Kainz, Gerd; Michor, Peter W. Natural transformations in differential geometry. (English) Zbl 0654.58001 Czech. Math. J. 37(112), 584-607 (1987). It is proved that any product-preserving functor from the category of smooth manifolds to itself (satisfying some supplementary conditions) is given by prolongation by a “Weil algebra”, as in A. Weil [Théorie des points proches sur les variétés différentiables, Colloq. Int. Cent. Nat. Rech. Sci. 52, 111-117 (1953; Zbl 0053.249). A similar result was found independently by D. Eck [Product preserving functors on smooth manifolds, J. Pure Appl. Algebra 42, 133- 140 (1986; Zbl 0606.58006)]. The authors utilize this algebraic viewpoint to make some alternative calculations on relations between Lie brackets, covariant differentiation etc. for vector fields. They do not put the theory in a categorical context, where the Weil algebras represent infinitesimal manifolds in their own right, as in, say, E. Dubuc \([C^{\infty}\)-schemes, Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003)]. Reviewer: A.Kock Cited in 2 ReviewsCited in 20 Documents MSC: 58A05 Differentiable manifolds, foundations 18F15 Abstract manifolds and fiber bundles (category-theoretic aspects) Keywords:product-preserving functor; Weil algebra; Lie brackets; covariant differentiation Citations:Zbl 0053.249; Zbl 0606.58006; Zbl 0483.58003 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] H. Federer: Geometric measure theory. Grundlehren Band 153, Springer Verlag 1969. · Zbl 0176.00801 [2] D. B. A. Epstein W. P. Thurston: Transformation groups and natural bundles. Proc. London Math. Soc. (III), 38 (1979), 219-236. · Zbl 0409.58001 · doi:10.1112/plms/s3-38.2.219 [3] A. Kock: Synthetic differential geometry. London Math. Society, Lecture Note Series 51, 1981. · Zbl 0487.18006 · doi:10.1016/0022-4049(81)90048-7 [4] I. Kolář: Natural transformations of the second tangent functor into itself. Arch. Math. (Brno) 4, 1984. · Zbl 0578.58004 [5] A. Kriegl: Die richtigen Räume für Analysis im Unendlichdimensionalen. Monatshefte Math. 94 (1982), 109-124. · Zbl 0489.46035 · doi:10.1007/BF01301929 [6] I. Moerdijk K. E. Reyes: The tangent functor category revisited. Preprint, Amsterdam 1983. · Zbl 0623.58002 [7] I. Moerdijk C. E. Reyes: \(C^\infty \)-rings. Preprint Montréal 1984. [8] B. L. Reinhart: Differential geometry of foliations. Ergebnisse 99, Springer-Verlag 1983. · Zbl 0506.53018 [9] I. Rosický: Abstract tangent functors. Preprint Brno 1984. · Zbl 0561.18008 [10] S. Šwierczkowski: A description of the tangent functor category. Coll. Math. 31 (1974). [11] C. L. Terng: Natural vector bundles and natural differential operators. Amer. J. Math. 100, 775-828. · Zbl 0422.58001 · doi:10.2307/2373910 [12] A. Vanžurová: On geometry of the third tangent bundle. Acta Univ. Olom. 82 (1985). · Zbl 0619.53013 [13] A. Weil: Théorie des points proches sur les variétés differentiables. in Colloq. Top et Geo. Diff., Strassbourg 1953, 111-117. · Zbl 0053.24903 [14] J. E. White: The method of iterated tangents with applications in local Riemannian geometry. Pitman 1982. · Zbl 0478.58002 [15] David J. Eck: Product preserving functors on smooth manifolds. preprint 1985. · Zbl 0606.58006 · doi:10.1016/0022-4049(86)90076-9 [16] O. O. Luciano: Categories of multiplicative functors and Morimoto’s Conjecture. Preprint 1986. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.