## Manifolds over algebras and their application to the geometry of jet bundles.(English. Russian original)Zbl 0804.53041

Russ. Math. Surv. 48, No. 2, 75-104 (1993); translation from Usp. Mat. Nauk 48, No. 2(290), 75-106 (1993).
The author systematically discusses all fundamental concepts and results concerning manifolds over associative commutative finite-dimensional algebras in the real domain. The paper is self-contained and may be used both for references and for an introduction into the subject.
Let $$A$$ be an algebra as stated above, $$L$$ a finite-dimensional $$A$$- module, $$M$$ a smooth manifold. A diffeomorphism $$h: U\subset M\to V\subset L$$ (where $$U$$, $$V$$ are open subsets) is called an $$L$$-chart. A collection of $$L$$-charts $$h_ \alpha$$ on a covering $$\bigcup U_ \alpha= M$$ such that the tangent maps of $$h_ \alpha\circ h^{-1}_ \beta$$ are isomorphisms of $$L$$ determines the structure under consideration. The paper includes many topics: definitions of all needed concepts, study of structures induced on leaves of foliation if $$L$$ has a submodule, the particular case of manifolds over local algebras ($$A$$-near points in the sense of A. Weil, $$A$$-jets), the $$A$$-prolongation functors, transversal $$A$$-strutures in foliated manifolds, related pseudogroup structures, theory of geometrical objects ($$A$$-frames and Lie groups $$L^ r_ n(A)$$), in particular the covariant derivatives and higher- order connections.
The list of 60 references is incomplete. See, e.g., I. Kolář, P. W. Michor and J. Slovák [Natural operations in differential geometry (Springer Verlag, Berlin 1993; Zbl 0782.53013)].
Reviewer: J.Chrastina (Brno)

### MSC:

 53C10 $$G$$-structures 53C12 Foliations (differential geometric aspects) 58A20 Jets in global analysis

Zbl 0782.53013
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