Result 358, Differential geometry

A three-manifold without conjugate points or nonpositive curvature

Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Can a space avoid the refocusing of nearby straightest paths without allowing an everywhere nonpositively curved geometry? This manuscript reports a three-dimensional example that separates these two ways of constraining a space's geometry.

What changes?

The construction is a closed, connected, orientable smooth three-manifold: a compact three-dimensional space without boundary, with a consistent orientation. It admits a smooth Riemannian metric, a rule for measuring lengths and angles, without conjugate points. This means no infinitesimal refocusing along geodesics, the geometry's straightest paths. Yet it admits no smooth Riemannian metric of nonpositive sectional curvature, meaning curvature is zero or negative in every tangent two-dimensional plane. The claim excludes every such metric, not just the constructed one.

What does that help mathematicians do?

The reported example shows that the existence of a metric without conjugate points does not force the existence of a nonpositively curved metric, already in dimension three. Researchers therefore cannot generally replace the first assumption with the second, even by changing the metric on the same manifold. This distinguishes a condition about geodesic behavior from a stronger requirement on curvature, at the level of which spaces admit them.

Are there practical applications?

Its immediate value is foundational: it helps delimit which geometric assumptions can legitimately support a theorem about three-manifolds. In particular, a proposed argument that passes from absence of conjugate points to existence of nonpositive curvature cannot work in full generality. The supplied material describes a mathematical counterexample, not a practical technology or computational method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric

September 24, 2026 22 pages Main result formalized in Lean

We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026,
  author = {{OpenAI}},
  title = {{A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf}{OAI:A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/358.md.

A three-manifold without conjugate points or nonpositive curvature

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric

Scope

The formalized result separates absence of conjugate points from nonpositive sectional curvature. It constructs a closed connected orientable smooth three-manifold with a smooth Riemannian metric having no conjugate points, while the same manifold admits no smooth metric of everywhere nonpositive sectional curvature. The no-conjugate-points assertion ranges over all geodesics and tangent-bundle Jacobi fields, including constant geodesics and unbounded time intervals. The separate no-focal-points and CAT(0) consequences are not included.

Comparator links

Result	Comparator statement
No conjugate points without a nonpositively curved metric	ConjugatePoints.lean