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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 5

# A class of four-dimensional CR submanifolds in six dimensional nearly Kähler manifolds

Miroslava Antić
Published Online: 2018-10-20 | DOI: https://doi.org/10.1515/ms-2017-0175

## Abstract

We investigate four-dimensional CR submanifolds in six-dimensional strict nearly Kähler manifolds. We construct a moving frame that nicely corresponds to their CR structure and use it to investigate CR submanifolds that admit a special type of doubly twisted product structure. Moreover, we single out a class of CR submanifolds containing this type of doubly twisted submanifolds.

Further, in a particular case of the sphere ${\mathbb{S}}^{6}\left(1\right)$, we show that the two families of four-dimensional CR submanifolds, those that admit a three-dimensional geodesic distribution and those ruled by totally geodesic spheres ${\mathbb{S}}^{3}$ coincide, and give their classification, which as a subfamily contains a family of doubly twisted CR submanifolds.

MSC 2010: 53B25; 53C42; 53C25

This research was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, project 174012.

## References

• [1]

ANTIĆ M.: A class of four-dimensional CR submanifolds of the sphere ${\mathbb{S}}^{\mathrm{6}}\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{\right)}$, J. Geom. Phys. 110 (2016), 78–89.Google Scholar

• [2]

ANTIĆ M.: Four-dimensional CR submanifolds of the sphere ${\mathbb{S}}^{\mathrm{6}}\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{\right)}$ with two-dimensional nullity distribution, J. Math. Anal. Appl. 445 (2017), 1–12.Google Scholar

• [3]

BEJANCU A.: Geometry of CR-Submanifolds, D. Reidel Publ., Dordrecht, 1986.Google Scholar

• [4]

BOLTON, J.—DILLEN, F.—DIOOS, B.—VRANCKEN, L.: Almost complex surfaces in the nearly Kähler ${\mathbb{S}}^{\mathrm{3}}\mathrm{×}{\mathbb{S}}^{\mathrm{3}}$, Tôhoku Math. J. 67 (2015), 1–17.Google Scholar

• [5]

BOLTON, J.—VRANCKEN, L.—WOODWARD, L. M.: On almost complex curves in the nearly Kähler 6-sphere, Q. J. Math. 45 (1994), 407–427.Google Scholar

• [6]

BUTRUILLE, J. B.: Homogeneous nearly Kähler manifolds. In: Handbook of PseudoRiemannian Geometry and Supersymmetry, RMA Lect. Math. Theor. Phys. 16, Eur. Math. Soc., Zürich, 2010, pp. 399–423.Google Scholar

• [7]

DILLEN, F.—VERSTRAELEN, L.—VRANCKEN, L.: Classification of totally real 3-dimensional submanifolds of ${\mathbb{S}}^{\mathrm{6}}\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{\right)}$ with $K\mathrm{\ge }\mathrm{1}\mathrm{/}\mathrm{16}$, J. Math. Soc. Japan 42 (1990), 565–584.Google Scholar

• [8]

EJIRI, E.: Totally real submanifolds in a 6-sphere, Proc. Am. Math. Soc. 83 (1981), 759–763.Google Scholar

• [9]

GRAY, A.: The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), 233–248.Google Scholar

• [10]

GRAY, A.—HERVELLA, L. M.: The sixteen classes of almost Hermitian manifold and their linear invariants, Ann. Math. Pura App. 123 (1980), 35–58.Google Scholar

• [11]

HARVEY, R.—LAWSON, H. B.: Calibrated Geometries, Acta Math. 148 (1982), 47–157.Google Scholar

• [12]

NAGY, P. A.: Nearly Kähler geometry and Riemannian foliations, Asian J. Math. 6 (2002), 481–504.Google Scholar

• [13]

PODESTÀ, F.—SPIRO, A.: 6-dimensional nearly Kähler manifolds of cohomogeneity one, J. Geom. Phys. 60, (2010), 156–164.Google Scholar

• [14]

PONGE, R.—RECKZIEGEL, H.: Twisted products in pseudo-Riemannian geometry, Geom. Dedicata 48 (1993), 15–25.Google Scholar

• [15]

SEKIGAWA, K.: Some CR-submanifolds in a 6-dimensional sphere, Tensor, N. S. 41 (1984), 13–20.Google Scholar

• [16]

UDDIN, S.: On doubly warped and doubly twisted product submanifolds, Int. Electron. J. Geom. 3 (2010), 35–39.Google Scholar

• [17]

ZHANG, Y.—DIOOS, B.—HU, Z.—VRANCKEN, L.—WANG, X.: Lagrangian submanifolds in the 6-dimensional nearly Kähler manifolds with parallel second fundamental form, J. Geom. Phys. 108 (2016), 21–37.Google Scholar

Accepted: 2017-09-22

Published Online: 2018-10-20

Published in Print: 2018-10-25

Communicated by Július Korbaš

Citation Information: Mathematica Slovaca, Volume 68, Issue 5, Pages 1129–1140, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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