Research Article
BibTex RIS Cite
Year 2021, , 125 - 131, 15.04.2021
https://doi.org/10.36890/iejg.790910

Abstract

References

  • [1] Antić, M.: Ruled three-dimensional CR submanifolds of the sphere S6(1). Publications de l’institute mathematique 101 (115), 25–35, (2017).
  • [2] Antić, 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] Djorić, 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).

Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$

Year 2021, , 125 - 131, 15.04.2021
https://doi.org/10.36890/iejg.790910

Abstract

It is well known that the sphere $S^6(1)$ admits an almost complex structure $J$ which is nearly K\"{a}hler. A submanifold $M$ of an almost Hermitian manifold is called a CR submanifold if it admits a differentiable almost complex distribution $\mathcal{D}_1$ such that its orthogonal complement is a totally real distribution. In this case the normal bundle of the submanifold also splits into two distributions $\mathcal{D}_3$, which is almost complex, and a totally real complement. In the case of the proper three-dimensional CR submanifold of a six-dimensional manifold the distribution $\mathcal{D}_3$ is non-trivial. Here, we investigate three-dimensional CR submanifolds of the sphere $S^6(1)$ admitting an umbilic direction orthogonal to $\mathcal{D}_3$ and show that such submanifolds do not exist.  

References

  • [1] Antić, M.: Ruled three-dimensional CR submanifolds of the sphere S6(1). Publications de l’institute mathematique 101 (115), 25–35, (2017).
  • [2] Antić, 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] Djorić, 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).
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Miroslava Antic 0000-0002-2111-7174

Djordje Kocic 0000-0003-2255-2992

Publication Date April 15, 2021
Acceptance Date November 22, 2020
Published in Issue Year 2021

Cite

APA Antic, M., & Kocic, D. (2021). Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$. International Electronic Journal of Geometry, 14(1), 125-131. https://doi.org/10.36890/iejg.790910
AMA Antic M, Kocic D. Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$. Int. Electron. J. Geom. April 2021;14(1):125-131. doi:10.36890/iejg.790910
Chicago Antic, Miroslava, and Djordje Kocic. “Three-Dimensional CR Submanifolds in $S^6(1)$ With Umbilical Direction Normal to $\mathcal{D}_3$”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 125-31. https://doi.org/10.36890/iejg.790910.
EndNote Antic M, Kocic D (April 1, 2021) Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$. International Electronic Journal of Geometry 14 1 125–131.
IEEE M. Antic and D. Kocic, “Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 125–131, 2021, doi: 10.36890/iejg.790910.
ISNAD Antic, Miroslava - Kocic, Djordje. “Three-Dimensional CR Submanifolds in $S^6(1)$ With Umbilical Direction Normal to $\mathcal{D}_3$”. International Electronic Journal of Geometry 14/1 (April 2021), 125-131. https://doi.org/10.36890/iejg.790910.
JAMA Antic M, Kocic D. Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$. Int. Electron. J. Geom. 2021;14:125–131.
MLA Antic, Miroslava and Djordje Kocic. “Three-Dimensional CR Submanifolds in $S^6(1)$ With Umbilical Direction Normal to $\mathcal{D}_3$”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 125-31, doi:10.36890/iejg.790910.
Vancouver Antic M, Kocic D. Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$. Int. Electron. J. Geom. 2021;14(1):125-31.