Dipartimento di Matematica ed Informatica,

Facolt\'a di Scienze Matematiche,

Fisiche e Naturali,

Universit\'a degli Studi di Salerno,

Via Ponte del Melillo,

84084 Fisciano (SA),

E-mail: vinograd@unisa.it

**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.