Tensor Decompositions and Their Properties

Patrik Peška (), Marek Jukl and Josef Mikeš
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Patrik Peška: Department Algebra and Geometry, Faculty of Science, Palacky University in Olomouc, 17. Listopadu 12, 77146 Olomouc, Czech Republic
Marek Jukl: Department Algebra and Geometry, Faculty of Science, Palacky University in Olomouc, 17. Listopadu 12, 77146 Olomouc, Czech Republic
Josef Mikeš: Department Algebra and Geometry, Faculty of Science, Palacky University in Olomouc, 17. Listopadu 12, 77146 Olomouc, Czech Republic

Mathematics, 2023, vol. 11, issue 17, 1-13

Abstract: In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ ), σ = 1 , 2 , 3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

Keywords: tensor; decomposition; coordinate transformation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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