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Higher-order preconnections in synthetic differential geometry of jet bundles. (English) Zbl 1071.58004

Summary: In our previous papers [H. Nishimura, Bull. Belg. Math. Soc. – Simon Stevin 8, No. 4, 639–650 (2001; Zbl 1030.58001) and Beitr. Algebra Geom. 44, No. 2, 471–481 (2003; Zbl 1044.58006)] 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 show how to translate repeated 1-jets into higher-order preconnections. Finally, we demonstrate that our repetitive and non-repetitive approaches to jet bundles tally, as far as formal manifolds are concerned.

MSC:

58A20 Jets in global analysis
51K10 Synthetic differential geometry
58A03 Topos-theoretic approach to differentiable manifolds
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