On holomorphic hypercomplex connections (Q1945465)

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.