Quantum geometry of connected state manifolds: When diabolic points act as bridges between eigenstate manifolds

Parametric Hamiltonians often exhibit point-like spectral degeneracies, called diabolic points (DPs) or conical intersections, which can lead to singularities in the Provost-Vallee metric of eigenstate manifolds. We regularise the metric by a transformation to polar coordinates and develop a formalism in which DPs act as bridges between adjacent eigenstate manifolds, glueing them into a single connected state manifold (CSM). We characterise the topology of this structure and extend the local rules for nodal lines governing the Berry phase to global ones. On the CSM, the class of geodesics can be enlarged by those passing through the DPs. They can be stable with respect to their initial direction, and can be shorter than those constrained to a single eigenstate manifold, providing new geodesic shortcuts. Further, the CSM provides a new mechanism for Berry phase computation along paths traversing the DPs.