I am studying measure-theoretic and functional analytic notions of surfaces since my background comes from physics I am wondering whether there is a simiar concept such as Lie groups in this setting. Indeed, it is standard to use Lie groups and symmetries to obtain solutions of PDEs.
Is there a standard notion of symmetry for measure theoretic surfaces? Is it possible to define some analogues of differential invariants in the setting of varifolds or of currents (linear functionals on the space of differential forms), where we are far from having a differential structure? I hope the question is not too vague and that it does make sense.