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Strong infinitesimal linearity, with applications to strong difference and affine connections. (English) Zbl 0564.18009
The notions of strong infinitesimal linearity, first considered by F. Bergeron [Objet infinitésimalement linéaire dans un modèle adapté de géométrie différentielle synthétique, Rapport de recherches, DMS 80-12, Univ. de Montréal (1980)], and of strong difference [cf.: I. Kolař, Tensor, New Ser. 38, 98-102 (1982; Zbl 0512.58002); J. E. White, The method of iterated tangents with applications in local Riemannian geometry (1982; Zbl 0478.58002)] are used in the context of synthetic differential geometry to provide with simple proofs, independent of coordinates, certain equalities on affine connections, and to get a version of the Ambrose-Palais-Singer theorem [see A. Kock, Var. Publ. Ser., Aarhus Univ. 35, 192-202 (1983; Zbl 0552.58001); also M. Bunge and P. Sawyer, Cah. Topologie Géom. Différ. 25, 221-258 (1984)]. For motivation, one of the properties of a commutative K-algebra object R in a topos E, which satisfies the central axiom \(1^ W\), proposed by F. W. Lawvere and the first named author, is taken.
Reviewer: G.F.Nassopoulos

MSC:
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
51K10 Synthetic differential geometry
18B25 Topoi
53C05 Connections (general theory)
58A20 Jets in global analysis
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References:
[1] 1 F. Bergeron , Objet infinitésimalement linéaire dans un modèle adapté de GDS , in: Géométrie Différentielle Synthétique , ed. G. Reyes, Rapport de Recherches . DMS 80-12, Univ. de Montréal : 1980 .
[2] 2 M. Bunge & P. Sawyer , On connections, geodesics and sprays in synthetic differential geometry , Cahiers Top. et Géom. Diff. , this issue ; preliminary version in: Category Theoretic Methods in Geometry , Aarhus Var. Publ. Series 35 ( 1983 ). MR 739541 | Zbl 0568.18006 · Zbl 0568.18006
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