Positive Curvature on S² × S²
This presentation explains a breakthrough construction of a metric with strictly positive sectional curvature on S² × S², resolving a classical four-dimensional existence problem. The construction begins with the nonnegatively curved Cheeger–Müter metric, which has isolated zero-curvature planes, and uses a carefully designed three-stage perturbation to eliminate all remaining degeneracy without destroying smoothness or positivity elsewhere.Script
For over half a century, differential geometers wondered whether the product of two two-spheres could be given strictly positive curvature. The Hsiang–Kleiner theorem ruled out any positively curved metric with continuous symmetry, making the problem deceptively hard.
The authors start with the Cheeger–Müter metric, which already has nonnegative curvature. At every point away from two special curves, exactly one two-plane has zero curvature while all others curve positively. These zero-curvature planes form a smooth four-dimensional locus inside the eight-dimensional space of all two-planes.
The perturbation proceeds in three stages. The first-order term is designed so its curvature variation vanishes on the distinguished zero planes. The second-order correction makes curvature nonnegative everywhere, strictly positive except along a special two-dimensional torus. The third-order term then removes the final degeneracy on that torus.
After the second-order correction, only one geometric family of planes remains flat: those over a torus where both sphere coordinates are perpendicular to a chosen vertical axis. The second-order curvature coefficient equals a positive constant times one minus the squared vertical component, so it vanishes precisely on this torus and is positive everywhere else.
The third-order coefficient on the torus is not constant; it oscillates. The authors use the Laplacian to subtract the oscillatory part, leaving only its positive average. This conformal trick converts a sign-changing function into a uniform positive constant, ensuring that a small perturbation parameter of the correct sign makes every plane positively curved.
The result is existential but definitive: the product of two two-spheres admits positive sectional curvature, answering a question open since the mid-twentieth century. To explore more cutting-edge geometry and create videos from the latest research, visit EmergentMind.com.
