A three-manifold without conjugate points

A closed three-manifold with a Riemannian metric without conjugate points that admits no metric of nonpositive sectional curvature. Two holed-torus-times-circle pieces are glued so that the boundary classes satisfy h_2 = h_1 + f_1 and f_2 = h_1; shifts of axes and a Pythagorean length formula in spaces with the CN inequality show that no nonpositively curved metric exists, and a concave cometric on the neck, with an index-form analysis of every visit of a geodesic to the neck, excludes conjugate points. The course is an exposition of OpenAI's proof (2026).

Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Conjugate points and index forms
2. The manifold and its fundamental group
3. Spaces with the CN inequality
4. Axes, shifts and the two-piece obstruction
5. A concave cometric and convex collars
6. Neck passages and the index form
7. Returning passages and the main theorem

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Integral monodromy and the fundamental group (Complex threefolds from marked torus families)
• Comparison theorems, cut loci and curvature and topology (Bundles, curvature and holonomy)
• Completeness and the Hopf–Rinow theorem (Bundles, curvature and holonomy)
• Jacobi fields, conjugate points and the Morse index theorem (Bundles, curvature and holonomy)
• Jacobi fields, the second variation and the Cartan-Hadamard theorem (course DG-FND); van Kampen's theorem (course CG-S6); the fundamental group of the circle and covering spaces (core course Algebraic Topology).