Isometry germs and related structures

We define an Isometry germ at any given event $x$ of space-time as a vector field $\xi$ defined in a neighborhood of $x$ such that the Lie derivative of both the metric and the Riemannian connection are zero at this event. Two isometry germs can be said to be equivalent if their values and the values of their first derivatives coincide at $x$. The corresponding quotient space can be endowed with a structure of a bracket algebra which is a deformation of de Sitter's Lie algebra. Each isometry germ defines also a local stationary frame of reference, the consideration of the family of adapted coordinate transformations between any two of them leading to a local novel structure that generalizes the Lorentz group.