On connections, geodesics and sprays in synthetic differential geometry. (English) Zbl 0568.18006

Var. Publ. Ser., Aarhus Univ. 35, 68-124 (1983).
The authors present the notions of ”connection” and ”spray” in very simple coordinate free synthetic terms, and give a synthetic proof of the ”Ambrose-Palais-Singer” theorem on the bijective correspondence between sprays and symmetric connections. [The synthetic connection notion considered is that of the reviewer and G. E. Reyes, ibid. 30, 158- 195 (1979; Zbl 0418.18008).]
To get the correspondence, they impose some ad hoc axioms; subsequent simplifications by various other authors have eliminated the need for these axioms, cf. the reviewer and R. Lavendhomme [Cah. Topologie Géom. Différ. 25, 311-324 (1984; Zbl 0564.18009)] and Chapter V of a forthcoming monograph of I. Moerdijk and G. E. Reyes. This implies even an extension of the classical theorem to certain infinite dimensional manifolds.
For the entire collection see [Zbl 0517.00005].
Reviewer: A.Kock


18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
18B25 Topoi
53B05 Linear and affine connections
51K10 Synthetic differential geometry