Vanishing Hachtroudi Curvature and Local Equivalence to the Heisenberg Pseudosphere

Abstract

To any completely integrable second-order system of real or complex partial differential equations:

$$\begin{aligned} y_{x^{k_1}x^{k_2}} = F_{k_1,k_2} \left( x^1,\dots ,x^n,\,y,\,y_{x^1},\ldots ,y_{x^n} \right) \end{aligned}$$

with $1 \leqslant k_1,\, k_2 \leqslant n$ and with $F_{ k_1, k_2} = F_{ k_2, k_1}$ in $\underline{n \geqslant 2}$ independent variables $(x^1, \ldots , x^n)$ and in one dependent variable y, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials to the associated submanifold of solutions, what the vanishing of the Hachtroudi curvature gives can be precisely translated to characterize when both families of Segre varieties and of conjugate Segre varieties associated to a Levi nondegenerate real analytic hypersurface M in $\mathbb {C}^n$ ($n \geqslant 3$) can be straightened to be affine complex (conjugate) lines. In continuation to a previous paper devoted to the quite distinct $\mathbb {C}^2$-case, this then characterizes in an effective way those hypersurfaces of $\mathbb {C}^{n+1}$ in higher complex dimension $n + 1\geqslant 3$ that are locally biholomorphic to a piece of the $(2n + 1)$-dimensional Heisenberg quadric, without any special assumption on their defining equations.