Questions tagged [jet-bundles]
The jet-bundles tag has no usage guidance.
61
questions
0
votes
0
answers
12
views
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 ...
- 637
2
votes
0
answers
43
views
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_\...
- 51
9
votes
1
answer
132
views
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 ...
- 248
1
vote
1
answer
71
views
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 ...
- 345
2
votes
0
answers
36
views
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-...
- 551
4
votes
1
answer
118
views
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 ...
- 43
1
vote
1
answer
60
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
$$j_0^r t_{f(x)}\circ j_0^r (t_{-f(x)}\circ ...
- 3,631
4
votes
1
answer
143
views
Contangent space as a jet space, (inconsistency ?), Renteln
In Renteln's, Manifolds, Tensors and Forms, p. 81, The cotangent space as a jet space$^*$, we have the following definitions
Let $f:M \to \mathbb R$ be a smooth function, $p \in M$, and $\{x^i\}$ ...
- 3,865
1
vote
1
answer
74
views
The variety of $\mathbb{C}[t]_{< d}$-points on a variety
Let $X \subseteq \mathbb{C}^n$ be an affine variety defined by $f_i(x_1, \ldots, x_n)=0, 1 \le i \le m$. I am interested in the points $(g_1, \ldots, g_n) \in \mathbb{C}[t]^n$ where $\deg(g_i) < d$ ...
- 111
8
votes
1
answer
210
views
High-order derivatives are independent of the chart
I am studing Ehresmann's jet bundles on manifolds and I came up with a (maybe silly) question.
In order to make it easy I skip the details of the definition and go directly to the part that I don't ...
- 362
2
votes
0
answers
65
views
calculate the kernel of a germ map
I have a question, more about how to calculate the kernel of a certain map:
Let the ring $\varepsilon_n=\lbrace f\colon (\mathbb{R}^{n},0)\longrightarrow \mathbb{R}: f \quad\textit{is map germ}\...
- 411
2
votes
1
answer
126
views
A Formula of Jet Bundles in Gelfand's Book
In Gelfand's book Discriminants, resultants, and multidimensional determinants he gives a formula:
$$J_{1}(L)\cong J_{1}(\mathcal{O}_{X})\otimes L$$
Here $L$ is a line bundle on some complex variety.
...
- 81
1
vote
0
answers
18
views
Proof that equality of derivative of section is coordinate independent
Lemma 4.1.1 in D.J Saunders 's book (The geometry of jet bundles) says:
Let $(E,\pi,M)$ be a bundle, and let $p$ $\in M$. Suppose that
$\phi$ and $\psi$ are section that satisfy $\phi(p)=\psi(p)$. Let ...
1
vote
1
answer
124
views
Does the natural exact sequence of the holomorphic jet bundle spilt?
Let $X$ be a complex manifold. Let us consider the jet bundle of the trivial line bundle on $X$. We denote it as $J_{1}(\mathbb{C})$. We have the short exact sequence:
$$0\rightarrow\Omega_{X}\...
- 81
1
vote
1
answer
67
views
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 ...
- 5,760
2
votes
0
answers
72
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 ...
- 10.4k
3
votes
1
answer
56
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\...
- 368
0
votes
0
answers
45
views
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 $J^1B$ ...
- 11
0
votes
1
answer
124
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^{...
3
votes
0
answers
68
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:
...
- 195
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 ...
- 7,397
1
vote
1
answer
65
views
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 ...
- 23
5
votes
1
answer
523
views
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 ...
- 3,603
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 ...
- 71.6k
3
votes
1
answer
115
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 ...
2
votes
0
answers
22
views
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, ...
- 21
1
vote
0
answers
116
views
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 ...
3
votes
1
answer
155
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 $\phi(p)=\psi(p)$. ...
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 ...
- 6,230
0
votes
0
answers
28
views
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{\...
- 5,510
1
vote
0
answers
88
views
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 ...
- 119
1
vote
0
answers
112
views
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 ...
- 1,834
3
votes
0
answers
96
views
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 ...
- 6,230
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 ...
- 429
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 ...
- 185
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(...
- 10.3k
4
votes
0
answers
118
views
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 ...
- 23.5k
4
votes
0
answers
127
views
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 ...
- 13.9k
4
votes
0
answers
375
views
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 ...
- 810
5
votes
1
answer
93
views
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 ...
- 23.5k
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 ...
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,...
- 9,782
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 $...
1
vote
1
answer
151
views
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 ...
- 233
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)$ ...
- 674
1
vote
1
answer
72
views
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 ...
- 9,782
1
vote
1
answer
136
views
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 ...
- 1,491
2
votes
0
answers
53
views
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 \...
- 1,491
1
vote
1
answer
231
views
$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) \...
- 3,473
1
vote
1
answer
62
views
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 $...
- 5,446