We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.

Dynamical phase transition (DPT) characterizes the abrupt change of dynamical properties in nonequilibrium quantum many-body systems. It has been demonstrated that extra fluctuating modes besides conventional order parameter field can drastically equilibrium transitions. However, counterpart phenomena DPTs have rarely explored. Here, we study DPT Dirac system after a sudden quench, and find fer...