  Time-dependent $\mathcal{PT}$-symmetric quantum mechanics is featured by a
varying inner-product metric and has stimulated a number of interesting studies
beyond conventional quantum mechanics. In this paper, we explore geometric
aspects of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We not
only find a geometric phase factor emerging naturally from cyclic evolutions of
$\mathcal{PT}$-symmetric systems, but also formulate a series of differential
geometry concepts, including connection, curvature, parallel transport, metric
tensor, and quantum geometric tensor. Our findings constitute a useful, perhaps
indispensible, tool to tackle physical problems involving
$\mathcal{PT}$-symmetric systems with time-varying system's parameters. To
exemplify the application of our findings, we show that the unconventional
geometrical phase [Phys. Rev. Lett. 91, 187902 (2003)], consisting of a
geometric phase and a dynamical phase proportional to the geometric phase, can
be expressed as a single geometric phase identified in this work.
