XVI

Introduction

In Chapter 8 we give three applications of homogeneous spaces.

The first is about homogeneous Einstein metrics. These are Riemann-

ian metrics whose Ricci tensor is proportional to the metric. The

second refers to symplectic geometry, which is rooted in Hamilton's

laws of optics. Here we present a Hamiltonian system on generalized

flag manifolds. A Hamiltonian system is a special case of an inte-

grable system, which is a subject that has attracted much attention

recently. The third application deals with homogeneous geodesies in

homogeneous spaces. Geodesies are important not only in geometry,

being length minimizing curves, but also have important applications

in mechanics since, for example, the equation of motion of many sys-

tems reduces to the geodesic equation in an appropriate Riemannian

manifold. Here, we present some results about homogeneous spaces,

all of whose geodesies are homogeneous, that is, they are orbits of

one-parameter subgroups. These are usually known in the literature

as g.o. spaces.