Identifying topological invariants through symmetry principles

In this work, we propose a novel symmetry-based approach to identifying topological densities in gravitational theories. Starting from a generic Lagrangian constructed from quadratic curvature invariants, we systematically analyze the symmetries admitted by such actions. We find that requiring invariance under a particular class of transformations singles out a unique combination of coefficients: precisely that of the Gauss-Bonnet term. This result provides an alternative route to identifying topological densities, independent of standard geometric or cohomological constructions. We show that this symmetry criterion provides new insight into the structure of effective gravitational actions and the role of topological terms in theories of gravity.