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Pseudo-Hermitian 2-type Legendre surfaces in the unit sphere \(S^5\). (English) Zbl 1452.53016

Summary: In this paper, we show that it is Chen surfaces that non-minimal pseudo-Hermitian mass-symmetric 2-type Legendre surfaces in \(S^5\). Moreover, we show that pseudo-Hermitian mass-symmetric 2-type Legendre surfaces in \(S^5\) are locally the product of two pseudo-Hermitian circles.

MSC:

53B25 Local submanifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:

[1] C. Baikoussis and D. E. Blair, 2-type integral surfaces in S5(1), Tokyo J. Math. 14 (1991), no. 2, 345-356. https://doi.org/10.3836/tjm/1270130378 · Zbl 0763.53055 · doi:10.3836/tjm/1270130378
[2] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203, Birkhauser Boston, Inc., Boston, MA, 2002. https://doi.org/10.1007/978-1-4757-3604-5 · Zbl 1011.53001
[3] B.-Y. Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, 1, World Scientific Publishing Co., Singapore, 1984. https://doi.org/10.1142/0065 · Zbl 0537.53049
[4] J. T. Cho, Geometry of contact strongly pseudo-convex CR-manifolds, J. Korean Math. Soc. 43 (2006), no. 5, 1019-1045. https://doi.org/10.4134/JKMS.2006.43.5.1019 · Zbl 1119.53050 · doi:10.4134/JKMS.2006.43.5.1019
[5] J.-E. Lee, Laplacians and Legendre surfaces in pseudo-Hermitian geometry, Bull. Iranian Math. Soc. 44 (2018), no. 4, 899-913. https://doi.org/10.1007/s41980-018-0058-1 · Zbl 1426.53029 · doi:10.1007/s41980-018-0058-1
[6] N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), no. 1, 131-190. https://doi.org/10.4099/math1924.2.131 · Zbl 0346.32010 · doi:10.4099/math1924.2.131
[7] S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), no. 1, 349-379. https://doi.org/10.2307/2001446 · Zbl 0677.53043 · doi:10.1090/S0002-9947-1989-1000553-9
[8] S. · Zbl 0379.53016 · doi:10.4310/jdg/1214434345
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