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

Year 2021, Volume: 4 Issue: 3, 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

References

• 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/
• Z. Chen, J. Merker: On differential invariants of parabolic surfaces, Dissertationes Mathematicæ, 559 (2021), 110 pages, arxiv.org/abs/1908.07867/
• M. Fels, W. Kaup: CR manifolds of dimension 5: a Lie algebra approach, J. Reine Angew. Math., 604 (2007), 47–71.
• M. Fels, W. Kaup: Classification of Levi degenerate homogeneous CR-manifolds in dimension 5, Acta Math., 201 (2008), 1–82.
• 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/.
• 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.
• M. Freeman: Local biholomorphic straightening of real submanifolds, Annals of Mathematics, 106 (2) (1977), 319–352.
• 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.
• A. Isaev: Zero CR-Curvature Equations for Levi Degenerate Hypersurfaces via Pocchiola’s Invariant, arxiv.org/pdf/1809.03029/
• H. Jacobowitz: An introduction to CR structures, Math. Surveys and Monographs, 32, Amer. Math. Soc., Providence (1990).
• 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/
• C. Medori, A. Spiro: The equivalence problem for 5-dimensional Levi degenerate CR manifolds, Int. Math. Res. Not. IMRN, 2014 (20), 5602–5647.
• 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.
• J. Merker, P. Nurowski: On degenerate para-CR structures: Cartan reduction and homogeneous models, arxiv.org/abs/2003.08166/ (2020).
• J. Merker: Lie symmetries of partial differential equations and CR geometry, Journal of Mathematical Sciences (N.Y.), 154 (2008), 817–922.
• J. Merker: Equivalences of 5-dimensional CR manifolds, IV: Six ambiguity matrix groups (Initial G-structures), arxiv.org/abs/1312.1084/
• J. Merker: Equivalences of 5-dimensional CR-manifolds V: Six initial frames and coframes; Explicitness obstacles, arxiv.org/abs/1312.3581/
• 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.
• 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/
• P. Nurowski, J. Tafel: Symmetries of Cauchy-Riemann spaces, Lett. Math. Phys., 15 (1) (1988), 31–38.
• P. J. Olver: Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, (1995).
• 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).
• P. J. Olver, J. Pohjanpelto: Moving frames for Lie pseudo-groups, Canad. J. Math., 60 (6) (2008), 1336–1386.
• P. J. Olver, J. Pohjanpelto: Differential invariant algebras of Lie pseudo-groups, Adv. Math., 222 (5) (2009), 1746–1792.
• 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/