##
**Smooth manifolds over local algebras and Weil bundles.**
*(English.
Russian original)*
Zbl 1007.58001

J. Math. Sci., New York 108, No. 2, 249-294 (2002); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz.; 73, VINITI, Moscow 162-236 (2002).

The theory of smooth manifolds over local algebras belongs to the fields of geometry and topology of manifolds that have multiaffine structures defined by associative commutative algebras. This field of study is closely related to the geometry of jet bundles, the theory of differential-geometric objects, and the topology of foliations. The main idea in constructing the theory of smooth manifolds over algebras is the replacement of the field of real numbers \(\mathbb R\) and linear spaces over \(\mathbb R\) by a certain associative commutative finite-dimensional algebra \(\mathbb A\) over \(\mathbb R\) and \(\mathbb A\)-modules of finite dimension over \(\mathbb R\), respectively.

The general theory of manifolds over algebras was studied by A. P. Shirokov, J. Vanzura, V. V. Vishnevskij, G. I. Kruchkovich, and others. A. P. Shirokov [Tr. Geom. Semin. 1, 425-456 (1966; Zbl 0198.54601)] proved that structures of finite-dimensional manifolds over local algebras arise on tangent bundles and bundles of \(\mathbb A\)-closed points in the sense of A. Weil.

In this survey, the author presents the review of new trends in studies of the differential geometry and topology of smooth manifolds over local algebras.

The paper consists of seven sections. After presenting the introductory information in the first two sections, in section 3 the author considers the structure of an \(n\)-dimensional \(\mathbb A\)-smooth manifold on the Weil \(\mathbb A\)-jet bundle \(J^{\mathbb A}M_n\) over a real smooth manifold \(M_n\). The existence of the structure of an \(\mathbb A\)-smooth manifold on \(J^{\mathbb A}M_n\) leads to the appearance of the structural \(\mathbb A\)-affine differential group \(D_n(\mathbb A)\) of the bundle \(J^{\mathbb A}M_n\), and the principal bundle \(B({\mathbb A})M_n\), called the \(\mathbb A\)-affine frame bundle on \(M_n\) associated with \(J^{\mathbb A}M_n\), with the structural group \(D_n(\mathbb A)\). The author studies connections in bundles with the structural group \(D_n(\mathbb A)\) and presents an interpretation of the object of torsion of a connection of higher order in terms of the theory of manifolds over algebras. The existence of natural foliations on \(\mathbb A\)-smooth manifolds allows to associate various holonomy representations with them. These representations are discussed in the next section. Section 5 is devoted to bi-graded \(\widehat d\)-cohomologies of \(\mathbb A\)-smooth manifolds and resolutions of sheaves of \(\mathbb A\)-smooth forms with values in quotients algebras and ideals. In section 6 the author considers a generalization of the Molino construction of \(d_F\)-cohomologies of tensorial forms on a fiber principal bundle to the case of \(\widehat d\)-cohomologies of tensorial forms on an \(\mathbb A\)-smooth principal bundle \(P^{\mathbb A}\) and studies \(\mathbb A\)-smooth connections in \(P^{\mathbb A}\). In the last section \(\mathbb A\)-affine Ehresmann connections on \(\mathbb A\)-smooth manifolds are introduced.

The general theory of manifolds over algebras was studied by A. P. Shirokov, J. Vanzura, V. V. Vishnevskij, G. I. Kruchkovich, and others. A. P. Shirokov [Tr. Geom. Semin. 1, 425-456 (1966; Zbl 0198.54601)] proved that structures of finite-dimensional manifolds over local algebras arise on tangent bundles and bundles of \(\mathbb A\)-closed points in the sense of A. Weil.

In this survey, the author presents the review of new trends in studies of the differential geometry and topology of smooth manifolds over local algebras.

The paper consists of seven sections. After presenting the introductory information in the first two sections, in section 3 the author considers the structure of an \(n\)-dimensional \(\mathbb A\)-smooth manifold on the Weil \(\mathbb A\)-jet bundle \(J^{\mathbb A}M_n\) over a real smooth manifold \(M_n\). The existence of the structure of an \(\mathbb A\)-smooth manifold on \(J^{\mathbb A}M_n\) leads to the appearance of the structural \(\mathbb A\)-affine differential group \(D_n(\mathbb A)\) of the bundle \(J^{\mathbb A}M_n\), and the principal bundle \(B({\mathbb A})M_n\), called the \(\mathbb A\)-affine frame bundle on \(M_n\) associated with \(J^{\mathbb A}M_n\), with the structural group \(D_n(\mathbb A)\). The author studies connections in bundles with the structural group \(D_n(\mathbb A)\) and presents an interpretation of the object of torsion of a connection of higher order in terms of the theory of manifolds over algebras. The existence of natural foliations on \(\mathbb A\)-smooth manifolds allows to associate various holonomy representations with them. These representations are discussed in the next section. Section 5 is devoted to bi-graded \(\widehat d\)-cohomologies of \(\mathbb A\)-smooth manifolds and resolutions of sheaves of \(\mathbb A\)-smooth forms with values in quotients algebras and ideals. In section 6 the author considers a generalization of the Molino construction of \(d_F\)-cohomologies of tensorial forms on a fiber principal bundle to the case of \(\widehat d\)-cohomologies of tensorial forms on an \(\mathbb A\)-smooth principal bundle \(P^{\mathbb A}\) and studies \(\mathbb A\)-smooth connections in \(P^{\mathbb A}\). In the last section \(\mathbb A\)-affine Ehresmann connections on \(\mathbb A\)-smooth manifolds are introduced.

Reviewer: Andrew Bucki (Oklahoma City)

### MSC:

58A05 | Differentiable manifolds, foundations |

58A20 | Jets in global analysis |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58B05 | Homotopy and topological questions for infinite-dimensional manifolds |

53C05 | Connections (general theory) |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |