On holomorphic hypercomplex connections
type: Item
instance of: scholarly article
title: On holomorphic hypercomplex connections
zbMATH Open document ID: 1264.53039
published in: Advances in Applied Clifford Algebras
publication date: 2013-04-08T00:00:00Z
review text: A hypercomplex algebra \(\mathcal{U}_m\) is a real \(m\)-dimensional associative algebra with unit \(1\). If there exists an isomorphism between the algebra \(\mathcal{U}_m\) and a subalgebra \(\Pi\) in the algebra of \((1,1)\) tensor fields (affinors) on a smooth manifold \(M\), one says that \(M\) is endowed with a hypercomplex structure. The authors study some properties of integrable commutative hypercomplex structures endowed with a holomorphic torsion free connection whose curvature tensor field satisfies a purity condition with respect to the covariantly constant structure affinors. For the hypercomplex functions the Scheffers conditions of holomorphicity are considered. Then there presented the algebraic \(\Pi\)-structures on manifolds, studying their regularity and integrability. The properties of pure tensors with respect to a regular \(\Pi\)-structure and of the \(\mathcal{U}\)-holomorphic tensors are studied in real coordinates. For the pure connections one studies their torsion tensors. The \(\mathcal{U}\)-holomorphic hypercomplex connections are studied in the real model. Then some properties of the pure curvature tensors of such connections are considered.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C26
zbMATH DE Number: 6151438
zbMATH Keywords: pure tensors and connections
zbMATH Keywords: holomorphic tensors and connections
zbMATH Keywords: Tachibana operator
zbMATH Keywords: Norden metric
zbMATH Keywords: regular representations
author: A. Salimov
author: Nejmi Ceng{i}z
author: Manouchehr Behboudi Asl
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
full work available at URL: s00006-012-0339-3
OpenAlex ID: W2053541781
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DOI: 10.1007/S00006-012-0339-3
description: scientific article; zbMATH DE number 6151438