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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 $j^kf(x):X \to J^k_x(X,Y)$ which can be understood a section into the jet space, then how can I picture $T_{j^kf(x)}J^k(X,Y)$.

Intuitively I would say that this should be related with the space $J^{k-1}(X,Y)$ but I can't get my head around this. If somebody knows a good source for this topic I would be really thankful.

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This will be a bit hand-wavy, but the moral of the story is that $T_{J^k(X,Y)}$ represents the infinitesimal deformations of Taylor series of maps from $X$ to $Y$, and that all you really need to know to understand what's going on is already contained in the situation of maps of one real variable.

The tangent space around a point of any manifold is morally speaking the space of infinitesimal deformations around that point. That is, given $x \in X$, the tangent space $T_{X,x}$ tells us in what directions we can move $x$ "a little bit".

Now, a $k$-jet is the manifold way of talking about Taylor series up to the $k$-th degree. Your $j^k f$ then corresponds to the $k$-th Taylor series of $f$ around each point. If we restrict ourselves to one real variable (that is, assume for the moment that $\dim X = \dim Y = 1$), then our picture is something like this: $$ j^k f(x) = a_0 + a_1 x + \cdots + a_k x^k/k!. $$ We can deform each coefficient $a_j$ independently, and the tangent space of the space of $k$-jets thus has a "direction" for each of the $a_j$, that is, for each deformation of each degree of derivative of $f$ (which is a roundabout way of saying that its dimension is $k+1$).

Coming again to the general case of $\dim X = n$ and $\dim Y = m$, we see pretty much the same situation, except the extra dimensions in the manifolds give us more directions into which deform each derivative of our function, so the dimension of the tangent space will be higher.

You can use this description to write down a basis for the tangent space of $k$-jets at a point starting from local coordinates for $X$ and $Y$, if you feel like that would help.

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  • $\begingroup$ I appreciate your answer but can you explain it more formal? Or can you give a reference for your knowledge. I know the geometric and the algebraic definition of the jet space but I can't deduce the structure of the tangent space. And what exactly do you mean by a deformation? I know a deformation of vector fields but I guess that is not what you mean. $\endgroup$ – JDoe Sep 29 '16 at 23:18
  • $\begingroup$ Can you please give me a reference for what you described here? A book with more detailed info would be great. Thanks! $\endgroup$ – JDoe Oct 11 '16 at 10:47

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