OAI.ThreeManifold.main_theorem

The theorem states that the proposition MainStatement holds, i.e. there exists a closed three-dimensional manifold with the following properties. The manifold M is a Hausdorff, second countable, compact, connected topological space, equipped with a charted-space structure modeled on ℝ³ (functions Fin 3 → ℝ) and a smooth (C^∞) manifold structure, such that every pair of atlas charts has transition maps whose Jacobian determinant is positive wherever defined, so the atlas is oriented. Moreover, M carries a smooth Riemannian metric g with no conjugate points, yet M carries no smooth Riemannian metric of nonpositive sectional curvature. Here, no conjugate points means: for every open interval U of ℝ, every smooth geodesic γ on U (one whose coordinate expression satisfies the geodesic equation, defined through Christoffel symbols computed from g in each atlas chart), and every smooth Jacobi field J along γ on U (satisfying the Jacobi equation, with covariant derivative and curvature term computed in charts), if J vanishes at two distinct points of U then J vanishes at every point of U. Nonpositive sectional curvature is expressed chartwise: at every point of every chart and for all coordinate vectors u and v, the g-inner product of the curvature term R(u,v)v with u is at most zero.