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

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

References

1. Z. Chen, W. G. Foo, J. Merker and T. A. Ta: Normal forms for rigid C2;1 hypersurfaces $M_5\subset \mathbb{C}_3$, Taiwanese Journal of Mathematics, 25 (2) (2021), 333–364, arxiv.org/abs/1912.01655/
2. Z. Chen, J. Merker: On differential invariants of parabolic surfaces, Dissertationes Mathematicæ, 559 (2021), 110 pages, arxiv.org/abs/1908.07867/
3. M. Fels, W. Kaup: CR manifolds of dimension 5: a Lie algebra approach, J. Reine Angew. Math., 604 (2007), 47–71.
4. M. Fels, W. Kaup: Classification of Levi degenerate homogeneous CR-manifolds in dimension 5, Acta Math., 201 (2008), 1–82.
5. W. G. Foo, J. Merker and T. A. Ta: Rigid equivalences of 5-dimensional 2-nondegenerate rigid real hypersurfaces $M_5\subset \mathbb{C}_3$ of constant Levi rank 1, Michigan Math. J., to appear, arxiv.org/abs/1904.02562/.
6. M. Freeman: Real submanifolds with degenerate Levi form. Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Williams Coll., Williamstown, Mass., 1975), Part 1, pp. 141–147. Amer. Math. Soc., Providence, R.I., 1977.
7. M. Freeman: Local biholomorphic straightening of real submanifolds, Annals of Mathematics, 106 (2) (1977), 319–352.
8. H. Gaussier, J. Merker: A new example of uniformly Levi degenerate hypersurface in $\mathbb{C}_3$, Ark. Mat., 41 (1) (2003), 85–94. Erratum: 45 (2) (2007), 269–271.

9. A. Isaev: Zero CR-Curvature Equations for Levi Degenerate Hypersurfaces via Pocchiola’s Invariant, arxiv.org/pdf/1809.03029/
10. H. Jacobowitz: An introduction to CR structures, Math. Surveys and Monographs, 32, Amer. Math. Soc., Providence (1990).
11. S. Lie (Author), J. Merker (Editor): Theory of Transformation Groups I. General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation, Springer-Verlag, Berlin, Heidelberg (2015), arxiv.org/abs/1003.3202/
12. C. Medori, A. Spiro: The equivalence problem for 5-dimensional Levi degenerate CR manifolds, Int. Math. Res. Not. IMRN, 2014 (20), 5602–5647.
13. C. Medori, A. Spiro: Structure equations of Levi degenerate CR hypersurfaces of uniform type, Rend. Semin. Mat. Univ. Politec. Torino, 73 (1–2) (2015), 127–150.
14. J. Merker, P. Nurowski: On degenerate para-CR structures: Cartan reduction and homogeneous models, arxiv.org/abs/2003.08166/ (2020).
15. J. Merker: Lie symmetries of partial differential equations and CR geometry, Journal of Mathematical Sciences (N.Y.), 154 (2008), 817–922.
16. J. Merker: Equivalences of 5-dimensional CR manifolds, IV: Six ambiguity matrix groups (Initial G-structures), arxiv.org/abs/1312.1084/
17. J. Merker: Equivalences of 5-dimensional CR-manifolds V: Six initial frames and coframes; Explicitness obstacles, arxiv.org/abs/1312.3581/
18. J. Merker, S. Pocchiola: Explicit absolute parallelism for 2-nondegenerate real hypersurfaces $M_5\subset \mathbb{C}_3$ of constant Levi rank 1, Journal of Geometric Analysis, 30 (2020), 2689–2730, 10.1007/s12220-018-9988-3. Addendum: 3233–3242, 10.1007/s12220-019-00195-2.
19. J. Merker, S. Pocchiola and M. Sabzevari: Equivalences of 5-dimensional CR manifolds, II: General classes I, II, III1, III2, IV1, IV2, 5 figures, 95 pages, arxiv.org/abs/1311.5669/
20. P. Nurowski, J. Tafel: Symmetries of Cauchy-Riemann spaces, Lett. Math. Phys., 15 (1) (1988), 31–38.
21. P. J. Olver: Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, (1995).
22. P. J. Olver: Normal forms for submanifolds under group actions, Symmetries, differential equations and applications, 1–25. Springer Proc. Math. Stat. 266, Springer, Cham, (2018).
23. P. J. Olver, J. Pohjanpelto: Moving frames for Lie pseudo-groups, Canad. J. Math., 60 (6) (2008), 1336–1386.
24. P. J. Olver, J. Pohjanpelto: Differential invariant algebras of Lie pseudo-groups, Adv. Math., 222 (5) (2009), 1746–1792.
25. S. Pocchiola: Explicit absolute parallelism for 2-nondegenerate real hypersurfaces $M_5\subset \mathbb{C}_3$ of constant Levi rank 1, arxiv.org/abs/1312.6400/