In Chapter 1 we introduced the group of affine transformations, which consists of those transformations of an affine space which preserve its affine structure. In Chapter 4 we discussed the idea of volume, and picked out from among all affine transformations the subgroup of those which preserve the volume form, namely those with unimodular linear part. In Chapter 7 we introduced another structure on affine space: a metric. We now examine the transformations which preserve this structure. They are called isometries.
An isometry of an affine space is necessarily an affine transformation. This may be deduced from the precise definition, as we shall show, and need not be imposed as part of it. Isometries form a group. Particular examples of isometry groups which are important and may be familiar are the Euclidean group, the group of isometries of Euclidean space ε3; and the Poincaré group, which is the group of isometries of Minkowski space ε1,3. Each of these groups is intimately linked to a group of linear transformations of the underlying vector space, namely the group preserving its scalar product. Such groups are called orthogonal groups (though in the case of ε1,3 the appropriate group is more frequently called the Lorentz group).
Any one-parameter group of isometries induces a vector field on the affine metric space on which it acts. This vector field is called an infinitesimal isometry.