A compact normalized-null-cone obstruction

Definitions and mathematical statement

Let $M$ be a smooth closed manifold of dimension $n\ge2$, and let $g$ be a smooth semi-Riemannian metric of indefinite signature $(p,q)$ with $p,q\ge1$. A null geodesic is a nonconstant affinely parametrized geodesic $\gamma$ with $g(\dot\gamma,\dot\gamma)=0$. Null completeness means that every such geodesic is defined for all $t\in{\mathbb{R}}$; geodesic completeness, assumed in the original question, is stronger.

Literal Question 9.2.1. Can such a complete $(M,g)$ admit a nonzero smooth one-form $\eta$ for which \[ \forall\gamma\ \exists a_\gamma\in{\mathbb{R}}\setminus\{0\}\ \exists b_\gamma\in{\mathbb{R}}\ \forall t\in{\mathbb{R}}, \qquad \eta_{\gamma(t)}(\dot\gamma(t))=a_\gamma t+b_\gamma, \] where $\gamma$ ranges over all nonconstant affinely parametrized null geodesics? The constants may depend on the geodesic. "Grows linearly in both directions" is understood by this displayed affine formula, not as positivity in both time directions. The source-verification field identifies the OCR string $C_1\,6=0$ as $C_1\ne0$.

Supplied negative-answer claim. The aid asserts that this property is impossible already under null completeness and for $\eta$ of class $C^1$. To state its compactness reduction, choose any auxiliary Riemannian metric $h$ and put \[ \mathcal N_h=\{(x,v)\in TM:g_x(v,v)=0,\ |v|_h=1\}, \qquad A_\eta(x,v)=(\nabla_v\eta)(v). \] The claimed theorem says that, if evaluation of $\eta$ is affine on every complete null geodesic, then $A_\eta$ vanishes at some point of $\mathcal N_h$. Equivalently, at least one such geodesic has zero slope. The aid describes the obstruction using compactness of $\mathcal N_h$, a uniform positive lower bound for $|A_\eta|$ if it has no zeros, and conservation of $A_\eta(\gamma,\dot\gamma)$ along each affine-evaluation orbit. This is a supplied candidate theorem, not a verified literature citation.

Projective-rigidity context and quantifier limitation. The source states the semi-Riemannian projective Lichnerowicz--Obata conjecture: if a connected Lie group acts on a complete $(M^n,g)$, $n\ge2$, by diffeomorphisms preserving unparametrized geodesics, then its action is isometric or $g$ has constant sectional curvature. The source reports the Riemannian case in [Matveev2005, Matveev2007] and the closed two-dimensional case in [Matveev2012b], while noting difficulties in indefinite dimension at least three.

For a projective vector field, identified with its metric-dual one-form $\eta$, the source quotes a second-order projective-field equation involving an auxiliary one-form $\lambda$. Its null contraction is asserted to imply \[ \frac{{\,\mathrm{d}}^2}{{\,\mathrm{d}} t^2}\eta(\dot\gamma(t))=0. \] The printed extraction of the full indexed equation is damaged; its precise derivative and trace conventions are not reconstructed here. The subsequent conformal-field argument requires zero slope on every null geodesic, which would give \[ \nabla_i\eta_j+\nabla_j\eta_i=\rho g_{ij} \] for some function $\rho$. The negative answer to the literal existence question supplies only some zero-slope null direction for a given affine-evaluation form. Therefore the source's implication to full projective rigidity does not follow from that negative answer alone. Its text also says "positive" at one point where its surrounding discussion says "negative"; both inconsistencies are retained as qualifications. The final step invokes Weyl's assertion that a transformation both projective and conformal is a homothety; on a closed manifold a global positive homothety factor is one by volume preservation.

The source additionally asks whether the velocity of almost every geodesic stays in a bounded subset of $TM$ for complete semi-Riemannian metrics on closed manifolds. On compact $M$, boundedness can be specified as $\sup_{t\in{\mathbb{R}}}|\dot\gamma(t)|_h<\infty$. A measure on initial conditions is not specified, and an ambient almost-everywhere assertion cannot simply be substituted for a null-cone assertion. The source presents this as another possible route to its conjecture, not as an established theorem.

The extract ends with the start of Section 9.3, communicated by H. Baum: closed Riemannian manifolds are geodesically complete, whereas closed manifolds with indefinite metrics can be incomplete. It supplies no further complete Section 9.3 problem.

Short English statement

The literal question demands a nonzero affine slope on every complete null geodesic. The supplied aid claims that compactness rules this out even with only null completeness. This gives a zero-slope direction, which is insufficient for the source's proposed deduction that all projective vector fields are conformal.

Sources

• Local UnsolvedMath registry, permanent ID 20001925, source code catalogue entry; canonical aim-geometry-notes.json, notes[262].
• Keith Burns and Vladimir Matveev, Open problems and questions about geodesics, Question 9.2.1, AIM source https://aimath.org/pastworkshops/geodesicsproblems.pdf. The record reports verification at printed p. 22, PDF page 21, including $C_1\ne0$; this is source-reported verification.
• Supplied labels [Matveev2005], [Matveev2007], [Matveev2012b], and [Weyl1921]; no full bibliographic entries are supplied. Section 9.3 is attributed to H. Baum.
• Research attempt 1, registry update 2026-08-14: full solution, problem status at run "refuted", rigor "proved", novelty confidence low. The aid is explicitly machine-generated, not peer reviewed, and must be independently verified before citation.