Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfaces $ M_5 ⊂ \mathbb{C} $

Year 2021, , 318 - 377, 16.09.2021

Jöel Merker , Wei Foo

https://doi.org/10.33205/cma.943426

Cited By: 1

Abstract

The class ${\text{IV}}_2$ of $2$-nondegenerate constant Levi rank $1$ hypersurfaces $M^5\subset C^3$ 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 $M_{LC}^5$ over the real light cone in $R^3$. When either $W_0\equiv ̸0$ or $J_0\equiv ̸0$, by normalization of certain two group parameters $c$ and $e$, 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 $\left\{e\right\}$-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 $\left\{e\right\}$-structure are also expressed explicitly in terms of $W_0$ and $J_0$.

Keywords

Levi degenerate CR manifolds, 2-nondegeneracy, G-structures, Cartan method of equivalence, Cartan Lemma, Pocchiola invariants

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