Higher-order preconnections in synthetic differential geometry of jet bundles

Hirokazu Nishimura

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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