Holonomicity in synthetic differential geometry of jet bundles

Hirokazu Nishimura

Abstract: In the repetitive approach to the theory of jet bundles there are three methods of repetition, which yield non-holonomic, semi-holonomic, and holonomic jet bundles respectively. However the classical approach to holonomic jet bundles failed to be truly repetitive, for it must resort to such a non-repetitive coordinate-dependent construction as Taylor expansion. The principal objective in this paper is to give a purely repetitive definition of holonomicity by using microsquares (double tangents), which spells the flatness of the Cartan connection on holonomic infinite jet bundles without effort. The definition applies not only to formal manifolds but to microlinear spaces in general, enhancing the applicability of the theory of jet bundles considerably. The definition is shown to degenerate into the classical one in case of formal manifolds.

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