# 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 ...
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_\...
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 ...
1 vote
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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 ...
36 views

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 $J^1B$ ...
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^{...
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: ...
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 ...
1 vote
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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 ...
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 ...
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### 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 ...
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 ...
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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, ...
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 ...
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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 $\phi(p)=\psi(p)$. ...
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 ...
### 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 \$...