Abstract: A quantum state that depends on a parameter is a commonly studied structure in quantum physics. Examples include the ground state of a Hamiltonian with a parameter or Bloch states as functions of the quasimomentum. The change in the state as the parameter varies can be characterized by such geometric objects as the Berry phase or the quantum distance which has led to many insights in the understanding of quantum systems. In the talk, I will present a general framework for describing the geometric invariants of a manifold of quantum states that produces all known objects and also leads to some new ones. As the main tool we will use the Bargmann invariant that captures the relation between triples of quantum states within the manifold. When the states in the Bargmann invariant are taken close to each other one recovers the Berry curvature ω and quantum metric g at the leading order of the expansion in the separation between the states. This defines the intrinsic geometry of the manifold of states. We show that the higher orders in this expansion are functionally independent of ω and g, giving rise to new local gauge-invariant objects which probe the extrinsic properties of the manifold. Lastly, I will show how our formalism is useful for understanding material properties given a band structure and also might be applied to the study of adiabatic state preparation. The talk is based on arXiv:2205.15353.