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 $L:M\rightarrow \mathbb {R}$ on an $n$-dimensional manifold $M$ can induce canonically a map $A_M(L):T^* T^{(r)}M\rightarrow \mathbb {R}$ for $r$ a fixed natural number. He proves the following result: “Let $A: T^{(0,0)}\rightarrow T^{(0,0)}(T^* T^{(r)})$ be a natural operator for $n$-manifolds. If $n\ge 3$ then there exists a uniquely determined smooth map $H: \mathbb {R}^{S(r)}\times \mathbb {R}\rightarrow \mathbb {R}$ such that $A= A^{(H)}$.”The conclusion is that all natural functions on $T^* T^{(r)}$ for $n$-manifolds $(n\ge 3)$ are of the form $\lbrace H\circ (\lambda ^{\langle 0,1\rangle }_M,\dots , \lambda ^{\langle r,0\rangle }_M)\rbrace $, where $H\in C^\infty (\mathbb {R}^r)$ is a function of $r$ variables.

Détail

MR MR1463522 | Zbl 0909.58002

EUDML-ID : urn:eudml:doc:220233
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