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Zbl 0910.53014
Cabras, Antonella; Kolář, Ivan
On the iterated absolute differentiation on some functional bundles. (English)
[J] Arch. Math., Brno 33, No.1-2, 23-35 (1997). ISSN 0044-8753; ISSN 1212-5059/e

Let $Y_1\to M$ and $Y_2\to M$ be two fibered manifolds over the same base and let $\Cal F(Y_1,Y_2)=\underset {x\in M}\to \bigcup C^\infty (Y_{1x},Y_{2x})$ be the functional bundle of all smooth maps from a fiber of $Y_1$ into a fiber of $Y_2$ over the same base point. The aim of the present paper is to study some properties of connections on $\Cal F(Y_1,Y_2)$. \par The authors first introduce the concept of a vertical prolongation $\Cal V\Gamma $ of a connection $\Gamma $ on $\Cal F(Y_1,Y_2)$. Further, if $s:M\to \Cal F(Y_1,Y_2)$ is an arbitrary section and $\bigtriangledown _\Gamma s$ its absolute differential with respect to $\Gamma $, then the iterated absolute differential $\bigtriangledown ^2_{\Gamma ,\Lambda }s$ is defined as the absolute differential of $\bigtriangledown _\Gamma s$ with respect to the tensor product $\Cal V\Gamma \otimes \Lambda ^*$, where $\Lambda ^*$ is an arbitrary linear connection on the cotangent bundle $T^*M$. \par If $Y_2$ is a vector bundle and $\Gamma $ is a finite order connection (nonlinear in general), then it is proved that the alternation of $\bigtriangledown ^2_{\Gamma ,\Lambda }s$ satisfies a modification of the Ricci identity for nonlinear connections. \par Finally, it is proved that the general Ricci identity holds also in the case of an arbitrary finite order connection $\Gamma $ on $\Cal F(Y_1,Y_2)$. In this case, the tensor alternation is replaced by a more sophisticated operation. At this occasion, the general concept of the jet space $J^r(N,\Cal F(Y_1,Y_2))$ is defined and the corresponding spaces of iterated jets are studied.
[M.Doupovec (Brno)]

MSC 2000:: *53C05 Connections, general theory
58A20 Jets

Keywords: bundle of smooth maps; connection on a functional bundle; iterated absolute differentiation; iterated 2-jet; Ricci identity; nonlinear connection

Cited in: Zbl 0969.53010

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