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Tensor Decompositions and Their Properties

Author

Listed:
  • 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)

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.

Suggested Citation

  • Patrik Peška & Marek Jukl & Josef Mikeš, 2023. "Tensor Decompositions and Their Properties," Mathematics, MDPI, vol. 11(17), pages 1-13, August.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3638-:d:1223311
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