Generalization of conformal-disformal transformations of the metric in scalar-tensor theories

We study new classes of metric transformations in the context of scalar-tensor theories, which involve both higher derivatives of the scalar field and derivatives of the metric itself. In general, such transformations are not invertible as they involve derivatives of the metric, which typically leads to instability due to Ostrogradsky ghosts. We show, however, that a certain class of this type of transformations is invertible: We construct new examples of invertible conformal (and also disformal) transformations with higher derivatives. Finally, we make use of these new transformations to construct extended mimetic theories of gravity, and we study their properties in the context of cosmology.