2-step Nilmanifolds Research Articles

ANOSOV DIFFEOMORPHISMS ON A CLASS OF 2-STEP NILMANIFOLDS

In this paper we study nilmanifolds which are modeled on a quotient of a free 2-step nilpotent Lie group by a 1-dimensional subgroup. In fact we obtain a very easy criterion to decide whether or not such a nilmanifold admits an Anosov diffeomorphism.

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 translated geodesics in those groups. A good part of the paper focuses on the existence of closed timelike geodesics in compact Lorentz 2-step nilmanifolds. Other related results, corollaries, and examples are also presented.

On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold N / Γ , N/\Gamma , where N N is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and Γ \Gamma a discrete subgroup of N N acting on N N by left translations. For this purpose, we shall first show that if N N is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric g , g, then the restriction of g g to the center Z Z of N N is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group H 3 H_{3} . If ( N , g ) \left ( N,g\right ) is one such group, we prove that no timelike geodesic in ( N , g ) \left ( N,g\right ) can be translated by an element of N . N. By the way, we rediscover that the Heisenberg Lie group H 2 k + 1 H_{2k+1} admits a flat left-invariant Lorentz metric if and only if k = 1. k=1.

Closed geodesics in compact nilmanifolds

We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU(2).

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 2-step nilmanifolds. The Jacobi equations are also used to classify Riemannian foliations of 3-dimensional nilmanifolds.

Symplectic rigidity of geodesic flows on two-step nilmanifolds

We show that if two 2-step Riemannian nilmanifolds have symplectically conjugate geodesic flows, then they must be isometric. By 2-step Riemannian nilmanifold, we mean a Riemannian manifold of the form (T⧹ N, g), where N is a 2-step nilpotent Lie group, Γ is a cocompact discrete subgroup of N, and g is a metric whose pullback to N is left invariant.

Smoothly closed geodesics on 2-step nilmanifolds

Smoothly closed geodesics on 2-step nilmanifolds