Research Article
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Year 2021, Volume: 4 Issue: 3, 318 - 377, 16.09.2021
https://doi.org/10.33205/cma.943426

Abstract

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/

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

Abstract

The class IV2\sf IV2 of 22-nondegenerate constant Levi rank 11 hypersurfaces M5C3M5⊂C3 is governed by Pocchiola's two primary invariants W0W0 and J0J0. Their vanishing characterizes equivalence of such a hypersurface M5M5 to the tube M5LCMLC5 over the real light cone in R3R3. When either W0≢0W0≢0 or J0≢0J0≢0, by normalization of certain two group parameters cc and ee, an invariant coframe can be built on M5M5, showing that the dimension of the CR automorphism group drops from 1010 to 55. This paper constructs an explicit {e}{e}-structure in case W0W0 and J0J0 do not necessarily vanish. Furthermore, Pocchiola's calculations hidden on a computer now appear in details, especially the determination of a secondary invariant RR, expressed in terms of the first jet of W0W0. All other secondary invariants of the {e}{e}-structure are also expressed explicitly in terms of W0W0 and J0J0.

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/
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Jöel Merker 0000-0003-2653-2147

Wei Foo This is me 0000-0002-9022-8177

Publication Date September 16, 2021
Published in Issue Year 2021 Volume: 4 Issue: 3

Cite

APA Merker, J., & Foo, W. (2021). Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfaces $ M_5 ⊂ \mathbb{C} $. Constructive Mathematical Analysis, 4(3), 318-377. https://doi.org/10.33205/cma.943426
AMA Merker J, Foo W. Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfaces $ M_5 ⊂ \mathbb{C} $. CMA. September 2021;4(3):318-377. doi:10.33205/cma.943426
Chicago Merker, Jöel, and Wei Foo. “Differentiall $ {e} $-Structures for Equivalences of $ 2 $-Nondegenerate Levi Rank $ 1 $ Hypersurfaces $ M_5 ⊂ \mathbb{C} $”. Constructive Mathematical Analysis 4, no. 3 (September 2021): 318-77. https://doi.org/10.33205/cma.943426.
EndNote Merker J, Foo W (September 1, 2021) Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfaces $ M_5 ⊂ \mathbb{C} $. Constructive Mathematical Analysis 4 3 318–377.
IEEE J. Merker and W. Foo, “Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfaces $ M_5 ⊂ \mathbb{C} $”, CMA, vol. 4, no. 3, pp. 318–377, 2021, doi: 10.33205/cma.943426.
ISNAD Merker, Jöel - Foo, Wei. “Differentiall $ {e} $-Structures for Equivalences of $ 2 $-Nondegenerate Levi Rank $ 1 $ Hypersurfaces $ M_5 ⊂ \mathbb{C} $”. Constructive Mathematical Analysis 4/3 (September 2021), 318-377. https://doi.org/10.33205/cma.943426.
JAMA Merker J, Foo W. Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfaces $ M_5 ⊂ \mathbb{C} $. CMA. 2021;4:318–377.
MLA Merker, Jöel and Wei Foo. “Differentiall $ {e} $-Structures for Equivalences of $ 2 $-Nondegenerate Levi Rank $ 1 $ Hypersurfaces $ M_5 ⊂ \mathbb{C} $”. Constructive Mathematical Analysis, vol. 4, no. 3, 2021, pp. 318-77, doi:10.33205/cma.943426.
Vancouver Merker J, Foo W. Differentiall $ {e} $-structures for equivalences of $ 2 $-nondegenerate Levi rank $ 1 $ hypersurfaces $ M_5 ⊂ \mathbb{C} $. CMA. 2021;4(3):318-77.