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Differential geometry, Lie groups and symmetric spaces over general base fields and rings. (English) Zbl 1144.58002

Mem. Am. Math. Soc. 900, 202 p. (2008).
When we talk about differential prolongations, we have an iterative process in mind where the tangent functor \(T\) causes a correspondence between the manifolds and tangent bundles (floors), and another between the mappings and tangent mappings (differentials): \(M\to T^kM\), \(f\to T^kf\). We distinguish two approaches in this process, geometric and synthetic [see J. E. White, The method of iterated tangents with applications in local Riemannian geometry. Boston-London-Melbourne: Pitman Advanced Publishing Program (1982; Zbl 0478.58002); M. Rahula, New problems in differential geometry. Singapore: World Scientific (1993; Zbl 0795.53002); R. Lavendhomme, Lectures on naive synthetic differential geometry. Louvain-La-Neuve: Institut de Mathématique (1987; Zbl 0688.18006) and W. Bertram, H. Glöckner and K.-H. Neeb, Expo. Math. 22, No. 3, 213–282 (2004; Zbl 1099.58006)].
In the geometric approach, we consider the floor \(T^kM\) as a \(k\)-multiple sector-bundle and the scalar linear functions in floors as sector-forms, after White. We have solutions, integrals and (infinitesimal) symmetries for a distribution \(\Delta\), and we interpret the connection in the bundle as a structure of type \(\Delta_h\oplus\Delta_v\). In this structure the objects decompose into invariant subobjects which (as covariant derivatives) are nessessary for a description of laws of movements.
The book under consideration treats the differential operators in infinite jets over arbitrary fields and rings. The connection in a multilinear bundle is considered as a Dombowski splitting of exact sequence. The apparatus for multilinear functions on the floors is developed with the help of dual numbers \((\varepsilon^2= 0)\). Invertible jets form a basis for tangent groups \(T^kG\), working on the floors \(T^kM\). In short, the whole arsenal for differential prolongations is perfectly presented in synthetic coordinate-free manner. In sum, one can speak of a very large united Lie-Cartan calculus, where Lie derivatives realize the linearization of transformations and flows, and sector-forms (miscellaneously symmetrized) generalize well-known exterior Cartan’s forms.
See also A. Kock, Synthetic differential geometry. Cambridge etc.: Cambridge University Press (1981; Zbl 0466.51008) and I. Kolář, P. W. Michor and J. Slovák, Natural operations in differential geometry. Kiev: TIMPANI (2001; Zbl 1084.53001).

MSC:

58A05 Differentiable manifolds, foundations
58A20 Jets in global analysis
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
51K10 Synthetic differential geometry
53C35 Differential geometry of symmetric spaces
53B05 Linear and affine connections

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