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Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups

Mohammed Guediri, Mona Bin-Asfour (2014)

Archivum Mathematicum

The purpose of this paper is to investigate Ricci-flatness of left-invariant Lorentzian metrics on 2-step nilpotent Lie groups. We first show that if , is a Ricci-flat left-invariant Lorentzian metric on a 2-step nilpotent Lie group N , then the restriction of , to the center of the Lie algebra of N is degenerate. We then characterize the 2-step nilpotent Lie groups which can be endowed with a Ricci-flat left-invariant Lorentzian metric, and we deduce from this that a Heisenberg Lie group H 2 n + 1 can be...

Riemannian semisymmetric almost Kenmotsu manifolds and nullity distributions

Yaning Wang, Ximin Liu (2014)

Annales Polonici Mathematici

We consider an almost Kenmotsu manifold M 2 n + 1 with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that M 2 n + 1 is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that M 2 n + 1 is ξ-Riemannian-semisymmetric. Moreover, if M 2 n + 1 is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove that M 2 n + 1 is...

Rigidity of Einstein manifolds and generalized quasi-Einstein manifolds

Yi Hua Deng, Li Ping Luo, Li Jun Zhou (2015)

Annales Polonici Mathematici

We discuss the rigidity of Einstein manifolds and generalized quasi-Einstein manifolds. We improve a pinching condition used in a theorem on the rigidity of compact Einstein manifolds. Under an additional condition, we confirm a conjecture on the rigidity of compact Einstein manifolds. In addition, we prove that every closed generalized quasi-Einstein manifold is an Einstein manifold provided μ = -1/(n-2), λ ≤ 0 and β ≤ 0.

Rigidity of noncompact manifolds with cyclic parallel Ricci curvature

Yi Hua Deng (2014)

Annales Polonici Mathematici

We prove that if M is a complete noncompact Riemannian manifold whose Ricci tensor is cyclic parallel and whose scalar curvature is nonpositive, then M is Einstein, provided the Sobolev constant is positive and an integral inequality is satisfied.

Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

David M J. Calderbank, Henrik Pedersen (2000)

Annales de l'institut Fourier

We study the Jones and Tod correspondence between selfdual conformal 4 -manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3 -manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence...

Semi-parallel CR submanifolds in a complex space form

Mayuko Kon (2011)

Colloquium Mathematicae

We show that there is no proper CR submanifold with semi-flat normal connection and semi-parallel second fundamental form in a complex space form with non-zero constant holomorphic sectional curvature such that the dimension of the holomorphic tangent space is greater than 2.

Currently displaying 361 – 380 of 479