Geodesics without focusing on a manifold that forbids nonpositive curvature

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Family 358 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

On a negatively curved surface, nearby geodesics tend to spread apart. On a sphere, geodesics leaving the north pole meet again at the south pole. The second behavior has an infinitesimal version called the appearance of conjugate points.

Nonpositive sectional curvature prevents conjugate points. But does the converse hold at the level of the underlying manifold? If a closed manifold has some metric without conjugate points, must it also have some possibly different metric of nonpositive sectional curvature?

These are two existence questions about the same topological space. Producing a metric with a small region of positive curvature does not answer them, because that manifold might still admit another negatively curved metric.

Family 358 claims a negative answer in dimension three. It constructs a closed orientable three-manifold with a smooth metric without conjugate points, while its fundamental group obstructs every nonpositively curved metric. The geometry of its geodesics is therefore less restrictive than the existence of a nonpositive-curvature geometry on the same manifold.

2. Background and essential definitions

Conjugate points measure infinitesimal refocusing

A Riemannian metric assigns inner products to tangent spaces and thereby defines lengths and geodesics. Along a geodesic $\gamma(s)$, an infinitesimal variation through neighboring geodesics is described by a Jacobi field $J$, satisfying

$$D_s^2J+R(J,\dot\gamma)\dot\gamma=0.$$

Here $D_s$ differentiates vector fields along the path while respecting the metric, and $R$ is the curvature tensor. The equation is a geometric version of a linear second-order oscillator equation.

Two points on the geodesic are conjugate when a nonzero Jacobi field vanishes at both times. Intuitively, a family of initially diverging geodesics has refocused to first order. “No conjugate points” excludes this on every geodesic.

For a normal field, meaning $J\perp\dot\gamma$, nonpositive sectional curvature gives

$$\frac12\frac{d^2}{ds^2}|J|^2 =|D_sJ|^2-\langle R(J,\dot\gamma)\dot\gamma,J\rangle\ge0.$$

This convexity explains the implication from nonpositive curvature. Without a pointwise curvature sign, however, one may still control the total effect along whole geodesic segments.

The stronger condition no focal points requires a nonzero normal Jacobi field with $J(0)=0$ to satisfy

$$\frac{d}{ds}|J(s)|^2>0\qquad(s>0).$$

This demands persistent growth, not merely the absence of a second zero.

The manifold is assembled from two familiar pieces

Let $\Sigma$ be a torus with the interior of one disk removed. It is a compact surface with one boundary circle. Each product

$$N_i=\Sigma\times S^1$$

is a three-dimensional manifold whose boundary is a torus.

On that boundary, let $h_i$ denote the loop coming from $\partial\Sigma$, and $f_i$ the loop in the circle factor. These generate the torus’s fundamental group $\mathbb Z^2$. The construction glues the two boundary tori using

$$h_2=h_1+f_1,\qquad f_2=h_1.$$

The additive notation refers to loops in this abelian torus group. The gluing matrix has determinant $-1$, reversing boundary orientation so that the resulting closed manifold is orientable. Such a manifold assembled from circle-fibered pieces is called a graph manifold; “graph” refers to its decomposition, not to the graph of a function.

3. Development history and previous partial results

Gulliver’s 1975 work, On the variety of manifolds without conjugate points, showed that a metric without conjugate points can contain regions of positive curvature. Those examples were obtained by modifying negatively curved metrics. They separated two properties of a particular metric, but the underlying manifolds still possessed negatively curved metrics.

The stronger existence question appears in Ivanov and Kapovitch and in Burns and Matveev’s survey of questions about geodesics. Ivanov and Kapovitch specifically identified three-dimensional graph manifolds, including gluings of punctured-torus products, as important test cases.

For closed surfaces the existence question has a positive answer. A metric without conjugate points forces the relevant universal-cover and topological restrictions, giving nonpositive Euler characteristic; uniformization then supplies a flat or negatively curved metric. Thus dimension three is the first place where a counterexample of the present kind can occur.

