Mathematical Validation of Kalimuthu's Configurations as Degenerate Spherical Manifolds

Abstract

We provide a rigorous differential-geometric analysis of two spherical configurations proposed by Sennimalai Kalimuthu, where the sum of interior angles of a figure formed by three geodesics on a unit sphere S² attains 360° (2π) and 540° (3π). While such figures satisfy the global Gauss-Bonnet theorem Σθi = π + Area (T), we prove they are not regular geodesic triangles in the sense of do Carmo. Using the metric tensor ds² = dθ² + sin²θ dφ², analysis of boundary tangent vectors, and Jacobian rank evaluation of the spherical chart, we show that these configurations correspond to a π-lune and a hemisphere with coincident boundaries. At the polar vertex, the coordinate basis vector ∂φ vanishes, and the Jacobian rank collapses from 2 to 1, demonstrating a degenerate topological state where the two-dimensional interior fails to be embedded as a regular 2-simplex. The Gaussian curvature K=1 remains intrinsically regular, proving the degeneracy is a chart-dependent boundary singularity, not a curvature singularity. These configurations serve as explicit, pedagogical examples of collapsed and degenerate manifolds.