Rigidity theorems for cone structures

Abstract: Cone structures on differentiable manifolds are fields of cones in the tangent bundle. A cone structure is isotrivial, modeled on a fixed immersed submanifold of the projective space, if at each point the projectivised cone is projectively equivalent to that submanifold. We consider cone structures arising from ordinary differential equations via a canonical construction. We prove that isotrivial cone structures in this class, modeled on generic curves in the $n$-dimensional projective space or ruled surfaces in the three-dimensional projective space, are flat. We also discuss applications to causal geometries in four dimensions and dispersionless Lax systems.

• The paper proves that generic isotrivial cone structures of equation type, modeled on curves in projective space or ruled surfaces in P³, are flat and arise from linear ODEs.
• The authors convert equation-type conditions into overdetermined systems for frame structure functions, then use model-jet genericity and explicit polynomial witnesses to eliminate all curvature except conformal gauge freedom.
• The results constrain nontrivial examples in anti-self-dual causal geometry and dispersionless Lax theory, while identifying rational normal curves, Cayley structures, and special homogeneous surfaces as important exceptional cases.

Overview and main result

A cone structure on a smooth $n$-manifold $M$ is a smooth field of immersed submanifolds $C_x \subset P(T_xM)$, equivalently a field of cones in $TM$. Such a structure is isotrivial if all fibers $C_x$ are projectively equivalent to a fixed model submanifold $C_0 \subset P^{n-1}$. The paper by Frelik and Kryński establishes a rigidity theorem for isotrivial cone structures that arise from ordinary differential equations via a canonical construction. The main theorem states:

There exist open dense subsets $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$ (curves in projective space, for $n > 3$) and $S_2^3 \subset \mathfrak{R}^3_2$ (ruled surfaces in $P^3$) such that any isotrivial cone structure modeled on a model from these subsets, arising respectively from a scalar ODE or from a system of two second-order ODEs, is flat, and the corresponding equation or system is linear.

Here $M$0 denotes the space of $M$1-dimensional immersed submanifolds of $M$2 with the Whitney $M$3-topology, and $M$4 the ruled surfaces. Flatness means local diffeomorphism to a translated fixed cone in linear space; the theorem holds in both real and complex categories.

The proof strategy is uniform across both cases: isotriviality provides an adapted frame $M$5 on $M$6, unique up to conformal rescaling, whose structure functions $M$7 encode all curvature of the structure. The equation-type condition translates into a linear system for the $M$8 whose coefficients depend only on the model $M$9 and the spectral parameter $C_x \subset P(T_xM)$0. Since each equation can be differentiated arbitrarily many times in $C_x \subset P(T_xM)$1, one obtains an overdetermined system evaluated at a fixed $C_x \subset P(T_xM)$2. Genericity of $C_x \subset P(T_xM)$3 forces this system to have only the trivial solution modulo conformal transformations, which is equivalent to flatness. The generic subsets are open dense because non-trivial solvability is a closed polynomial condition on jets of $C_x \subset P(T_xM)$4, and explicit example curves/surfaces witnessing triviality are constructed.

Cone structures of equation type

The class of structures under study is defined through the pair $C_x \subset P(T_xM)$5 of tautological distributions on $C_x \subset P(T_xM)$6: $C_x \subset P(T_xM)$7 is the rank-one vertical distribution along the ruling parameter, and $C_x \subset P(T_xM)$8 has rank $C_x \subset P(T_xM)$9. A structure is non-degenerate if the derived flag $TM$0 grows as $TM$1, forcing $TM$2. It is of equation type if $TM$3 is locally equivalent to the Cartan distribution on a jet space $TM$4, so that $TM$5 is identified with a total derivative vector field encoding a system $TM$6, uniquely up to contact transformations.

Two characterizations anchor the analysis. For curve-type structures ($TM$7, $TM$8), equation type is equivalent to $TM$9 for all $C_x$0 — a Goursat-type condition on the filtration. Notably, for $C_x$1 this condition is void (the Engel distribution is generic), which is why the theorem requires $C_x$2. For ruled-surface structures ($C_x$3, $C_x$4), equation type is equivalent to the existence of an integrable rank-two subdistribution $C_x$5 with $C_x$6; choosing such a $C_x$7 realizes $C_x$8 as a three-dimensional path geometry.

Rigidity for curves in $C_x$9

For a scalar ODE ($C_0 \subset P^{n-1}$0), the model curve admits a parametrization $C_0 \subset P^{n-1}$1, and isotriviality yields frames $C_0 \subset P^{n-1}$2 with $C_0 \subset P^{n-1}$3 parametrized by $C_0 \subset P^{n-1}$4. Assuming $C_0 \subset P^{n-1}$5 has no symmetry, the frames are unique up to conformal rescaling, under which the structure functions transform as $C_0 \subset P^{n-1}$6 — a transformation invariance that extends to arbitrary "gauge" terms $C_0 \subset P^{n-1}$7 without changing the constraint equations.

The equation-type condition becomes the linear system

$C_0 \subset P^{n-1}$8

where $C_0 \subset P^{n-1}$9 is the inverse of the Wronskian matrix $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$0, depending only on $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$1. The key observation is that $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$2-differentiation produces arbitrarily many independent equations with constant coefficients after evaluation at $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$3, with coefficients depending on arbitrarily high jets of $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$4.

Genericity is witnessed by an explicit polynomial curve built from exponents $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$5 (all exceeding $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$6) chosen so that relevant degree combinations are pairwise distinct. A degree count shows that coefficients of distinct unknowns $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$7 appear at distinct polynomial orders, immediately killing all $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$8 with $S_1^{n-1} \subset \mathfrak{V}^{n-1}_1$9; the remaining $n > 3$0 are then pinned down uniquely up to the two-parameter conformal freedom by one additional equation contributed by the leading $n > 3$1 term. The conclusion is that the frame is conformally flat, hence the structure is flat and the generating ODE is linear.

A remark gives a concrete algebraic characterization of the exceptional set in dimension four: existence of a nontrivial solution forces a quintic relation among the $n > 3$2 and their derivatives up to order $n > 3$3, interpretable as containment of the osculating (tangential) variety of $n > 3$4 in a degree-five hypersurface — a highly restrictive condition. The authors concede that a complete characterization of non-generic curves remains open.

By contrast, the $n > 3$5-structure case (rational normal curves) is genuinely exceptional: there the system is underdetermined — in dimension four, 8 equations for 24 unknown functions — and its solvability is equivalent to vanishing of Bryant's torsion, consistent with the known abundance of non-flat examples.

Rigidity for ruled surfaces in $n > 3$6

For systems of two second-order ODEs ($n > 3$7, $n > 3$8), the model is a ruled surface given by a $n > 3$9-valued function $S_2^3 \subset \mathfrak{R}^3_2$0 in an affine Grassmannian chart. The integrability condition of Proposition 2 is implemented by seeking vector fields

$S_2^3 \subset \mathfrak{R}^3_2$1

spanning an integrable $S_2^3 \subset \mathfrak{R}^3_2$2. The condition $S_2^3 \subset \mathfrak{R}^3_2$3 determines $S_2^3 \subset \mathfrak{R}^3_2$4 explicitly by linear algebra (using Montgomery's lemma that every generic rank-three distribution contains a unique rank-two subdistribution), leaving a single scalar PDE from the vertical component:

$S_2^3 \subset \mathfrak{R}^3_2$5

Crucially, this equation separates $S_2^3 \subset \mathfrak{R}^3_2$6-dependent from $S_2^3 \subset \mathfrak{R}^3_2$7-dependent terms, is linear in the derivatives $S_2^3 \subset \mathfrak{R}^3_2$8, quadratic in the $S_2^3 \subset \mathfrak{R}^3_2$9, and contains no linear terms. Prolongation in $P^3$0 therefore yields an overdetermined linear system for the vector of quadratics $P^3$1 and first derivatives $P^3$2, with coefficients depending on high jets of the surface.

The witness surface is constructed from a $P^3$3-valued polynomial with widely separated exponent blocks $P^3$4 ensuring pairwise distinct exponent combinations. The authors report using Wolfram Mathematica code to analyze the resulting coefficient equations: twelve distinguished quadratic terms $P^3$5 occupy unique degrees and vanish outright; cross-terms $P^3$6 with $P^3$7 are likewise eliminated. The surviving relations reduce, together with the Jacobi identities, to derivative equations of the form $P^3$8 and quadratic constraints forcing $P^3$9. The latter imply that all $M$00 equal $M$01 for a single function $M$02 — verified via the Poincaré lemma applied to closed 1-forms built from the dual coframe — so the only freedom is conformal, completing the proof of flatness.

Non-flat exceptions do exist: Cayley structures, and structures modeled on certain homogeneous surfaces in $M$03 studied recently by Kryński. The authors state they expect the main theorem to hold for general $M$04 and $M$05, noting the proof strategy appears to extend but that the required witness construction has not been checked in general.

Intrinsic torsion and characteristic connections

The paper recasts the equation-type condition in the language of essential torsion. The filtration $M$06 reduces the frame bundle of $M$07 to a principal $M$08-bundle with $M$09 upper triangular. The essential torsion — the torsion of any $M$10-connection modulo the image of the Spencer operator $M$11 — has components $M$12 with $M$13, and Proposition 3 asserts that equation type is equivalent to the only nonzero components satisfying $M$14. For the surface case, the structure group is a block-upper-triangular subgroup of $M$15, the essential torsion is seven-dimensional, and equation type amounts to vanishing of $M$16 for $M$17 plus invertibility of the $M$18 cell $M$19.

In the final section, the characteristic connection of Hwang is recovered a posteriori. Working on the six-dimensional $M$20 with distributions $M$21 and the tautological $M$22, a line subbundle $M$23 with $M$24 and $M$25 is a characteristic connection, whose existence implies flatness. Under the theorem's hypotheses, the Lax fields $M$26 lose their vertical components and the frame commutes, so $M$27 spans such an $M$28: the characteristic connection is read off directly from the Lax pair.

Applications

Three application domains motivate the result. First, in causal geometry, four-dimensional cone structures generalize conformal metrics; in split signature, the ODE-type condition is equivalent to integrability of the $M$29-planes, i.e., to anti-self-duality of the metric. The theorem thus says that for a generic cone, every isotrivial ASD causal spacetime modeled on it is flat — sharply restricting potential nontrivial examples beyond the metric and Cayley cases. Second, in the theory of dispersionless Lax pairs $M$30 on $M$31, commutativity $M$32 induces an integrable cone structure on $M$33; for $M$34 with suitable non-degeneracy this structure is of equation type. Consequently, for a generic model surface the associated Lax system admits only trivial solutions when $M$35 solves it — supporting the claim that meaningful dispersionless integrable systems depending non-polynomially on the spectral parameter are rare. Third, the result parallels rigidity theorems for isotrivial VMRT structures of complete intersection type due to Fu–Hwang, connecting the analysis to the algebraic geometry of varieties of minimal rational tangents.

Limitations and open questions

Several qualifications are stated explicitly. The generic sets are characterized existentially: the proofs exhibit witness models but do not give a complete intrinsic description of the complement, except partially in dimension four via the quintic osculating-cone condition. The extension of Theorem 1 to general $M$36 and $M$37 is conjectural — the method appears to generalize, but the witness construction has not been carried out. The genericity argument relies on the models having no symmetry, and the overdetermined linear systems must be shown to have full rank for the specific witnesses, a computation performed partly by computer algebra in the surface case. Whether the rigidity dichotomy can be sharpened to classify all non-generic models (beyond rational normal curves, Cayley structures, and the homogeneous surfaces of Kryński) remains open.

Conclusion

The paper proves that isotrivial cone structures of equation type are rigid for generic models: generically, such a structure arising from ODEs is flat and the underlying differential equation is linear. The mechanism — separation of $M$38- and $M$39-dependence producing an overdetermined jet system killed by genericity of the model — applies uniformly to curves in $M$40 and ruled surfaces in $M$41, and is reformulated intrinsically via essential torsion and connected to Hwang's characteristic connections. The consequences constrain isotrivial ASD causal geometries and dispersionless Lax systems alike, indicating that nontrivial examples require special, non-generic model cones.