On symmetrization of jets

Włodzimierz M. Mikulski

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 1, page 157-168
  • ISSN: 0011-4642

Abstract

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Let F = F ( A , H , t ) and F 1 = F ( A 1 , H 1 , t 1 ) be fiber product preserving bundle functors on the category ℱℳ m of fibred manifolds Y with m -dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism ( A , H , t ) → ( A 1 , H 1 , t 1 ) to be a G L ( m ) -invariant algebra homomorphism ν : A → A 1 with t 1 = ν ∘ t . The main result is that there exists an ℱℳ m -natural transformation F Y → F 1 Y depending on a classical linear connection on the base of Y if and only if there exists a quasi-morphism ( A , H , t ) → ( A 1 , H 1 , t 1 ) . As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.

How to cite

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Mikulski, Włodzimierz M.. "On symmetrization of jets." Czechoslovak Mathematical Journal 61.1 (2011): 157-168. <http://eudml.org/doc/196715>.

@article{Mikulski2011,
abstract = {Let $F=F^\{(A,H,t)\}$ and $F^1=F^\{(A^1,H^1,t^1)\}$ be fiber product preserving bundle functors on the category $\mathcal \{FM\}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\rightarrow (A^1,H^1,t^1)$ to be a $GL(m)$-invariant algebra homomorphism $\nu \colon A\rightarrow A^1$ with $t^1=\nu \circ t$. The main result is that there exists an $\mathcal \{FM\}_m$-natural transformation $FY\rightarrow F^1Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\rightarrow (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.},
author = {Mikulski, Włodzimierz M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {jets; higher order connections; Ehresmann prolongation; Weil functors; bundle functors; natural operators; jet; higher order connection; Ehresmann prolongation; Weil functor; bundle functor; natural operator},
language = {eng},
number = {1},
pages = {157-168},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On symmetrization of jets},
url = {http://eudml.org/doc/196715},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Mikulski, Włodzimierz M.
TI - On symmetrization of jets
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 1
SP - 157
EP - 168
AB - Let $F=F^{(A,H,t)}$ and $F^1=F^{(A^1,H^1,t^1)}$ be fiber product preserving bundle functors on the category $\mathcal {FM}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\rightarrow (A^1,H^1,t^1)$ to be a $GL(m)$-invariant algebra homomorphism $\nu \colon A\rightarrow A^1$ with $t^1=\nu \circ t$. The main result is that there exists an $\mathcal {FM}_m$-natural transformation $FY\rightarrow F^1Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\rightarrow (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.
LA - eng
KW - jets; higher order connections; Ehresmann prolongation; Weil functors; bundle functors; natural operators; jet; higher order connection; Ehresmann prolongation; Weil functor; bundle functor; natural operator
UR - http://eudml.org/doc/196715
ER -

References

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