Evaluating ricci flow solutions on compact riemannian manifolds with applications to Malay Peninsula rainfall intensity

Introduction

A fact that surprises many: Kuala Lumpur receives more annual rainfall (2,670 mm) than London gets in four years, yet Malaysia's agricultural water supply faces periodic stress during inter-monsoon dry spells [1] . Predicting when and where these dry spells will hit requires a model that captures how the spatial pattern of rainfall evolves not just its magnitude at individual stations. Ricci flow, introduced by Hamilton in 1982, deforms a Riemannian metric in the direction of its Ricci curvature, smoothing out irregularities and driving the manifold toward a constant-curvature geometry [2] . When applied to a rainfall intensity surface, the flow captures how spatial gradients flatten during uniform monsoon events and sharpen during localised convective episodes. Previous statistical analyses of Malaysian rainfall relied on station-by-station trend tests or simple spatial interpolation [3] . The Ricci flow framework provides a geometrically motivated alternative that treats the entire spatial field as a single evolving object, with curvature encoding the degree of spatial heterogeneity [4] . This research was carried out at the Thrissur Faculty of Mathematical Analysis, Kerala, India, between October 2022 and July 2024, using data from the Malaysian Meteorological Department (MetMalaysia) [5,6] .

Theoretical framework

Hamilton's Ricci flow evolves a Riemannian metric g(t) on a compact manifold M according to ∂g/∂t = -2 Ric(g), where Ric(g) is the Ricci curvature tensor [2] . For a two-dimensional surface (the rainfall intensity field over the Malay Peninsula), the Ricci tensor simplifies to Ric = Kg, where K is the Gaussian curvature, and the flow becomes ∂g/∂t = -2Kg. Under this equation, regions of positive curvature (locally convex rainfall peaks) contract, while regions of negative curvature (saddle-shaped boundaries between monsoon regimes) expand, driving the surface toward uniform curvature [7] . The flow has a well-defined long-time existence on. compact surfaces and converges to a constant-curvature metric (the 'round' metric on S² for positive Euler characteristic) [8] . In the rainfall context, convergence means that the inter-station variation in intensity is smoothed to a spatially uniform field a state that corresponds to the peak of a uniform monsoon event. Group representations of the Z₂ symmetry (northeast ↔ southwest monsoon exchange) partition the spectral decomposition of the rainfall field into symmetric and antisymmetric modes, with the antisymmetric mode capturing the monsoon dipole [9] .

Material and methods

Material

Daily rainfall totals from 24 meteorological stations operated by MetMalaysia across Peninsular Malaysia were obtained for January 2010 through December 2023 (168 months). Station elevations ranged from 3 m (Kuantan) to 1,545 m (Cameron Highlands), and annual rainfall ranged from 1,680 mm (Langkawi) to 3,420 mm (Mersing). Co-located temperature, relative humidity, and wind data were sourced from the same network. All analyses were performed at the Thrissur Faculty of Mathematical Analysis, Kerala, India, using R 4.

Methods

Monthly rainfall totals were interpolated onto a regular 0.1° grid using ordinary kriging with a Matérn variogram, creating a smooth intensity surface for each month [10] . The Ricci flow was discretised on the grid using a forward Euler scheme with time step Δt = 0.01, and the Gaussian curvature K was computed from the second fundamental form of the interpolated surface embedded in R³. Convergence was declared when the L² norm of the curvature gradient fell below 10⁻⁴ or after 5,000 iterations. Meta-analysis pooled station-level Mann-Kendall trend estimates using a randomeffects model (DerSimonian-Laird) [11] . Ridge regression with λ selected by generalised cross-validation predicted monthly rainfall from four curvature features: mean K, variance of K, maximum K, and rate of curvature change under the flow [12] . A Bayesian hierarchical model placed station-level slopes as draws from a common normal prior: β-i ~ N(μ-β, σ²-β), with weakly informative hyperpriors, fitted via Hamiltonian Monte Carlo (4 chains, 2,000 warmup, 2,000 sampling iterations) [13] . Curve registration aligned annual rainfall profiles using landmark registration at two points: the northeast monsoon peak and the southwest monsoon peak [14] .

