Malakhaltsev, M. A. On a type of manifolds over algebra of dual numbers. (English) Zbl 0932.53027 Tr. Geom. Semin. 21, 70-79 (1991). The paper considers manifolds over the algebra \(R(\varepsilon)\) of dual numbers (for the general theory of manifolds over algebras, see V. V. Vishnevskij, A. P. Shirokov and V. V. Shurygin [‘Spaces over algebras’ (Russian) (Kazan University) (1985; Zbl 0592.53001)]). The author proves that if the canonical foliation of a compact \(2n\)-dimensional manifold \(M\) over \(R(\varepsilon)\) is a fiber bundle \(\pi : M \to B\) with fiber \(F\), then (1) \(F\) is an \(n\)-dimensional torus \(T^n\); (2) the structure group of the frame bundle \(L(B)\) of \(B\) reduces to the discrete group \(GL(n,\mathbb{Z})\); (3) the bundle \(\pi\) is associated with a principal bundle with the group \(A(T^n)\) of affine transformations of \(T^n\). Conversely, if \(\pi : M \to B\) is a bundle with properties (1)–(3), then on \(M\) there exists a structure of manifold over \(R(\varepsilon)\). With the use of these results, the author constructs a countable set \(\{J_k\}\) of structures over \(R(\varepsilon)\) on \(T^2\) such that \((T^2, J_k) \not\cong (T^2, J_l)\), \(k \neq l\). Reviewer: M.Malakhal’tsev (Kazan’) Cited in 2 Documents MSC: 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:manifold over algebra; integrable almost tangent structure; discrete structure group; parallelizable manifold Citations:Zbl 0592.53001 PDFBibTeX XMLCite \textit{M. A. Malakhaltsev}, Tr. Geom. Semin. 21, 70--79 (1991; Zbl 0932.53027) Full Text: EuDML