# Questions tagged [jets]

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### 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 $\gamma:(-1,1)\to M$ ...
1 vote
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### 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 ...
1 vote
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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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 $J^... 3 votes 0 answers 305 views ### Multivalued solution of PDE${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 $for$\... 709 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$^k$. ...
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### 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 ...
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### 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 ...
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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 $g: M \rightarrow ... 3 votes 1 answer 566 views ### Jet spaces between non Hausdorff manifolds I found it very hard to find literature about smooth manifolds that are not required to be Hausdorff. In particular I'm interested in their local properties: 1.) Are the$r$-th order jet bundles$J^r(...
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 $M$. ...
### 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 ...