example if I have a map $f:X \to Y$, where $X$ and $Y$ are manifolds and I have the k-jet extension $j^kf(x):X \to J^k_x(X,Y)$ ...
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1 vote
1 answer
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Coordinate-free description of the kernel of the jet projection?

Given a vector bundle $E \to M$ we can take the $r$-jet prolongation $J^r E \to M$. This is equipped with a vector bundle morphism called the jet-projection $\pi^r_{r-1}: J^r E \to J^{r-1} E$. It is ...
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  • 9,782
1 vote
1 answer
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Question about the proof of the Malgrange Preparation Theorem

I have a question about the proof of what is sometimes called the generalized Malgrange preparation theorem. This proof is in both Brocker and Lander's "Differentiable Germs and Catastrophes" and ...
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  • 1,491
2 votes
0 answers
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For what sets $E \subset \mathbb R^n$ is there a $C^m$ field of polynomials on $E$ with vanishing constant term?

First some definitions, since the notation is somewhat non-standard: let $\mathcal P(m,n)$ denote the vector space of polynomials with real coefficients of degree at most $m$ in $n$ variables. Let $E \...
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  • 1,491
1 vote
1 answer
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$r$-jet of a smooth function and its fiber bundle.

Let $M$ be a smooth manifold of dimension $n$. Let $E$ denote the bundle of germs of smooth functions on $M$. For every stalk $E_x$ we can define the ideal $$I_x^k=\{ f \in\mathcal{C}^{\infty}(M) \...
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  • 3,473
1 vote
1 answer
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elements of $Z$ can be written uniquely as $z=p d^{n+1}x+p_A^\mu dy^A \wedge d^nx_\mu$

Let $X$ be an oriented $n+1$-dimensional manifold which coordinates on it are denoted $x^\mu$, $\mu =0,1,...,n$ and let $\pi_{XY}:Y\to X$ be a finite dimensional fiber bundle and fiber coordinates on $...
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