On the Geometry in the Large of Einstein-like Manifolds
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 d...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/13/2208 |
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| author | Josef Mikeš Lenka Rýparová Sergey Stepanov Irina Tsyganok |
| author_facet | Josef Mikeš Lenka Rýparová Sergey Stepanov Irina Tsyganok |
| author_sort | Josef Mikeš |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T12:46:47Z |
| format | Article |
| id | doaj.art-5fd5dccef65a42fd86531f095ad06858 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T12:46:47Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5fd5dccef65a42fd86531f095ad068582023-11-30T22:11:24ZengMDPI AGMathematics2227-73902022-06-011013220810.3390/math10132208On the Geometry in the Large of Einstein-like ManifoldsJosef Mikeš0Lenka Rýparová1Sergey Stepanov2Irina Tsyganok3Department of Algebra and Geometry, Palacký University Olomouc, 77147 Olomouc, Czech RepublicDepartment of Algebra and Geometry, Palacký University Olomouc, 77147 Olomouc, Czech RepublicDepartment of Mathematics, Finance University, 125468 Moscow, RussiaDepartment of Mathematics, Finance University, 125468 Moscow, RussiaGray 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.https://www.mdpi.com/2227-7390/10/13/2208Einstein-like manifoldBochner methodSampson LaplacianBourguignon Laplacianvanishing theorem |
| spellingShingle | Josef Mikeš Lenka Rýparová Sergey Stepanov Irina Tsyganok On the Geometry in the Large of Einstein-like Manifolds Mathematics Einstein-like manifold Bochner method Sampson Laplacian Bourguignon Laplacian vanishing theorem |
| title | On the Geometry in the Large of Einstein-like Manifolds |
| title_full | On the Geometry in the Large of Einstein-like Manifolds |
| title_fullStr | On the Geometry in the Large of Einstein-like Manifolds |
| title_full_unstemmed | On the Geometry in the Large of Einstein-like Manifolds |
| title_short | On the Geometry in the Large of Einstein-like Manifolds |
| title_sort | on the geometry in the large of einstein like manifolds |
| topic | Einstein-like manifold Bochner method Sampson Laplacian Bourguignon Laplacian vanishing theorem |
| url | https://www.mdpi.com/2227-7390/10/13/2208 |
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