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.Download a zipped PostScript file of this joint paper with L. A. Cordero :
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.
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.
We present examples of geodesics, geodesic surfaces, and conjugate loci in the 3-dimensional Heisenberg group with signatures (+ - +) and (- - +).
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.
The following page shows surfaces comprised of geodesics emanating from the identity element in the 3-dimensional Heisenberg group with basis e1 , e2 , 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.
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 .