The generalized flame surface density (FSD) transport conditional on local flow topologies in premixed turbulent flames has been analyzed based on a detailed chemistry direct numerical simulation database of statistically planar turbulent hydrogen-air premixed flames with an equivalence ratio of 0.7 representing the corrugated flamelets, thin reaction zones and broken reaction zones regimes of combustion. The local flow topologies have been categorized by the values of the three invariants of the velocity gradient tensor and the statistical behaviors of the generalized FSD and different terms of its transport equation conditional on these flow topologies have been analyzed in detail for different choices of the reaction progress variable. The qualitative behavior of the different terms of the generalized FSD transport equation has been found to be similar for different choices of reaction progress variable but the statistical behaviors of the tangential strain rate term and its components have been found to be affected by the regime of combustion. The topologies, which exist for all values of dilatation rate, contribute significantly to the generalized FSD transport in premixed turbulent flames for all regimes of combustion. An unstable nodal flow topology, which is representative of a counter-flow configuration, has been found to be a dominant contributor to the FSD transport for all regimes of combustion irrespective of the choice of reaction progress variable. Moreover, a focal topology which is obtained only for positive values of dilatation rate, has been found to contribute significantly, especially to the curvature and propagation terms of the FSD transport equation for all regimes of combustion including the broken reaction zones regime. However, the contributions of the flow topologies to the turbulent transport and tangential strain rate term, which are obtained only for positive dilatation rates, have been found to weaken from the corrugated flamelets to the broken reaction zones regime.