CHAPTER 9 - Generalized Heisenberg groups

Summary

In chapter 7 we considered in detail the Heisenberg group. It is well-known that this group plays an important role in physics, for example in quantum mechanics and in the theory of contact transformations. It is also used extensively in harmonic analysis. Sometimes it provides nice examples in Riemannian geometry. We refer to [16], [31] where the Heisenberg group is used in connection with the problem of characterizing spaces by means of the volume of small geodesic spheres. See also [40].

Two properties of the Heisenberg group are important to be noted here. On the one hand it is an example of a 2-step nilpotent group. On the other hand we have shown in Theorem 7.2 that it is a naturally reductive homogeneous space. In this chapter we first show how the first property leads to a nice generalization, namely to the so-called generalized Heisenberg groups or groups of type H (see [21], [22], [23], [50]). But we will also show that some of the properties of the Heisenberg group do not hold for this larger class. More precisely we will show that there are groups of type H which are not naturally reductive. To do this we mainly concentrate on a remarkable six-dimensional example of Kaplan [23] but we also provide a different proof using only the methods of these notes. Finally we show that this six-dimensional manifold provides an example for some open problems related to the work of D'Atri and Nickerson [9], [10], [11].