I Question on tangent space and jet spaces

[mnb96]

Hello,

I am reading some material related to jet spaces, which at first glance seem to be a generalization of the concept of tangent space.
I am confused about what is the correct definition of a jet space. In particular, given a map $f: X \rightarrow Y$ between two manifolds, what is the k-jet of f at x?

It seems to me that some authors sometimes define it as: $$J^k_xf := \left [ \partial_\alpha f \right(x) ]$$ where $|\alpha|\leq k$ and the multi-indices $\alpha$ contains the indices of the variables with respect to which differentiate. But other authors define it as: $$J^k_xf := \left [x,\, \partial_\alpha f \right(x) ]$$
In other words, they sometimes arbitrarily "attach" also the variable x to the same quantity in the first definition. Which of the two is the correct one?

For example, is the 1-jet space of $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ at x given by $$J^1_xf=\left[f(x), f_1(x),\ldots,f_m(x)\right]$$ or $$J^1_xf=\left[x,\,f(x), f_1(x),\ldots,f_m(x)\right]$$?

 

[jambaugh]

I've only dipped my toe into Olver's books on symmetry groups and differential equations so am not yet clear on jet spaces but they are not, as I understand it, fiber bundles. One first constructs a Jet space as a manifold rather than a fiber bundle then determines a specific fibration. So for a functional relation y=f(x) which we may identify with a submanifold of the product space X\times Y the jet spaces are constructed by extending this embedding with variables which, when we later impose constraints, will be identified with the derivatives.
$$ X\times Y \to X\times Y \times Y'\times Y''...$$
The (typically but not necessary functional) relations defining curves in the original product space then by extension define curves in the extended space once one imposes the derivative identifying constraints.

That is, how I understood Peter Olver to define things in Applications of Lie Groups to Differential Equations and others.

And now my understanding gets too fuzzy to continue constructively. There is presumably a fibration of this manifold forming the Jet bundle defined by the matching up the derivatives of curves in the initial X\times Y space to the extended space. This is something I've wanted to sit down for a quiet year and absorb.

 

[mnb96]

Thanks jambaugh for your help, and especially for pointing me to Olver's work.

I checked Olver's book "Classical invariant theory", and I found there the same confusing "double definition" that seems to propagate in other texts as well. See excerpt below:

In the first highlighted sentence he clearly defines the jet of a function as $$\left[ f_\alpha(x) \right]$$ (according to my notation), while in the second highlighted sentence he apparently (re)defines the jet of f as $$\left[x, f_\alpha(x) \right]$$

So, are jets defined with that x attached to the prolongation of f or no?

 

[martinbn]

It depends whether you mean the jet itself or the point in the jet bundle. Similarly as a tangent vector $v$ at a point $x$ or $(x,v)$.

 

[mnb96]

I see...so basically I have to interpret the text according to the following analogies:

prolongation (or jet) of f at x ↔ tangent vector of f at x
jet space of f at x ↔ tangent space of f at x
jet bundle of f ↔ tangent bundle of f

This would explain the reason for the "double definition". The first definition (without the attached x) would represent an element of the jet space, while the second one (with the x) represents an element of the jet bundle.

If that's the case, then the confusion arises from using the same term "jet" to denote an element of the jet space or an element of the jet bundle.