##
**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)].

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 |