Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles

Mikulski, W. M.

Proceedings of the 15th Winter School "Geometry and Physics", GDML_Books, (1996), p. [199]-206 / Harvested from

Résumé

The author studies the problem how a map $L : M \to ℝ$ on an $n$ -dimensional manifold $M$ can induce canonically a map $A_{M} (L) : T^{*} T^{(r)} M \to ℝ$ for $r$ a fixed natural number. He proves the following result: “Let $A : T^{(0, 0)} \to T^{(0, 0)} (T^{*} T^{(r)})$ be a natural operator for $n$ -manifolds. If $n \geq 3$ then there exists a uniquely determined smooth map $H : ℝ^{S (r)} \times ℝ \to ℝ$ such that $A = A^{(H)}$ .”The conclusion is that all natural functions on $T^{*} T^{(r)}$ for $n$ -manifolds $(n \geq 3)$ are of the form ${H \circ (λ_{M}^{〈 0, 1 〉}, \dots, λ_{M}^{〈 r, 0 〉})}$ , where $H \in C^{\infty} (ℝ^{r})$ is a function of $r$ variables.

MR MR1463522 | Zbl 0909.58002

EUDML-ID : urn:eudml:doc:220233
Mots clés:

@article{701587,
     title = {Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles},
     booktitle = {Proceedings of the 15th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1996},
     pages = {[199]-206},
     mrnumber = {MR1463522},
     zbl = {0909.58002},
     url = {http://dml.mathdoc.fr/item/701587}
}

Mikulski, W. M. Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles, dans Proceedings of the 15th Winter School "Geometry and Physics", GDML_Books,  (1996), pp. [199]-206. http://gdmltest.u-ga.fr/item/701587/