{
  "id": "a-three-manifold-without-conjugate-points",
  "title": "A three-manifold without conjugate points",
  "description": "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).",
  "units": [
    {
      "id": "conjugate-points-and-index-forms",
      "title": "Conjugate points and index forms",
      "reader": "conjugate-points-and-index-forms.html",
      "source": "src/conjugate-points-and-index-forms.md",
      "source_sha256": "54429d6fa3ad1f073d1b829c317091f596e7408c2f451f2b13e853ba5626c4f3",
      "status": "author_self_checked"
    },
    {
      "id": "the-manifold-and-its-fundamental-group",
      "title": "The manifold and its fundamental group",
      "reader": "the-manifold-and-its-fundamental-group.html",
      "source": "src/the-manifold-and-its-fundamental-group.md",
      "source_sha256": "e7605b70f743ddd00a576abf794456f54b405972031344c901fd384de24dff4f",
      "status": "author_self_checked"
    },
    {
      "id": "spaces-with-the-cn-inequality",
      "title": "Spaces with the CN inequality",
      "reader": "spaces-with-the-cn-inequality.html",
      "source": "src/spaces-with-the-cn-inequality.md",
      "source_sha256": "18f52b7af70a5cf99d27accec1da8fd44ca6bca2e340d9753a406b3dc1e2c29f",
      "status": "author_self_checked"
    },
    {
      "id": "axes-shifts-and-the-two-piece-obstruction",
      "title": "Axes, shifts and the two-piece obstruction",
      "reader": "axes-shifts-and-the-two-piece-obstruction.html",
      "source": "src/axes-shifts-and-the-two-piece-obstruction.md",
      "source_sha256": "67773414c7a2d0dce00fb8421cb152e650d8e21c27b27213d14ac5081167bf37",
      "status": "author_self_checked"
    },
    {
      "id": "a-concave-cometric-and-convex-collars",
      "title": "A concave cometric and convex collars",
      "reader": "a-concave-cometric-and-convex-collars.html",
      "source": "src/a-concave-cometric-and-convex-collars.md",
      "source_sha256": "4edbb1d896a95dfa96daf3912d720dcae51150ae123cbc34a6d992d31205a4fa",
      "status": "author_self_checked"
    },
    {
      "id": "neck-passages-and-the-index-form",
      "title": "Neck passages and the index form",
      "reader": "neck-passages-and-the-index-form.html",
      "source": "src/neck-passages-and-the-index-form.md",
      "source_sha256": "c40320ad9d3cb0a70d6b7b39a66d214e8db034c9b2a689b1f9f56388b6e2bc9d",
      "status": "author_self_checked"
    },
    {
      "id": "returning-passages-and-the-main-theorem",
      "title": "Returning passages and the main theorem",
      "reader": "returning-passages-and-the-main-theorem.html",
      "source": "src/returning-passages-and-the-main-theorem.md",
      "source_sha256": "1dc26e6b1ef693bf3534f466c2e97eb7314fc196880e128b1266be5f006341db",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course DG-FND, lesson Jacobi fields, conjugate points and the Morse index theorem: the Jacobi equation (Theorem A.1), the second variation of energy (Theorem A.4), nonpositive curvature has no conjugate points (Corollary F.3), the Morse index theorem (Theorem D.1, exercises only)",
    "course DG-FND, lesson Comparison theorems, cut loci and curvature and topology: the Cartan-Hadamard theorem (Theorem C.1) and the strong convexity of squared distance with the inequality (C.4) (Theorem C.2)",
    "course DG-FND, lesson Completeness and the Hopf-Rinow theorem",
    "course CG-S6, lesson Integral monodromy and the fundamental group: Lemma 3.1 (van Kampen for two open sets)",
    "core course Algebraic Topology (D60): Roberts's notes, Theorem 6 (the fundamental group of the circle) and Theorem 9 (classification of covering spaces)",
    "the named hypothesis (MCH) of the fourth lesson, used only for Corollary 4.3 (locally CAT(0) metrics), and Theorem 7.1 of Ivanov and Kapovitch, used only for the remark on focal points; the main theorem uses neither"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}