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", Publisher: Circolo Matematico di Palermo(Palermo), page [199]-206

Abstract

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

How to cite

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Mikulski, W. M.. "Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles." Proceedings of the 15th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1996. [199]-206. <http://eudml.org/doc/220233>.

@inProceedings{Mikulski1996,
abstract = {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.},
author = {Mikulski, W. M.},
booktitle = {Proceedings of the 15th Winter School "Geometry and Physics"},
keywords = {Proceedings; Geometry; Physics; Winter school; Srni (Czech Republic)},
location = {Palermo},
pages = {[199]-206},
publisher = {Circolo Matematico di Palermo},
title = {Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles},
url = {http://eudml.org/doc/220233},
year = {1996},
}

TY - CLSWK
AU - Mikulski, W. M.
TI - Natural operators lifting functions to cotangent bundles of linear higher order tangent bundles
T2 - Proceedings of the 15th Winter School "Geometry and Physics"
PY - 1996
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [199]
EP - 206
AB - 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.
KW - Proceedings; Geometry; Physics; Winter school; Srni (Czech Republic)
UR - http://eudml.org/doc/220233
ER -

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