# Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\rightarrow M$$ of the vector bundles of $E$-valued totally symmetric rank-$k$ covariant tensors for all $k=0,\ldots,r$ (here $\vee^1 T^*M=T^*M$ and $\vee^0 T^*M\rightarrow M=$ the trivial line bundle $\mathrm{pr}_1:M\times\mathbb{R}\rightarrow M$ of scalars, so that $\vee^0T^*M\otimes E=E$) may be chosen at the level of $r$-th order jet prolongations $j^r s$ of smooth sections $s\in\Gamma(E)=\{s'\in C^\infty(M,E)\ |\ \pi\circ s'=\mathrm{id}_M\}$ of $E$ by means of symmetrized iterated covariant derivatives of $s$. This is used to write linear partial differential operators between spaces of smooth sections of vector bundles in a global form (see below).

To do so, we start by fixing a covariant derivative on $E$ and a (say, torsion-free) covariant derivative on $M$, and then extend them to a covariant derivative on $E$-valued covariant tensor fields on $M$ by means of the Leibniz rule for tensor products of smooth sections. We denote this common extension by $\nabla$. Given $s\in\Gamma(E)$, we define the $k$-th order iterated covariant derivative $\nabla^k s$ of $s$ with respect to $\nabla$ recursively as $$\nabla^0s=s\ ,\,\nabla^1 s=\nabla s\ ,\,\nabla^{k+1}s=\nabla(\nabla^k s)\ ,$$ so that $\nabla^k s\in\Gamma(\otimes^k T^*M\otimes E)$. The totally symmetric part of $\nabla^k s$ (which, of course, belongs to $\Gamma(\vee^k T^*M\otimes E)$) will be denoted by $D^k_\nabla s$.

One can show that the map $$j^r s \mapsto i_\nabla(j^r s)\doteq(s,D_\nabla s=\nabla s,\ldots,D^r_\nabla s)$$ is a $C^\infty(M)$-linear isomorphism between the $C^\infty(M)$-modules $\Gamma(J^r E)$ and $\Gamma(\oplus^r_{k=0}\vee^kT^*M\otimes E)$, thus establishing an instance of the bundle isomorphism stated in the first paragraph. Now, given two vector bundles $\pi:E\rightarrow M$, $\pi':F\rightarrow M$, a linear partial differential operator $P$ of type $E\rightarrow F$ and order $r\geq 0$ is the composite of the $r$-th order jet prolongation map $\Gamma(E)\ni s\mapsto j^r s\in\Gamma(J^r E)$ with a $C^\infty(M)$-linear map $p$ from $\Gamma(J^r E)$ into $\Gamma(F)$ (the "symbol map" of $P$). Using $i_\nabla$, one may write $$Ps=p(j^r s)=(p\circ i_\nabla^{-1})(s,D_\nabla s,\ldots,D^r_\nabla s)=\sum^r_{k=0}a_k D_\nabla^k s\ ,$$ where $a_k\in\Gamma(\vee^k T M\otimes E^*\otimes F)=$ the $k$-th order coefficient of $P$ with respect to $\nabla$ is uniquely determined by $P$ and $\nabla$ for each $k=0,\ldots,r$ (remark: uniqueness of $a_k$ no longer holds if the symmetry requirement is dropped). Moreover, $a_r$ (the "principal symbol" of $P$) does not depend on $\nabla$.

The earliest reference I could find on this statement is Chapter IV, Section 9 (more precisely, the Corollary to Theorem 7, pp. 90-91) of the book edited by Richard S. Palais, Seminar on the Atiyah-Singer Index Theorem (Princeton University Press, 1965). This chapter was written by Palais himself, and it has no references.

Question: Is this really the first published proof of this result? As stated is it due to Palais, or does it go back even farther?

• I had not known that the result dates back to Palais' work. I learned it from engineering.ucsb.edu/~saber.jafarpour/time_varying.pdf (see Lemma 2.1) and apparently those authors did not come across your reference either. – Umberto Lupo Jun 8 '17 at 10:48
• That is actually a very nice reference (despite lacking a bit of historical care, as you noticed and most people using/quoting the result do), thanks! – Pedro Lauridsen Ribeiro Jun 20 '17 at 14:40
• The jets are usually attributed to Ehresmann – Dmitri Zaitsev Nov 16 '17 at 22:16
• @DmitriZaitsev That's true, but that's not what I am asking. I've also tracked Ehresmann's pioneer work on jets and (arbitrary-order) connections, and couldn't find any trace of the result I've stated - to wit, relating jets to iterated (first-order) covariant derivatives. The book by Jafarpour and Lewis quoted by Umberto Lupo above traces related results to a Trans. AMS paper by Pohl (1966) whose preprint actually dates back to 1963, but there (as the authors themselves state) no such formula can be found explicitly. – Pedro Lauridsen Ribeiro Nov 16 '17 at 22:33
• Pohl's results were derived independently by Libermann (1963) (a student of Ehresmann) and Feldman (1963), but one also cannot find any explicit isomorphism there. There is a couple of references by Ehresmann (1955) and Libermann (1961) on higher-order connections I couldn't get access to (they are both in rather obscure proceedings volumes) and maybe there is something there closer to what I ask, but I really don't know. – Pedro Lauridsen Ribeiro Nov 16 '17 at 22:37