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On two tensor fields which are analogous to the curvature and torsion tensor fields. (English) Zbl 0885.53014

Let \(\xi\) be a vector bundle over a differentiable manifold \(M\). On the bundle \(\xi\) and on the tangent bundle \(TM\) are given connections \(\nabla\) and \(\widetilde{\nabla}\), respectively, and let \(R\), \(\widetilde{R}\) and \(T\) be the curvature tensors of \(\nabla\) and \(\widetilde{\nabla}\) and the torsion of \(\nabla\), respectively. Combining \(\nabla\), \(\widetilde{\nabla}\), \(R\), \(\widetilde{R}\) and \(T\), new tensor fields \(R^*\), \(\widetilde{R}^*\) and \(T^*\) are defined and their basic properties and geometric interpretations are established. The construction of \(R^*\) and \(\widetilde{R}^*\) is motivated by the theory of semi-symmetric manifolds. Bianchi identities for \(R\) and \(\widetilde{R}\) are obtained. All these notions are generalized to the case of a principal \(G\)-bundle in order to study commutativity of its restricted holonomy group under some assumptions.

MSC:

53B05 Linear and affine connections
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