The topology needed for a candidate was already available. Leeb constructed graph-manifold examples obstructing nonpositive curvature, and Kapovich–Leeb studied the corresponding obstruction to group actions on nonpositively curved spaces. Family 358 uses an instance of that known two-piece obstruction. Its main new claim is a metric without conjugate points on that same manifold.

There is also a useful boundary between nearby conditions. Ivanov and Kapovitch proved that a closed three-manifold admits a metric without focal points if and only if it admits a metric of nonpositive sectional curvature. Consequently, a counterexample for conjugate points would necessarily fail the stronger no-focal-points condition for every metric.

4. What the new manuscript claims

The main theorem asserts a closed, connected, orientable smooth three-manifold $M$ with two properties:

• It admits a $C^\infty$ Riemannian metric with no conjugate points.
• It admits no $C^\infty$ Riemannian metric with sectional curvature everywhere at most zero.

The manifold is the explicit two-piece gluing described above. The construction inserts a torus-shaped neck between the pieces; adding this collar does not change the diffeomorphism type.

The topological obstruction is stated in a stronger form. The group $\pi_1(M)$, consisting of homotopy classes of based loops, admits no proper cocompact isometric action on a nonempty proper $\mathrm{CAT}(0)$ space.

A CAT(0) space is a geodesic metric space whose triangles are no thicker than their Euclidean comparison triangles. “Proper” for the space means closed bounded sets are compact. A proper action has only finitely many group elements moving one compact set to meet another, and cocompactness means the quotient is compact.

As a consequence, the manifold admits no locally CAT(0) length metric inducing its topology. A length metric measures distance by infima of path lengths. Thus allowing certain nonsmooth notions of nonpositive curvature does not evade the obstruction.

Combining the main theorem with Ivanov–Kapovitch’s equivalence also excludes every metric without focal points on this manifold. The paper therefore separates no conjugate points from both nonpositive sectional curvature and no focal points at the level of metric existence.

5. Why the advance matters

Importance: 5/5, conditional on the claim. The result would answer a longstanding existence question in the smallest possible dimension and distinguish two broad classes of closed manifolds defined by geodesic behavior.

The central point is the quantifier over metrics. The counterexample would not merely show that one metric has some positive curvature. It would show that no replacement metric can make sectional curvature nonpositive, even though another global metric avoids infinitesimal refocusing everywhere.

The construction also illustrates the different strengths of local and integrated conditions. Sectional curvature restricts each tangent two-plane at each point. Absence of conjugate points constrains how curvature acts along an entire geodesic through the Jacobi equation. Carefully controlled positive contributions can coexist with the latter condition.

The obstruction belongs to topology and group theory; the positive existence statement belongs to differential geometry and variational estimates. Making both hold on the same explicit manifold is the essential advance claimed here.

6. The main idea

A short algebraic conflict rules out nonpositive curvature

Suppose the fundamental group acted properly and cocompactly on a proper CAT(0) space. The flat torus theorem makes the common $\mathbb Z^2$ subgroup act by translations on a Euclidean plane.

In each product piece, the circle fiber commutes with every loop in the surface factor. The surface boundary is a commutator of two such loops. The resulting group relations force the translation vector of the boundary loop to be perpendicular to that of the fiber.

Let $u$ and $v$ be the translation vectors for $h_1$ and $f_1$. The first product piece requires

$$\langle u,v\rangle=0.$$

The gluing makes the second pair of translation vectors $u+v$ and $u$. Its orthogonality requirement gives

$$0=\langle u+v,u\rangle=|u|^2.$$

But a generator of the translation lattice has a nonzero vector. This is the incompatibility behind the known obstruction. The manuscript includes the group-theoretic details to obtain the full CAT(0)-action statement.

