  Topological insulators have been studied intensively over the last decades.
Earlier research focused on Hermitian Hamiltonians, but recently, peculiar and
interesting properties were found by introducing non-Hermiticity. In this work,
we apply a quantum geometric approach to various Hermitian and non-Hermitian
versions of the Su-Schrieffer-Heeger (SSH) model. We find that this method
allows one to correctly identify different topological phases and topological
phase transitions for all SSH models, but only when using the metric tensor
containing both left and right eigenvectors. Whereas the quantum geometry of
Hermitian systems is Riemannian, introducing non-Hermiticity leads to
pseudo-Riemannian and complex geometries, thus significantly generalizing from
the quantum geometries studied thus far. One remarkable example of this is the
mathematical agreement between topological phase transition curves and
lightlike paths in general relativity, suggesting a possibility of simulating
space-times in non-Hermitian systems. We find that the metric in non-Hermitian
phases degenerates in such a way that it effectively reduces the dimensionality
of the quantum geometry by one. This implies that within linear response
theory, one can perturb the system by a particular change of parameters while
maintaining a zero excitation rate.
