Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

中文翻译：

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无色散可积层次结构和 GL(2, ℝ) 几何

准保形或总帐(2, ℝ) 几何n维流形米由度数的有理正态曲线域定义n– 投影余切丛中的 1 ℙ吨*米. 已知这种几何出现在具有消失的 Wünschmann (Doubrov-Wilczynski) 不变量的 ODE 的解空间上。在本文中，我们讨论了另一种天然来源总帐(2, ℝ) 结构，即 PDE 的无色散可积层次结构，例如无色散 Kadomtsev-Petviashvili (dKP) 层次结构。在后一种情况下，总帐(2, ℝ) 结构与层次结构的特征多样性（主要符号）相吻合。无色散层次结构提供了特别有趣的对合类的明确示例总帐(2, ℝ) 文献中研究的结构。因此，我们获得无扭转总帐(2, ℝ) Bryant [5] 的结构，出现在四维奇异全息的背景下，以及完全测地线总帐(2, ℝ) Krynski [33] 的结构。后者具有兼容的仿射连接（带扭转）和完全测地线的双参数族α-流形（来自无色散 Lax 方程），这使它们成为 Einstein-Weyl 几何的自然概括。我们的主要结果表明，内合总帐(2, ℝ) 结构由一个无色散可积系统控制，该系统的一般局部解取决于 2n– 3 个变量的 4 个任意函数。这建立了 Wünschmann 条件系统的可积性。