Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 45, No. 2, pp. 677-696 (2004)
Higher-order preconnections in synthetic differential geometry of jet bundles
Hirokazu NishimuraInstitute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan e-mail: email@example.com
Abstract: In our previous papers (Nishimura [2001 and 2003]) we dealt with jet bundles from a synthetic perch by regarding a 1-jet as something like a pinpointed (nonlinear) connection (called a preconnection) and then looking on higher-order jets as repeated 1-jets. In this paper we generalize our notion of preconnection to higher orders, which enables us to develop a non-repetitive but still synthetic approach to jet bundles. Both our repetitive and non-repetitive approaches are coordinate-free and applicable to microlinear spaces in general. In our non-repetitive 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 1-jets into higher-order preconnections. Finally we will demonstrate that our repetitive and non-repetitive 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
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Electronic version published on: 9 Sep 2004. This page was last modified: 4 May 2006.