$I-$Love$-$Curvature: Exploring compact stars' quasi-universal relation with curvature scalars

We investigate quasi-universal relations in neutron stars linking standard observables, such as tidal deformability ($\Lambda$) and normalized moment of inertia ($\bar{I}$), with normalized curvature scalars in general relativity. These curvature scalars include the Ricci scalar ($\mathcal{R}$), the Ricci tensor contraction ($\mathcal{J}$), the Weyl scalar ($\mathcal{W}$), and the Kretschmann scalar ($\mathcal{K}$). We systematically examine both piecewise polytropic and color-flavor-locked equations of state, finding: (1) significant correlations between both local (central and surface) and global (volume-averaged) curvature scalars with $\bar{I}$ and $\Lambda$; (2) especially strong correlations between surface and volume-averaged curvature scalars and both $\bar{I}$ and $\Lambda$; (3) a near equation-of-state-independent maximum for the normalized Ricci scalar, suggesting a link to the trace anomaly; and (4) new universal relations involving normalized central and volume-averaged pressure and energy density, which also correlate strongly with $\bar{I}$ and $\Lambda$. Using constraints from GW170817 and low-mass X-ray binaries, we demonstrate that $\Lambda$ measurements directly constrain both scalar curvature quantities and the interior properties of canonical-mass neutron stars. These findings agree with the literature on equation-of-state-dependent Bayesian inference estimates. Our identified relations thus provide an equation-of-state-insensitive connection between stellar observables, spacetime geometry, and the microphysics of compact stars.