On the Geometry of Hypersurfaces in $${\mathbb {S}}^{2} \times {\mathbb {S}}^{2}$$

Lu, Xiaoge; Wang, Peng; Wang, Xiaozhen

doi:10.1007/s00025-022-01787-1

Cited by 4 publications

(1 citation statement)

References 13 publications

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“…Recently, Gao, Hu, Ma and the fourth author [7] studied the Hopf hypersurfaces of S 2 × S 2 and the hypersurfaces with isometric Reeb flow. Lu, Wang and Wang [16] constructed some new examples of cohomogeneity one hypersurfaces in S 2 × S 2 with constant C. Nevertheless, many fundamental problems remain open.…”

Section: Introductionmentioning

confidence: 99%

Affine hypersurfaces with constant sectional curvature

Antić

Vrancken

et al. 2021

Pacific J. Math.

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In this paper, we classify the hypersurfaces of S 2 × S 2 with constant sectional curvature. By applying the so-called Tsinghua principle, which was first discovered by the first three authors in 2013 at Tsinghua University, we prove that the constant sectional curvature can only be 1 2 and the product angle function C defined by Urbano is identically zero. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in S 2 × S 2 with C = 0, and we establish a one-to-one correspondence between the involving minimal hypersurface and the famous "sinh-Gordon equation"As a byproduct, we give a complete classification of the hypersurfaces of S 2 × S 2 with constant mean curvature and constant product angle function C.

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Section: Introductionmentioning

confidence: 99%