- Constructive Mathematical Analysis
- Vol: 4 Issue: 3
- Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfac...
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
View : 12 | Download : 9
Publication Date : 2021-09-16
Article Type : Research
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