Build a neck with controlled inverse metric

Each product core is given nonpositive curvature. Between them, use coordinates $t\in[0,1]$ and $x\in\mathbb T^2$ and a neck metric

$$g=dt^2+dx^{\mathsf T}P(t)^{-1}dx.$$

The positive-definite matrix $P(t)$ is the cometric, the inverse of the metric matrix in torus directions. It is chosen concave in the matrix sense:

$$-P''(t)\ge0,$$

meaning that every vector evaluates this matrix to a nonnegative number.

The path has exact product forms near its ends, and smooth warped collars connect it to the cores. A small parameter controls its derivatives. Appropriate unequal torus periods allow a geometrically small shear to realize the fixed integral gluing map. The topology is not being changed continuously into a trivial gluing; the metric scales make the required shear manageable.

Estimate complete neck passages, including their endpoints

The relevant variational quantity is the index form along a unit-speed geodesic, with tangent $T$:

$$I(V)=\int\bigl(|D_sV|^2-\langle R(V,T)T,V\rangle\bigr)\,ds.$$

It measures the second variation of geodesic energy for a variation field $V$. On the nonpositively curved cores, the integrand is nonnegative. In the neck, the aim is a nonnegative integral over each whole visit, even if some pointwise curvature terms have the wrong sign.

Concavity of the cometric gives a nonnegative coordinate second variation. The intrinsic index form differs from it by endpoint and tangential terms. Those terms cannot simply be discarded, because the variation field generally does not vanish whenever a geodesic enters or leaves the neck.

Passages that cross monotonically, remain at constant height, or stay in simple endpoint collars can be treated separately. The delicate case is a geodesic entering the neck, turning, and returning to the same boundary. Its entire excursion must be controlled with arbitrary field values at both endpoints.

The manuscript uses a positive boundary contribution to control a component of the field’s endpoint change. Cauchy–Schwarz and a quadratic minimization combine that control with the coordinate energy. A strict timing margin for excursions that leave the endpoint collar makes the estimate uniform in entry angle, turning height and duration.

No upper bound on how long the excursion lasts is assumed. This matters because one fixed smooth metric must work for every complete geodesic, including nearly tangent or slowly returning ones.

Assemble local passage estimates into a global conclusion

The same small parameter is chosen for all types of passages. A complete geodesic may enter the neck infinitely many times, but the index integrand of a compactly supported variation field is absolutely integrable. The whole index integral is therefore the sum over its neck visits plus its contribution outside the neck.

Each full visit has a nonnegative intrinsic index form, and the cores contribute nonnegatively. The manuscript then uses the standard index-form argument to exclude any nonzero Jacobi field with two zeros. This establishes absence of conjugate points for the constructed metric, without requiring nonpositive curvature throughout the neck.

7. Scope, limitations, and open questions

The theorem is an existence result for a particular closed orientable three-manifold. It does not classify all graph manifolds admitting metrics without conjugate points, nor all possible metrics on this example.

Absence of conjugate points is not absence of focal points, monotonic divergence of every pair of geodesics, or a pointwise curvature bound. The stronger no-focal-points property is explicitly impossible on this manifold by the cited three-dimensional equivalence.

The result also does not assert that the constructed geodesic flow has stronger dynamical features such as uniform hyperbolicity. Those would require separate statements and estimates.

Natural further problems concern how widely the neck construction can be adapted, which gluings permit similar metrics, and how their geodesic dynamics reflect the topology. The construction and its uniform passage estimates are reported as claimed, without independent proof verification.

8. Sources and further reading

The source separates the known graph-manifold obstruction in Section 2 from its metric construction and global index estimates in Sections 3–6.

For the existence question and the no-focal-points distinction, see Ivanov–Kapovitch. For the underlying obstruction, see Leeb’s three-manifold paper. Gulliver’s On the variety of manifolds without conjugate points, Transactions of the AMS 210 (1975), 185–201, is the classical precursor separating properties of an individual metric.

Source manuscripts

• M1. OpenAI, A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.