Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 45, No. 2, pp. 677696 (2004) 

Higherorder preconnections in synthetic differential geometry of jet bundlesHirokazu NishimuraInstitute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 3058571, Japan email: logic@math.tsukuba.ac.jpAbstract: In our previous papers (Nishimura [2001 and 2003]) we dealt with jet bundles from a synthetic perch by regarding a 1jet as something like a pinpointed (nonlinear) connection (called a preconnection) and then looking on higherorder jets as repeated 1jets. In this paper we generalize our notion of preconnection to higher orders, which enables us to develop a nonrepetitive but still synthetic approach to jet bundles. Both our repetitive and nonrepetitive approaches are coordinatefree and applicable to microlinear spaces in general. In our nonrepetitive approach we can establish a theorem claiming that the $(n+1)$th jet space is an affine bundle over the $n$th jet space, while we have not been able to do so in our previous repetitive approach. We will show how to translate repeated 1jets into higherorder preconnections. Finally we will demonstrate that our repetitive and nonrepetitive approaches to jet bundles tally, as far as formal manifolds are concerned. Keywords: Synthetic differential geometry, jet bundle, preconnection, strong difference, repeated jets, formal manifold, formal bundle Classification (MSC2000): 51K10, 58A03, 58A20 Full text of the article:
Electronic version published on: 9 Sep 2004. This page was last modified: 4 May 2006.
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