102,394 research outputs found
Four-dimensional pseudo-Riemannian g.o. spaces and manifolds
No full textA g.o. manifold is a homogeneous pseudo-Riemannian manifold whose geodesics are all homogeneous, that is, they are orbits of a one-parameter group of isometries. A g.o. space
is a realization of a homogeneous pseudo-Riemannian manifold (M, g) as a coset space M = G/H, such that all the geodesics are homogeneous.
We prove that apart from the already classified non-reductive examples (Calvaruso et al., 2015), any four-dimensional pseudo-Riemannian g.o. manifold is naturally reductive.
To obtain this result, we shall also provide a complete description up to isometries of four-dimensional pseudo-Riemannian g.o. spaces, and show explicit realizations of the four-dimensional
pseudo-Riemannian naturally reductive spaces classified in Batat et al. (2015
From almost (para)-complex structures to affine structures on Lie groups
No full textLet G= H⋉ K denote a semidirect product Lie group with Lie algebra g= h⊕ k, where k is an ideal and h is a subalgebra of the same dimension as k. There exist some natural split isomorphisms S with S2= ± Id on g: given any linear isomorphism j: h→ k, we get the almost complex structure J(x, v) = (- j- 1v, jx) and the almost paracomplex structure E(x, v) = (j- 1v, jx). In this work we show that the integrability of the structures J and E above is equivalent to the existence of a left-invariant torsion-free connection ∇ on G such that ∇ J= 0 = ∇ E and also to the existence of an affine structure on H. Applications include complex, paracomplex and symplectic geometries.Fil: Calvaruso, Giovanni. Università del Salento; ItaliaFil: Ovando, Gabriela Paola. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentin
Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups
No full textLet (M, g) be a pseudo-Riemannian manifold. If M is compact, g is Riemannian and thetangent bundle TM is equipped with the Sasaki metric gs, parallel vector fields are the only harmonic maps from (M, g) to (TM, gs). On the other hand, if g is Lorentzian, then vector fields satisfying some harmonicity properties
need not be parallel. We investigate such properties for left-invariant vector fields onthree-dimensional Lorentzian Lie groups, obtaining several classification results and new
examples of critical points of energy functionals
On semi-direct extensions of the Heisenberg group
No full textAny S∈ sp(1 , R) induces canonically a derivation S of the Heisenberg Lie algebra h and so, a semi-direct extension GS= H⋊ exp (RS) of the Heisenberg Lie group H (Müller and Ricci in Invent Math 101: 545–582, 1990). We shall explicitly describe the connected, simply connected Lie group GS and a family ga of left-invariant (Lorentzian and Riemannian) metrics on GS, which generalize the case of the oscillator group. Both the Lie algebra and the analytic description will be used to investigate the geometry of (GS, ga) , with particular regard to the study of nontrivial Ricci solitons
Siklos spacetimes as homogeneous Ricci solitons
No full textWe consider a well-known class of homogeneous solutions of the Einstein–Maxwell equations found by Siklos and prove that all these spacetimes are nontrivial Ricci solitons. In particular, these examples show that several rigidity results for homogeneous Lorentzian gradient Ricci solitons do not extend to the non-gradient case
Harmonicity of vector fields on four-dimensional generalized symmetric spaces
No full textLet (M = G/H; g) denote a four-dimensional pseudo-Riemannian generalized symmetric space and g = m + h
the corresponding decomposition of the Lie algebra g of G. We completely determine the harmonicity properties
of vector fields belonging to m. In some cases, all these vector fields are critical points for the energy functional
restricted to vector fields. Vector fields defining harmonic maps are also classified, and the energy of these vector
fields is explicitly calculated
The Ricci soliton equation for homogeneous Siklos spacetimes
No full textWe complete the classification of Ricci solitons within all classes of homogeneous Siklos metrics
Einstein-Like Metrics on Three-Dimensional Non-unimodular Lorentzian Lie Groups
No full textWe revise the classification of Einstein-like left-invariant metrics on three-dimensional non-unimodular Lie groups. Because of the more general form of three-dimensional non-unimodular Lorentzian Lie algebras, further examples are found
Harmonicity of unit vector fields with respect to Riemannian g-natural metrics
No full textAbstractLet (M,g) be a compact Riemannian manifold and T1M its unit tangent sphere bundle. Unit vector fields defining harmonic maps from (M,g) to (T1M,g˜s), g˜s being the Sasaki metric on T1M, have been extensively studied. The Sasaki metric, and other well known Riemannian metrics on T1M, are particular examples of g-natural metrics. We equip T1M with an arbitrary Riemannian g-natural metric G˜, and investigate the harmonicity of a unit vector field V of M, thought as a map from (M,g) to (T1M,G˜). We then apply this study to characterize unit Killing vector fields and to investigate harmonicity properties of the Reeb vector field of a contact metric manifold
Two-step homogeneous geodesics in pseudo-Riemannian manifolds
No full textGiven a homogeneous pseudo-Riemannian space (G/H,⟨,⟩), a geodesic γ: I→ G/ H is said to be two-step homogeneous if it admits a parametrization t= φ(s) (s affine parameter) and vectors X, Y in the Lie algebra g, such that γ(t) = exp (tX) exp (tY) · o, for all t∈ φ(I). As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics ⟨,⟩ on the unimodular Lie group SL(2 , R) such that (SL(2,R),⟨,⟩) is a two-step g.o. space
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