Results on index forms, CN spaces, axes and the graph manifold

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The index-form criterion — Lemma 2.2. On a complete Riemannian manifold, if the index form of every compactly supported field along every complete unit-speed geodesic is nonnegative, there are no conjugate points.
• Assembly over visits — Proposition 2.3. If the sectional curvature is nonpositive outside an open set U and the index form of every compactly supported field is nonnegative on every visit of every geodesic to U, the metric has no conjugate points.
• A three-manifold without conjugate points and without nonpositive curvature — Theorem 3.1. There is a closed connected orientable smooth three-manifold with a metric without conjugate points that admits no metric of nonpositive sectional curvature.
• Normal forms in amalgamated products — Theorem 3.1. Every element of G_1 *_A G_2 is uniquely a s_1 ... s_n with a in A and alternating coset representatives; the factors embed and reduced alternating products are not in A.
• The fundamental group of the two-piece graph manifold — Proposition 4.1. pi_1(M) is an amalgam over Z^2 in which f_2 = h_1, h_2 = h_1 f_1, each h_i is a commutator in the centralizer of f_i, and h_1, f_1 generate Z^2.
• Hadamard manifolds satisfy the CN inequality — Proposition 1.4. A Hadamard manifold is a proper geodesic space with unique geodesic segments satisfying d(x,c_t)^2 <= (1-t)d(x,y)^2 + t d(x,z)^2 - t(1-t)d(y,z)^2.
• Nearest-point projections in CN spaces — Proposition 3.1. Projections onto complete convex subsets exist, are unique, satisfy the obtuse-angle inequality, do not increase distances and commute with isometries preserving the set.
• The Pythagorean identity for parallel lines — Proposition 4.1. Parallel geodesic lines in a CN space satisfy d(c(s),c'(u))^2 = D^2 + (s - u + sigma)^2 for constants D and sigma.
• Shifts between parallel lines add — Proposition 4.3. For pairwise parallel lines, sigma(c_1,c_3) = sigma(c_1,c_2) + sigma(c_2,c_3).
• Axes of hyperbolic isometries — Proposition 1.2. Every point of the minimal set of a hyperbolic isometry of a CN space lies on an axis, and l(g^n) = |n| l(g).
• The length formula — Proposition 2.4. If h commutes with a hyperbolic g, is semisimple and has shift zero, then l(g^m h^n)^2 = m^2 l(g)^2 + n^2 l(h)^2.
• Semisimplicity under proper cocompact actions — Proposition 3.1. In a proper cocompact action on a proper CN space every element is semisimple and every element of infinite order is hyperbolic.
• The two-piece obstruction — Theorem 4.1. The fundamental group of the two-piece graph manifold has no proper cocompact isometric action on a proper CAT(0) space.
• No metric of nonpositive curvature — Corollary 4.2. The two-piece graph manifold admits no Riemannian metric of nonpositive sectional curvature.
• A concave cometric — Lemma 2.1. For small delta the matrix path P is bounded, concave, and satisfies |P'p|^2 <= C delta p^T(-P'')p.
• The metric on the graph manifold — Proposition 4.1. The neck metric dt^2 + dx^T P^-1 dx, convex warped collars and flat cores form a smooth metric with nonpositive curvature outside the open neck.
• Endpoint correction — Lemma 3.1. The intrinsic index form equals the coordinate energy minus the endpoint term <T, Gamma(V,V)>, which is nonnegative at entries and exits when the field is tangent to the boundary.
• Passage classification — Lemma 4.1. A visit of a geodesic to the neck is the whole line at constant height, a monotone passage, or a returning passage with one turn.
• The returning estimate — Proposition 4.1. For small delta every returning passage has nonnegative normal index form, uniformly in all its parameters.
• The main theorem — Theorem 5.2. The constructed metric on the two-piece graph manifold has no conjugate points, and the manifold admits no metric of nonpositive curvature.

Reading order

Seven lessons, to be read in order; the third and fourth (CN spaces and isometries) and the fifth to seventh (the metric and the index form) can be read independently after the first two. The main theorem is proved in full in the programme. Corollary 4.3 of the fourth lesson (no locally CAT(0) metric) uses the named hypothesis (MCH), and the remark on focal points uses a theorem of Ivanov and Kapovitch.