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1
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0answers
55 views

Are affine maps (wrt to a connection), which preserve a tensor field, given by a PDE?

Let $(M, \nabla)$ be a manifold together with a connection on $TM$ and let $T$ be a tensor field on $M$. Suppose the pseudogroup $\Gamma$ of locally defined smooth maps, that simultanously preserve ...
3
votes
0answers
71 views

How to visualize the dual objects of jets of functions?

I work with a smooth $f: M \to \Bbb C$ and I would like to have an object mimicking the concept of "$k$-th order differential" from multivariate calculus. For various reasons that are not important ...
6
votes
1answer
261 views

Jets in synthetic differential geometry

As I understand it from Kostecki's notes, the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$ where $$D_k(n) = \{(x_1, \ldots, ...
6
votes
1answer
146 views

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 ...
4
votes
0answers
232 views

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 ...
2
votes
1answer
339 views

Induced Riemannian metric on Jet-Manifold

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 ...
2
votes
0answers
225 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 ...
4
votes
2answers
528 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$. ...
3
votes
1answer
978 views

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 ...
1
vote
1answer
539 views

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 ...
-1
votes
2answers
335 views

Inverse Problem for jet equations

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
1answer
486 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 ...
0
votes
0answers
240 views

Jet spaces for maps with constraints

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 $M$. ...
4
votes
2answers
1k views

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, ...