##
**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).

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).

Reviewer: Maido Rahula (Tartu)

### 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 |