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Double vector spaces. (English) Zbl 0701.58004

In differential geometry of higher order, one deals with some interesting algebraical structures possessing partial operations. Here we shall investigate the simplest of them - double vector spaces and their morphisms. The category \({\mathcal D}{\mathcal L}\) of double linear morphisms has been introduced by J. Pradines in C. R. Acad. Sci., Paris, Sér. A 278, 1523-1526 (1974; Zbl 0285.58002). Analogous investigations in double affine and affine-linear case have been studied by I. Kolář in Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys. 24, 883-887 (1976; Zbl 0354.58002). In this paper we shall show a slightly more general point of view. The category \({\mathcal D}{\mathcal L}\) will be described geometrically.

MSC:

58A20 Jets in global analysis
58A30 Vector distributions (subbundles of the tangent bundles)
55R65 Generalizations of fiber spaces and bundles in algebraic topology
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References:

[1] Pradines J.: Représentation des jets non holonomes par des morphismes vectoriels doubles soudés. C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1523-1526. · Zbl 0285.58002
[2] Kolář I.: On the Jet Prolongations of Smooth Categories. Bull. de l’Academie Polonaise des sciences, Sér. des sci. math., astr. et phys. - Vol. XXIV, No.10, 1976. · Zbl 0354.58002
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