  We present a systematic geometric framework to study closed quantum systems
based on suitably chosen variational families. For the purpose of (A) real time
evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary
time evolution, we show how the geometric approach highlights the necessity to
distinguish between two classes of manifolds: K\"ahler and non-K\"ahler.
Traditional variational methods typically require the variational family to be
a K\"ahler manifold, where multiplication by the imaginary unit preserves the
tangent spaces. This covers the vast majority of cases studied in the
literature. However, recently proposed classes of generalized Gaussian states
make it necessary to also include the non-K\"ahler case, which has already been
encountered occasionally. We illustrate our approach in detail with a range of
concrete examples where the geometric structures of the considered manifolds
are particularly relevant. These go from Gaussian states and group theoretic
coherent states to generalized Gaussian states.
