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I am trying to understand the jet bundles but currently I am stuck on the following questions:

Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) manifold $X$.

I know that the bundle $J_k(E)$ of k-jets of $E$ has the structure of a vector bundle over $X$.

I would like to know however what the rank of this vector bundle is.

Is $J_k(E)$ holomorphic in the case when $(E, \pi, X)$ is holomorphic?

Moreover, when $\pi: E\rightarrow X$ is a fiber bundle with structure group $G$, can we view $J_1(E)$ as the associated principal bundle $P$ associated to $E$ or am I wrong?

I have seen an interpretation of $J_1(E)$ as some sort of an "extended frame bundle" of E in the sense that its fiber consists of the set of all pairs comprising a basis of $T_pX$ $(T^{1, 0}_pX)$ and a basis of $E_p$, $p\in X$

P.S.: I am new here and I really hope that I don't annoy the experienced audience in this forum with trivialities. I would appreciate any help or suggestions or simply good references. Thank you in advance for your competent help.

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1 Answer 1

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There are two, intimately related, notions of jets associated with a vector bundle $\pi: E\to X$. The 1st is $J_k^{sect}(E)$, the space of $k$-jets of sections of $\pi:E\to X$, i.e. maps $s:X\to E$ such that $\pi\circ s=id_X$. The second is the space $J_k^{triv}(E)$ of $k$-jets at $0\in\mathbb C^n$ of local trivializations $U\times \mathbb C^r\to E$, where $U$ is some neighborhood of $0\in\mathbb C^n$. It is good to keep in mind both notions, as they both occur in the literature.

Begin with $J_k^{triv}(E)$. This is a fiber bundle over $X$, where the fiber at $x\in X$ is the $k$-jets at $0\in \mathbb C^n$, $j_k(\phi)_0$, of trivializations $\phi: U\times \mathbb C^r\to E$ over maps (embeddings) $U\to X$ such that $0\mapsto x$. Clearly, $j_0(\phi)_0$ is just a linear isomorphism $\mathbb C^r\to E_x$ , so $J_0^{triv}(E)$ is the frame bundle of $E$, a principal $GL(r)$-bundle, where $GL(r)$ is acting by pre-composition on isomorphisms $\mathbb C^r\to E_x$.

More generally, we can pre-compose any trivialization $\phi: U\times \mathbb C^r\to E$ with a bundle automorphism of $U\times \mathbb C^r$, over a diffeomorphism of $U$ that fixes $0$. Such a bundle automorphism is given by $(y, v)\mapsto (f(y), g(y)v)$, where $f$ is a diffeomorphism of $U$ that fixes $0$, and $g:U\to GL(r)$. Taking $k$-jets, $J_k^{triv}(E)$ becomes a principal $G_k$-bundle, where $G_k$ is the group of $k$-jets at $0$ of such bundle automorphisms.

The relation between the two notions of jets is the following: $J_k^{sect}(E)$ is the vector bundle associated to $J_k^{triv}(E)$ and a representation of $G_k$ on the space of $k$-jets at $0\in U$ of sections of $U\times \mathbb C^r\to U$ (I let you figure out the representation).

For $k=0$, $G_0=GL(r)$, $J_0^{sect}(E)$ is just $E$ itself and $J_0^{triv}(E)$ is the frame bundle of $E$.

For $k=1$, since $f(0)=0$, $j_1(f)_0$ is given by $df(0)\in GL(n)$. But $j_1(g)_0$ has two pieces of data: $g(0)\in GL(r)$ and $g(0)^{-1}dg(0):\mathbb C^n \to Mat_r(\mathbb C)$.

So $J_1(E)$ is a principal $G_1$-bundle over $X$, where $G_1$ is the semi-direct product of $GL(n)\times GL(r)$ with $(\mathbb C^n)^*\otimes Mat_r(\mathbb C)$, where $GL(n)$ acts on the 1st factor by the dual of the standard representation and $GL(r)$ on the second by conjugation. You can indeed think of $J_1^{triv}(E)$ as a bundle of "extended" frames for $E$, but an extended frame at $x\in X$ consists of more than just a pair of framings of $E_x$ and $T_xX$; there is also an extra piece, which you can describe invariantly as an $End(E_x)$ valued 1-form at $x$.

Note: I think the reference you give in your comment below is mistaken about this point; it says that $G_1=GL(n)\times GL(r)$. For $k>1$, $G_k$ is even more complicated.

To get the fiber dimension of $J_k^{sect}(E)$ inductively you can use the exact sequence of vector bundles $0\to S^k(T^* X)\otimes E\to J_k^{sect}(E)\to J_{k-1}^{sect}(E)\to 0.$ You get for example that $J_1^{sect}(E)$ is a vector bundle of rank $rn+r$.

For the fiber dimension of $J_k^{triv}(E)$ you can use a similar sequence (I let you fill in the details).

If $X$ is a complex manifold and $E\to X$ is a holomorphic vector bundle, than all bundles, maps etc are holomorphic as well.

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    $\begingroup$ Very informative response. In the paper arxiv.org/pdf/math/0702461v4.pdf on page 6 one views the first jet bundle $J_1(E)$ as a bundle of extended frames of $E$ (see my question for definition). In particular, $J_1(E)$ becomes $GL_n(\mathbb{C})\times GL_r(\mathbb{C})$-principal bundle, where $n=dimX$, $r=rkE$. I have a suggestion that in particular $J_1(E)$ is a ${1}\times GL_r(\mathbb{C})=GL_r(\mathbb{C})$-principal bundle and therefore "coincides" with the associated frame bundle of $E$. I am not completely sure however. $\endgroup$ Mar 23, 2014 at 14:59
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    $\begingroup$ OK, I looked at the reference and I realize I misunderstood your question (I think the same mistake occured when people answered a similar question you posted in mathoverflow). I have edited my answer. $\endgroup$
    – Gil Bor
    Mar 24, 2014 at 6:04
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    $\begingroup$ Yeah, I suspected that the jet bundles in my reference and in your answer are different. Your edit is very helpful. I will need some time to digest the different point of views on these objects. $\endgroup$ Mar 24, 2014 at 11:46
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    $\begingroup$ Decided to rewrite the whole answer. Thanks for for this question, was fun to think about it. $\endgroup$
    – Gil Bor
    Mar 25, 2014 at 17:48
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    $\begingroup$ Lots of thanks for your great effort. I will go trough the detail now. $\endgroup$ Mar 26, 2014 at 17:36

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