• Corpus ID: 17334581


  author={Luis A. Cordero and Phillip. E. Parker},
  journal={arXiv: Differential Geometry},
We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and make a study of lattices and periodic geodesics. Some major dierences from the Riemannian theory appear. There are many at groups ( 


We give a basic treatment of lattices Γ in these groups. Certain tori TF and TB provide the model fiber and the base for a submersion of Γ\N. This submersion may not be pseudoriemannian in the usual

Lorentz Geometry of 2-Step Nilpotent Lie Groups

We study the geometry of 2-step nilpotent Lie groups endowed with left-invariant Lorentz metrics. After integrating explicitly the geodesic equations, we discuss the problem of the existence of

PseudoH-type 2-step nilpotent Lie groups

PseudoH-type is a natural generalization of H-type to geometries with indefinite metric tensors. We give a complete determination of the conjugate locus including multiplicities. We also obtain a

Conjugate Loci of Pseudo-Riemannian 2-Step Nilpotent Lie Groups with Nondegenerate Center

We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is one-dimensional we obtain formulas in all cases (Theorem 2.5),

Pseudo-Riemannian Lie groups of modified H-type

We define a class of Riemannian and pseudo-Riemannian 2-step nilpotent Lie groups with nondegenerate centers (Definition 2.2) that generalize the H-type groups of Kaplan [8, 9, 10]. Examples are

Pseudo H-type 2-step Nilpotent Lie Groups

Pseudo H-type is a natural generalization of H-type to geometries with indefinite metric tensors. We give a complete determination of the conjugate locus including multiplicities. We also obtain a

Nilsolitons of H-type in the Lorentzian setting

It is known that all left-invariant pseudo-Riemannian metrics onH3 are algebraic Ricci solitons. We consider generalizations of RiemannianH-type, namely pseudoH-type and pH-type. We study algebraic

Pseudoriemannian Nilpotent Lie Groups

This is a survey article with a limited list of references (as required by the publisher) which appears in the Encyclopedia of Mathematical Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T.

On the Existence of a Codimension 1 Completely Integrable Totally Geodesic Distribution on a Pseudo-Riemannian Heisenberg Group

In this note we prove that the Heisenberg group with a left-invariant pseudo- Riemannian metric admits a completely integrable totally geodesic distribution of codimen- sion 1. This is on the

The timelike cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups

Abstract.We investigate the timelike cut locus and the locus of conjugate points in Lorentz 2-step nilpotent Lie groups. For these groups with a timelike center, we give some criteria for the



Cut and conjugate loci in two-step nilpotent Lie groups

We show that cut and conjugate loci coincide for a large class of nilpotent Lie groups with left-invariant metric. One consequently obtains lower bounds for the injectivity radius of nonsingular

Foundations of Differentiable Manifolds and Lie Groups

1 Manifolds.- 2 Tensors and Differential Forms.- 3 Lie Groups.- 4 Integration on Manifolds.- 5 Sheaves, Cohomology, and the de Rham Theorem.- 6 The Hodge Theorem.- Supplement to the Bibliography.-

Left-invariant Lorentz metrics on Lie groups

With J. Milnor [2] we consider a special class @ of solvable Lie groups. A non-commutative Lie group G belongs to @ if its Lie algebra g has the property that [x, y] is a linear combination of x and

Spaces of geodesics: products, coverings, connectedness

We continue our study of the space of geodesics of a manifold with linear connection. We obtain sufficient conditions for a product to have a space of geodesics which is a manifold. We investigate


In dimension three, there are only two signatures of metric tensors: Lorentzian and Riemannian. We find the possible pointwise symmetry groups of Lorentzian sectional curvatures considered as

Geometry of 2-step nilpotent groups with a left invariant metric. II

We obtain a partial description of the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group with a left invariant metric. We consider only the case that N is nonsingular;


We find the Riemann curvature tensors of all leftinvariant Lorentzian metrics on 3-dimensional Lie groups. MSC(1991): Primary 53C50; Secondary 53B30, 53C30. −−−−−−−−−−−−−−−−−−−−−−−−−−→Υ·

Sur les groupes de Lie nilpotents

SummaryDémonstration élémentaire du résultat suivant: si le groupe de Lie N est nilpotent et simplement connexe, et si σ est un automorphisme de N tel que σ* n'ait pas de valeur propre égale à1,

Curvatures of left invariant metrics on lie groups

The space of geodesics

Let M be a manifold with linear connection ▽. The space G(M) of all geodesics of M may be given a topological structure and may be realized as a quotient space of the reduced tangent bundle of M. The