Numerical integration of quantum time evolution in a curved manifold

The numerical integration of the Schr\"odinger equation by discretization time is explored for curved manifolds arising from finite representations based on evolving basis states. In particular, unitarity evolution assessed, in sense conservation mutual scalar products a set states, and with them orthonormality particle number. Although adequately represented known to give rise unitary spite curvature, discretized integrators easily break that conservation, thereby deteriorating their stability. Crank Nicolson algorithm, which offers Euclidian spaces independent time-step size $\mathrm{d}t$, can be generalised different ways. Here we compare previously proposed algorithm construction, albeit integrating wrong equation, faithful generalisation is, however, not strictly $\mathrm{d}t$.