Space Form Conjecture (pseudo-Riemannian constant curvature 1)

Establish the complete classification of signatures (p,q) for which there exists a compact, complete, pseudo-Riemannian manifold of signature (p,q) with constant sectional curvature 1, proving that this occurs if and only if (p,q) belongs to the specified list of allowed signatures.

Background

This is a central problem in non-Riemannian geometry and a signature-specific analogue of classical space form results, seeking compact models of constant curvature in indefinite metrics.

The conjecture ties existence to arithmetic constraints captured by the RadonHurwitz numbers and related topology of vector fields on spheres.