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On the Geometry in the Large of Einstein-like Manifolds

Author

Listed:
  • Josef Mikeš

    (Department of Algebra and Geometry, Palacký University Olomouc, 77147 Olomouc, Czech Republic)

  • Lenka Rýparová

    (Department of Algebra and Geometry, Palacký University Olomouc, 77147 Olomouc, Czech Republic
    Institute of Mathematics and Descriptive Geometry, Brno University of Technology, 60200 Brno, Czech Republic)

  • Sergey Stepanov

    (Department of Mathematics, Finance University, 125468 Moscow, Russia)

  • Irina Tsyganok

    (Department of Mathematics, Finance University, 125468 Moscow, Russia)

Abstract

Gray has presented the invariant orthogonal irreducible decomposition of the space of all covariant tensors of rank 3, obeying only the identities of the gradient of the Ricci tensor. This decomposition introduced the seven classes of Einstein-like manifolds, the Ricci tensors of which fulfill the defining condition of each subspace. The large-scale geometry of such manifolds has been studied by many geometers using the classical Bochner technique. However, the scope of this method is limited to compact Riemannian manifolds. In the present paper, we prove several Liouville-type theorems for certain classes of Einstein-like complete manifolds. This represents an illustration of the new possibilities of geometric analysis.

Suggested Citation

  • Josef Mikeš & Lenka Rýparová & Sergey Stepanov & Irina Tsyganok, 2022. "On the Geometry in the Large of Einstein-like Manifolds," Mathematics, MDPI, vol. 10(13), pages 1-10, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2208-:d:847035
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    References listed on IDEAS

    as
    1. Vladimir Rovenski & Sergey Stepanov & Irina Tsyganok, 2020. "The Bourguignon Laplacian and Harmonic Symmetric Bilinear Forms," Mathematics, MDPI, vol. 8(1), pages 1-9, January.
    2. Josef Mikeš & Vladimir Rovenski & Sergey Stepanov & Irina Tsyganok, 2021. "Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds," Mathematics, MDPI, vol. 9(9), pages 1-10, April.
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