  Studies have shown that the Hilbert spaces of non-Hermitian systems require
nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition
to time, can emerge naturally from a geometric formalism. Specifically, in this
formalism, Hamiltonians can be interpreted as a Christoffel symbol-like
operators, and the Schroedinger equation as a parallel transport in this
formalism. We then derive the evolution equations for the states and metrics
along the emergent dimensions and find that the curvature of the Hilbert space
bundle for any given closed system is locally flat. Finally, we show that the
fidelity susceptibilities and the Berry curvatures of states are related to
these emergent parallel transports.
