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Vukmirovic, Srdjan; Šukilović, Tijana

doi:10.37863/umzh.v72i5.645

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…All Riemannian geodesics are complete. In the pseudo-Riemannian case, only partial results are known: all metrics on 2-step nilpotent Lie groups are geodesically complete [13], on the real hyperbolic space R the only complete metrics are the Riemannian ones [21], etc. However, a more general result, or even the classification of groups that admit the pseudo-Riemannian complete metrics, is yet to come.…”

Section: Geometrical Propertiesmentioning

confidence: 99%

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Šukilović

2020

Theor appl mech (Belgr)

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In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.

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Section: Geometrical Propertiesmentioning

confidence: 99%

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Šukilović

2020

Theor appl mech (Belgr)

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“…All the left invariant Riemannian and Lorentzian metrics on Heisenberg group were classified in [24]. Pseudo-Riemannian metrics of Real hyperbolic space modelled as a Lie group, have been considered both by the variation of Lie brackets [14], and by the variation of inner products [25]. It has been shown in [15] that the only connected and simply-connected Lie groups admitting only one left-invariant Riemannian metric up to scaling and isometry are the Euclidean space, Real hyperbolic space and H 3 × R n (product of three-dimensional Heisenberg group and Euclidian space).…”

Section: Introductionmentioning

confidence: 99%

Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

Dekic¹,

Babic²,

Vukmirovic³

2021

Preprint

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It is well known that CH n has the structure of solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that all of these metrics are of constant negative scalar curvature and only one of them is Einstein (up to isometry and scaling). Finally, we present the relation between Ricci solitons on Heisenberg group and Einstein metric on CH n .

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Lie groups with all left-invariant semi-Riemannian metrics complete

Elshafei,

Ferreira,

Sánchez

et al. 2024

Trans. Amer. Math. Soc.

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For each left-invariant semi-Riemannian metric g g on a Lie group G G , we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of g g . When the adjoint representation of G G satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any g g . We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products K ⋉ ρ R n K \ltimes _\rho \mathbb {R}^n , where K K is the direct product of a compact and an abelian Lie group and ρ ( K ) \rho (K) is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the absence of linear growth and suggest new questions.

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Geodesic completeness of the left-invariant metrics on ℝ H n

Abstract: UDC 514 We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. We show that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case.

Cited by 3 publications

References 11 publications

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

Lie groups with all left-invariant semi-Riemannian metrics complete

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