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Manifolds modelled over free modules over the double numbers. (English) Zbl 1049.58002

Let \(\mathbb B:=\{z=x+jy\mid x,y\in \mathbb R,\;j^2=1\}\) be the algebra of double numbers. The authors introduce and study the concept of \(\mathbb B\)-differentiability, \(\mathbb B\)-analyticity and \(\mathbb B\)-holomorphy of a map \(F:U\to \mathbb B\) where \(U\subset \mathbb B^n\) is an open set. Next they turn to \(\mathbb B\)-manifolds and derive an important criterion for a \(2n\)-dimensional non-integrable \(\mathbb B\)-manifold to be a \(\mathbb B\)-manifold.

MSC:

58A05 Differentiable manifolds, foundations
30G35 Functions of hypercomplex variables and generalized variables
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