Differential $\{e\}$-structures for equivalences of $2$-nondegenerate Levi rank $1$ hypersurfaces $M^5 \subset \mathbb{C}^3$

Abstract

The class ${\sf IV}_2$ of $2$-nondegenerate constant Levi rank $1$ hypersurfaces $M^5 \subset \mathbb{C}^3$ is governed by Pocchiola's two primary invariants $W_0$ and $J_0$. Their vanishing characterizes equivalence of such a hypersurface $M^5$ to the tube $M_{\sf LC}^5$ over the real light cone in $\mathbb{R}^3$. When either $W_0 \not\equiv 0$ or $J_0 \not\equiv 0$, by normalization of certain two group parameters ${\sf c}$ and ${\sf e}$, an invariant coframe can be built on $M^5$, showing that the dimension of the CR automorphism group drops from $10$ to $5$. This paper constructs an explicit $\{e\}$-structure in case $W_0$ and $J_0$ do not necessarily vanish. Furthermore, Pocchiola's calculations hidden on a computer now appear in details, especially the determination of a secondary invariant $R$, expressed in terms of the first jet of $W_0$. All other secondary invariants of the $\{e\}$-structure are also expressed explicitly in terms of $W_0$ and $J_0$.