Explore the geometric invariants of quantum state manifolds in this 56-minute QuICS lecture. Delve into the Bargmann invariant as a tool for capturing relationships between quantum state triples within a manifold. Discover how expanding the Bargmann invariant for closely separated states reveals Berry curvature and quantum metric, defining intrinsic geometry. Examine higher-order expansions that yield new gauge-invariant objects probing extrinsic manifold properties. Learn about applications to material properties analysis through band structures and potential uses in adiabatic state preparation. Based on arXiv:2205.15353, this talk presents a comprehensive framework for characterizing parameter-dependent quantum states, offering insights beyond established concepts like Berry phase and quantum distance.