SavedAdd your journal

Journals / International Electronic Journal of Geometry / 2021 / Cilt: 14 - Sayı: 1

Three-Dimensional CR Submanifolds in S6(1) with Umbilical Direction Normal to D3

Miroslava ANTI´C
Djordje KOCI´C
Pages
125–131
Publication Date
2022-10-18
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In the case of the proper threedimensionalCR submanifold of a six-dimensional manifold the distribution D3 is non-trivial.Here, we investigate three-dimensional CR submanifolds of the sphere S6(1) admitting an umbilicdirection orthogonal to D3 and show that such submanifolds do not exist.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler.A submanifold M of an almost Hermitian manifold is called a CR submanifold if it admits adifferentiable almost complex distribution D1 such that its orthogonal complement is a totally realdistribution.
📊
Key Findings & Data:In this case the normal bundle of the submanifold also splits into two distributionsD3, which is almost complex, and a totally real complement.
💡
Conclusion & Impact:In the case of the proper threedimensionalCR submanifold of a six-dimensional manifold the distribution D3 is non-trivial.Here, we investigate three-dimensional CR submanifolds of the sphere S6(1) admitting an umbilicdirection orthogonal to D3 and show that such submanifolds do not exist.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler.A submanifold M of an almost Hermitian manifold is called a CR submanifold if it admits adifferentiable almost complex distribution D1 such that its orthogonal complement is a totally realdistribution. In this case the normal bundle of the submanifold also splits into two distributionsD3, which is almost complex, and a totally real complement. In the case of the proper threedimensionalCR submanifold of a six-dimensional manifold the distribution D3 is non-trivial.Here, we investigate three-dimensional CR submanifolds of the sphere S6(1) admitting an umbilicdirection orthogonal to D3 and show that such submanifolds do not exist.

🔖

Cite This Article

ANTI´C, M., KOCI´C, D. (2022). Three-Dimensional CR Submanifolds in S6(1) with Umbilical Direction Normal to D3. International Electronic Journal of Geometry, Cilt: 14 - Sayı: 1, 125–131. https://ajindex.com/articles/539572

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (17)

  1. Anti´c, M.: Ruled three-dimensional CR submanifolds of the sphere S6(1). Publications de l’institute mathematique 101 (115), 25–35, (2017)
  2. Anti´c, M., Vrancken, L.: Three-dimensional minimal CR submanifolds of the sphere S6(1) contained in a hyperplane. Mediterr. J. Math. 12, 1429–1449, (2015)
  3. Bejancu, A.: CR submanifold of a Kaehler manifold I. Proc. Amer.Math. Soc. 69(1), 135–142 (1978)
  4. Bejancu, A.: Geometry of CR-submanifolds, D. Reidel Publishing Company, Dordrecht (1986)
  5. Bejancu, A.: Umbilical CR-submanifolds of a Kaehler manifold. Rend. Mat. 12, 439–445 (1980)
  6. Blair, D.E., Chen, B.Y.: On CR-submanifolds of Hermitian manifolds. Israel J. Math. 34, 353–363 (1979)
  7. Butruille, J.-B.: Homogeneous nearly Kähler manifolds. In: Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lect. Math. Theor. Phys. 16, Eur. Math. Soc., Zürich, 399–423, (2010)
  8. Chen, B. Y.: Totally umbilical submanifolds of Kaehler manifolds. Arch. Math. 36, 83–91 (1981)
  9. Djori´c, M., Vrancken, L.: Three dimensional minimal CR submanifolds in S6 satisfying Chen’s equality. J. Geom. Phys. 56, 2279–2288, (2006)
  10. Ejiri, N.: Totally real submanifolds in a 6-sphere. Proc. Amer. Math. Soc. 83, 759–763, (1981)
  11. Gray, A.: Nearly Kähler manifolds. J. Diff. Geom. 4, 283–309, (1970)
  12. Gray, A.: Riemannian manifolds with geodesic symmetries of order 3. J. Diff. Geom. 7, 343–369, (1972)
  13. Gray, A.: Almost complex submanifolds of the six-sphere. Proc. Amer. Math. Soc. 20, 277–279, (1969)
  14. Hashimoto, H., Mashimo, K.: On some 3-dimensional CR-submanifolds in S6. Nagoya Math. J. 156, 171–185, (1999)
  15. Khan, K.A., Khan, V.A. and Hussain, S.I.: Totally Umbilical CR-submanifolds of Nearly Kähler Manifold. Geometriae Dedicata. 50, 47–51 (1994)
  16. Nagy, P.-A.:Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6, 481–504, (2002)
  17. Sekigawa, K.: Some CR-submanifolds in a 6-dimensional sphere. Tensor N.S. 41, 13–20, (1984).