×

On a type of manifolds over algebra of dual numbers. (English) Zbl 0932.53027

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 \((T^2, J_k) \not\cong (T^2, J_l)\), \(k \neq l\).

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citations:

Zbl 0592.53001
PDFBibTeX XMLCite
Full Text: EuDML