Symmetries, differential invariants and Einstein equations

Abstract:
Solutions of a system of PDEs are bearers of geometric structures whose types are determined by the PDE itself. In a sufficiently nonlinear situation geometric structures corresponding to different solutions are nonequivalent and such an equation is manifestly symmetry-less. So, the symmetry of a PDE depends on how large the deviation from this norm is. On the other hand, geometrical structures of the same type can be distinguished one from another by means of differential invariants, first of all, scalars. This way symmetries and scalar differential invariant become related. This relationship will be discussed and illustrated by Einstein equations.