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Time-dependent $\mathcal{PT}$-symmetric quantum mechanics is featured by avarying inner-product metric and has stimulated a number of interesting studiesbeyond conventional quantum mechanics. In this paper, we explore geometricaspects of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We notonly find a geometric phase factor emerging naturally from cyclic evolutions of$\mathcal{PT}$-symmetric systems, but also formulate a series of differentialgeometry concepts, including connection, curvature, parallel transport, metrictensor, and quantum geometric tensor. Our findings constitute a useful, perhapsindispensible, tool to tackle physical problems involving$\mathcal{PT}$-symmetric systems with time-varying system's parameters. Toexemplify the application of our findings, we show that the unconventionalgeometrical phase [Phys. Rev. Lett. 91, 187902 (2003)], consisting of ageometric phase and a dynamical phase proportional to the geometric phase, canbe expressed as a single geometric phase identified in this work.

Differential geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics