Curved Geometries in Persistent Homology: From Euclidean to AdS Metrics in Bow Echo Dynamics

We propose a geometry topological framework to analyze storm dynamics by coupling persistent homology with Anti-de Sitter (AdS)-inspired metrics. On radar images of a bow echo event, we compare Euclidean distance with three compressive AdS metrics (α = 0.01, 0.1, 0.3) via time-resolved H1 persistence diagrams for the arc and its internal cells. The moderate curvature setting (α=0.1) offers the best trade-off: it suppresses spurious cycles, preserves salient features, and stabilizes lifetime distributions. Consistently, the arc exhibits longer, more dispersed cycles (large-scale organizer), while cells show shorter, localized patterns (confined convection). Cross-correlations of H1 lifetimes reveal a temporal asymmetry: arc activation precedes cell activation. A differential indicator Δ(t) based on Wasserstein distances quantifies this divergence and aligns with the visual onset in radar, suggesting early warning potential. Results are demonstrated on a rapid Corsica bow echo; broader validation remains future work.