余秩二奇数秩分布及其最大第一Kronecker指标的h-原理
H-principle for corank two distributions of odd rank and maximal first Kronecker index
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中文总结 AI 辅助
本文研究余秩二奇数秩分布，证明其满足多参数C0-接近h-原理，并给出流形允许此类分布的完全拓扑刻画，推广了接触与偶接触情形。
中文摘要 AI 辅助
我们研究余秩为二的分布$D$，其秩为奇数$2N+1$，且其关联的斜对称形式束$\{d\alpha|_{D}(q)\mid \alpha\in\Omega^1(M),\\ \alpha|_{D}=0\}$在每一点都位于自然$\mathrm{GL}\bigl(D(q)\bigr)$-作用的通用轨道中；等价地，$D$具有最大第一Kronecker指标。在余秩二的情形下，此类分布是接触分布和偶接触分布的类比。我们证明相应的微分关系是充足的（ample），因此满足多参数$C^0$-接近的$h$-原理，并由此推导出一个$(2N+3)$维流形允许此类分布的完全拓扑刻画。该判据也适用于不可平行化的流形。据我们所知，目前仅对两类由开$\mathrm{Diff}$-不变关系定义的分布验证了充足性：偶接触分布（McDuff, 1987）以及任意环境维数下秩大于二且具有最大小增长向量的分布（Martínez-Aguinaga, 2026）；对于$N\ge2$，我们的类是其真开子类，通过束上的进一步条件区分。充足性的验证通过Kronecker--Weierstrass标准形（束在超平面上的限制）归结为计算固定次数的实二元形式对空间中结式超曲面补集的连通分量的凸包，这些分量由卷绕数（即经典Cauchy指标）分隔。这与McDuff的偶接触情形形成对比，后者中需移除的集合是薄的（即余维数至少为二），因此其补集自动连通，充足性立即成立。
英文摘要
We study corank-two distributions $D$ of odd rank $2N+1$ whose associated pencil of skew-symmetric forms $\{dα|_{D}(q)\mid α\inΩ^1(M),\ α|_{D}=0\}$ 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.
发表机构
- Texas A&M University（德克萨斯农工大学）
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