Boundary Limits of Spherical Triangles and Degenerate Configurations in Positive-Curvature Manifolds: A Clarification of Hemispherical and Zero-Area Limits

Abstract

This paper provides a corrected and rigorous mathematical framework for the boundary configurations discussed in Kalimuthu's work on extreme spherical geometries. We distinguish two distinct limiting cases on the unit sphere S² that have been conflated in previous formulations: (i) the zero-area limit where three vertices become collinear on a great circle arc with total length < π, for which the interior angular sum tends to π (180°) and area tends to zero, and (ii) the hemispherical boundary case where three great-circle arcs enclose a hemisphere, for which the angular sum equals 2π (360°) and the area equals 2πR². Using linear algebra, we show that dimensional collapse corresponds precisely to coplanarity of the vertex vectors v_A, v_B, v_C ∈ R³, i.e., det[v_A, v_B, v_C] = 0, which implies linear dependence. We re-derive the Gauss-Bonnet relation in the limit, provide a qualitative interpretation of Ricci flow ∂_t g_ij = -2 Ric_ij as transverse collapse, and present a simple numerical example. Finally, we discuss the relevance of this boundary analysis to polar singularities in GNSS and to closed FLRW models, with consistent notation and corrected field equations.