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Reachable sets for a class of contact sub-lorentzian metrics on ℝ³, and null non-smooth geodesics

Marek Grochowski (2008)

Banach Center Publications

We compute future timelike and nonspacelike reachable sets from the origin for a class of contact sub-Lorentzian metrics on ℝ³. Then we construct non-smooth (and therefore non-Hamiltonian) null geodesics for these metrics. As a consequence we deduce that the sub-Lorentzian distance from the origin is continuous at points belonging to the boundary of the reachable set.

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

Hyunjin Lee, Seonhui Kim, Young Jin Suh (2012)

Czechoslovak Mathematical Journal

In this paper, first we introduce a new notion of commuting condition that φ φ 1 A = A φ 1 φ between the shape operator A and the structure tensors φ and φ 1 for real hypersurfaces in G 2 ( m + 2 ) . Suprisingly, real hypersurfaces of type ( A ) , that is, a tube over a totally geodesic G 2 ( m + 1 ) in complex two plane Grassmannians G 2 ( m + 2 ) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G 2 ( m + 2 ) satisfying the commuting condition. Finally we get a characterization of Type ( A ) in terms of such commuting...

Revisiting linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Weiller F. C. Barboza, H. F. de Lima, M. A. Velásquez (2023)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we deal with n -dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space L p n + p of index p > 1 , which obeys some curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). Under appropriate hypothesis, we are able to prove that such a spacelike submanifold is either totally umbilical or isometric to an isoparametric...

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...

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