## Nilpotent Lie Groups

Pseudoriemannian Nilpotent Lie Groups, 30 June 2005, rev. 3 October 2005.
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. Françoise, G. L. Naber and Tsou S. T. Oxford: Elsevier, 2006. vol.4, pp.94--104.
Bibliography of pseudoRiemannian nilpotent Lie groups, 27 Jan 2010; updated 31 Mar 2013.
For my own convenience, and I hope for the community as well, I will maintain this with periodic updates (listings appreciated) for at least the next few years.
• PseudoRiemannian 2-step Nilpotent Lie Groups, preprint 10 May 1999, 62 pp. (Changed one word 15 Nov 2000.) [PDF]
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 differences from the Riemannian theory appear. There are many flat groups (versus  none), including Heisenberg groups. While still a semidirect product, the isometry group can be strictly larger than the obvious analogue. Everything is illustrated with explicit examples.

We introduce the notion of pH-type, which refines Kaplan's H-type and completes Ciatti's partial extension. We give a general construction for algebras of pH-type.

• Isometry Groups of pseudoRiemannian 2-step Nilpotent Lie Groups, Houston J. Math. 35 (2009) 49--72. [PDF]
While still a semidirect product, the isometry group can be strictly larger than the obvious Riemannian analogue Iaut. In fact, there are three relevant groups of isometries, IsplIautI, and Ispl < Iaut < I is possible when the center is degenerate. When the center is nondegenerate, Ispl = I.
• Lattices and Periodic Geodesics in pseudoRiemannian 2-step Nilpotent Lie Groups, Int. J. Geom. Methods Mod. Phys. 5 (2008) 79--99. [PDF]
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 sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.
• Conjugate Loci of PseudoRiemannian 2-step Nilpotent Lie Groups with Nondegenerate Center, Ann. Global Anal. Geom. 28 (2005) 1--18. [PDF]
We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is 1-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield some new information about the smoothness of the pseudoriemannian conjugate locus. We also obtain the multiplicities of all conjugate points.
• Examples of Conjugate Loci of PseudoRiemannian 2-step Nilpotent Lie Groups with Nondegenerate Center, in Recent Advances in Riemannian and Lorentzian Geometries, eds. K.L. Duggal and R. Sharma. Contemp. Math. 337. Providence: AMS, 2004. pp.91--108.
We present examples of geodesics, geodesic surfaces, and conjugate loci in the 3-dimensional Heisenberg group with signatures (+ - +) and (- - +).
• PseudoH-type 2-step Nilpotent Lie Groups, Houston J. Math. 31 (2005) 765--786. [PDF]
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 partial characterization in terms of the abundance of totally geodesic, 3-dimensional submanifolds.

### Geodesic Surfaces and Conjugate Loci in the Heisenberg Group

The following page shows surfaces comprised of geodesics emanating from the identity element in the 3-dimensional Heisenberg group with basis e, e, z . The first picture is with a Lorentzian metric of signature (- + -), the second set are with (- - +), and the third with (+ - +). To see "inside" the null cone, the third are rotated relative to the other two.

Geodesic Surfaces and Conjugate Loci

In all pictures, the curves radiating or emanating outward from the origin (0,0,0) are geodesics and the "concentric circles" transverse to them are curves of constant t -value, with geodesic parameter t .