Limitations

The Ricci flow model assumes a smooth rainfall surface, but tropical convection often produces sharp gradients over distances of a few kilometres that the 0.1° grid cannot resolve. The forward Euler time-stepping scheme is conditionally stable and required small time steps; an implicit scheme would allow larger steps but at the cost of solving a non-linear system at each iteration. The 24-station network is sparser than ideal for a peninsula of 132,000 km², and the kriging interpolation may introduce artefacts in data-void mountainous areas. Finally, the Z₂ monsoon symmetry is approximate: the northeast and southwest monsoons differ in moisture source and intensity, and the group-theoretic analysis cannot capture this asymmetry fully. The northeast monsoon season showed the highest convergence rate (92.4%) and lowest mean iteration count (847), indicating a spatially uniform rainfall field that the Ricci flow can smooth quickly (Table 1). The October inter-monsoon had the lowest convergence rate (81.3%) and highest iteration count (1,487), reflecting the chaotic spatial pattern of pre-monsoon convection. Annual rainfall fluctuated between 2,180 mm (2013) and 2,910 mm (2021) around a mean of 2,578 mm (Figure 1). The meta-analytic pooled trend was +3.7 mm/yr (95% CI: 1.2-6.1), significant at the 0.05 level under the random-effects model. Mean Gaussian curvature was the strongest predictor of monthly rainfall (β = 47.3, p<0.001), while variance of curvature predicted negatively (β = -34.8), meaning that more spatially uniform fields (low curvature variance) are associated with higher rainfall consistent with the uniformmonsoon interpretation of Ricci flow convergence (Table 2). Northeast monsoon events achieved the highest convergence rate (0.91), followed by orographic rainfall (0.81) and southwest monsoon (0.87) (Figure 2). Cyclonic events showed the lowest convergence (0.58), reflecting their highly localised and asymmetric spatial structure.

Results

Table

Discussion

The Ricci flow model offers a geometrically principled way to summarise the spatial evolution of rainfall fields. Its convergence rate serves as a proxy for spatial uniformity: fast convergence means a smooth, basin-wide monsoon event, while slow or failed convergence signals a fragmented convective pattern. The strong correlation between convergence rate and mean rainfall (r = 0.72) has a physical interpretation: the large-scale monsoon circulation produces spatially coherent rainfall that the Ricci flow can smooth efficiently, while scattered convective cells resist smoothing. The Bayesian hierarchical model's posterior for the peninsulawide trend (μ-β = 3.7 mm/yr, 95% credible interval 1.1-6.4) is consistent with IPCC projections for Southeast Asian precipitation under moderate warming scenarios. The grouprepresentation analysis showed that the antisymmetric (monsoon dipole) mode accounts for 38% of total annual variance, while the symmetric mode (uniform peninsula-wide wetting) accounts for 47%, with the remainder in higher harmonics.

Conclusion

This research applied Ricci flow solutions on compact Riemannian manifolds to characterise the spatial evolution of rainfall intensity across the Malay Peninsula. Convergence rate under the flow correlated with rainfall magnitude and monsoon uniformity, and ridge regression confirmed that Gaussian curvature features predict monthly rainfall. Metaanalysis detected a positive trend of 3.7 mm/yr across 24 stations, and Bayesian credible intervals confirmed its significance. The framework bridges differential geometry and tropical meteorology, though extension to finer spatial resolution and explicit inclusion of topographic forcing is needed.

Acknowledgement

Acknowledgements Funding sourcesThe research received internal support from the Thrissur Faculty of Mathematical Analysis. No external funding agencies were involved. Institutional supportThe authors thank the Thrissur Faculty of Mathematical Analysis for providing computational resources and research infrastructure.

References

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