Spherical Curves and Their Rigidity in Metric Spaces with Lower Curvature Bounds

Abstract

In this paper, we examine geometric relationships between metric spaces with curvature bounded below and their corresponding model spaces of constant curvature. Let \(\gamma\) be a closed spherical curve in a metric space whose curvature is bounded below by \(K\), lying at a distance \(r < \tfrac{\pi}{2\sqrt{K}}\) from a point. Let \(\gamma^{\prime}\) denote the circle of radius \(r\) centered at the corresponding point in the model space of constant curvature \(K\). Under suitable geometric equivalence conditions—namely, the preservation of pairwise distances between corresponding points of \(\gamma\) and \(\gamma^{\prime}\), the isometry of convex hulls of corresponding geodesic triangles, and the equality of arc lengths or total curvature—we show that the geodesic surface enclosed by \(\gamma\) is isometric to the region bounded by \(\gamma^{\prime}\). This result offers a foundational geometric characterization of metric spaces with curvature bounded below through their model counterparts and provides a framework for further study of total curvature, convexity, and isometric embeddings in such spaces.