On the Geometry in the Large of Einstein-like Manifolds

Josef Mikeš, Lenka Rýparová, Sergey Stepanov and Irina Tsyganok
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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
Sergey Stepanov: Department of Mathematics, Finance University, 125468 Moscow, Russia
Irina Tsyganok: Department of Mathematics, Finance University, 125468 Moscow, Russia

Mathematics, 2022, vol. 10, issue 13, 1-10

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.

Keywords: Einstein-like manifold; Bochner method; Sampson Laplacian; Bourguignon Laplacian; vanishing theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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