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

Authors : Jöel MERKER, Wei FOO

Pages : 318-377

Doi:10.33205/cma.943426

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Publication Date : 2021-09-16

Article Type : Research Paper

Abstract :The class IV 2 \sf IV2 of 2 2 -nondegenerate constant Levi rank 1 1 hypersurfaces M 5 ⊂ C 3 M5⊂C3 is governed by Pocchiola`s two primary invariants W 0 W0 and J 0 J0 . Their vanishing characterizes equivalence of such a hypersurface M 5 M5 to the tube M 5 L C MLC5 over the real light cone in R 3 R3 . When either W 0 ≢ 0 W0≢0 or J 0 ≢ 0 J0≢0 , by normalization of certain two group parameters c c and e e , an invariant coframe can be built on M 5 M5 , showing that the dimension of the CR automorphism group drops from 10 10 to 5 5 . This paper constructs an explicit { e } {e} -structure in case W 0 W0 and J 0 J0 do not necessarily vanish. Furthermore, Pocchiola`s calculations hidden on a computer now appear in details, especially the determination of a secondary invariant R R , expressed in terms of the first jet of W 0 W0 . All other secondary invariants of the { e } {e} -structure are also expressed explicitly in terms of W 0 W0 and J 0 J0 .
Keywords : Levi degenerate CR manifolds, 2 nondegeneracy, G structures, Cartan method of equivalence, Cartan Lemma, Pocchiola invariants