Three types of geometry: hyperbolic (negative curvature, saddle-shaped), flat (zero curvature, Euclidean), and spherical (positive curvature, round). These are the three Thurston geometries in dimension 3, and they do not interact — a manifold with one geometry is not continuously deformable into a manifold with another.
Except at the boundary. Consider a flat SU(2)-bundle over a hyperbolic surface. The total space is a 3-manifold carrying a flat metric derived from the product of the hyperbolic base and the flat connection. Now collapse the fibers: shrink the SU(2) fibers toward zero size while rescaling the metric to maintain finite diameter.
The limit is a spherical 3-manifold. A space constructed from hyperbolic geometry and flat connections degenerates, in the metric limit, to a space of positive curvature. The three geometries — hyperbolic, flat, and spherical — are connected through the collapsing process. The spherical manifold is the boundary point of a family of flat bundles over hyperbolic surfaces.
The mechanism is the curvature concentration that occurs during collapse. As the fibers shrink, the flat connection's holonomy becomes concentrated. In the limit, the concentrated holonomy generates the positive curvature of the spherical geometry. The curvature was latent in the holonomy of the flat connection and is released by the collapsing process.
The structural point: geometric types that appear discontinuous in the classification are continuous in the moduli space. The classification says “hyperbolic and spherical are different.” The collapsing theorem says “spherical is where hyperbolic goes when you squeeze it hard enough.” The boundary of one geometric type is another geometric type. The taxonomy is a snapshot; the moduli space is a landscape, and the geometric types are regions connected by paths.