On the geometry of lambda-symmetries, and PDEs reduction

Abstract

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions.