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H-principle for corank two distributions of odd rank and maximal first Kronecker index — arXiv:2609.38214

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.38214
License (as archived): CC0-1.0
Archived: 2026-10-07T14:11:32.107Z
SHA-256 of the archived text: c49ab785d77c87e0c1ccebaecd97dea3576d328b142d5e9ffef7b3f317c2d862
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H-principle for corank two distributions of odd rank and maximal first Kronecker index — arXiv:2609.38214 — arXiv (en), https://arxiv.org/abs/2609.38214 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_55f83000b442 (Metraipolis archive dc_55f83000b442, sha256 c49ab785d77c87e0c1ccebaecd97dea3576d328b142d5e9ffef7b3f317c2d862)

BibTeX

@misc{dc55f83000b442,
  title = {H-principle for corank two distributions of odd rank and maximal first Kronecker index — arXiv:2609.38214},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.38214},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_55f83000b442; sha256 c49ab785d77c87e0c1ccebaecd97dea3576d328b142d5e9ffef7b3f317c2d862},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.38214. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study corank-two distributions $D$ of odd rank $2N+1$ whose associated pencil of skew-symmetric forms $\{d\alpha|_{D}(q)\mid \alpha\in\Omega^1(M),\ \alpha|_{D}=0\}$ lies, at every point, in the generic orbit of the natural $\mathrm{GL}\bigl(D(q)\bigr)$-action; equivalently, $D$ has maximal first Kronecker index. In corank two, this class is the analogue of contact and even-contact distributions. We prove that the corresponding differential relation is ample and hence satisfies the multi-parametric $C^0$-close $h$-principle, and we deduce a complete topological characterization for a $(2N+3)$-dimensional manifold to admit such a distribution. The criterion is satisfied by non-parallelizable manifolds as well. To the best of our knowledge, ampleness has been verified so far for two classes of distributions defined by an open $\mathrm{Diff}$-invariant relation: even-contact distributions (McDuff, 1987) and distributions of rank greater than two with maximal small growth vector, in arbitrary ambient dimension (Martínez-Aguinaga, 2026); for $N\ge2$, our class is a proper open subclass of the latter, singled out by a further condition on the pencil. The verification of ampleness reduces, via Kronecker--Weierstrass normal forms of the restrictions of the pencil to hyperplanes, to computing the convex hulls of the connected components of the complement of the resultant hypersurface in the space of pairs of real binary forms of fixed degrees, components separated by a winding number, i.e., by the classical Cauchy index. This is in contrast with the even-contact case of McDuff, where the set to be removed is thin (i.e., of codimension at least two), so that its complement is automatically connected and ampleness is immediate.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-10-01 in math.OC).
Categorie arXiv: Differential Geometry (math.DG) ; Optimization and Control (math.OC); Symplectic Geometry (math.SG).
Pagina del lavoro: https://arxiv.org/abs/2609.38214 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T14:03:18.275Z dalla pagina https://arxiv.org/catchup/math.OC/2026-10-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).