On the nonexistence and rigidity for hypersurfaces of the homogeneous nearly Kähler <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>

Hu, Zejun; Moruz, Marilena; Vrancken, Luc; Yao, Zeke

doi:10.1016/j.difgeo.2021.101717

Cited by 7 publications

(3 citation statements)

References 29 publications

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“…Then, by (2.20), (2.21) and the above discussions, we have In order to prove Theorem 1.2 and Theorem 1.3, we still need the following useful formula due to K. Yano [19]. A simply proof is referred also to [13].…”

Section: Legendrian Submanifolds Of the Sasakian Space Form ñ 2n+1 (C)mentioning

confidence: 99%

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New characterizations of the Whitney spheres and the contact Whitney spheres

Hu¹,

Xing²

2021

Preprint

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In this paper, based on the classical K. Yano's formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction J H and the norm of the covariant differentiation of the second fundamental form h, where J is the almost complex structure and H is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure (ϕ, ξ, η, g), we also establish an optimal integral inequality involving the Ricci curvature in the direction ϕ H and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant.

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Section: Legendrian Submanifolds Of the Sasakian Space Form ñ 2n+1 (C)mentioning

confidence: 99%

“…Lemma 2.1 (cf. Lemma 5.1 of [13]). Let (M, g) be a Riemannian manifold with Levi-Civita connection ∇.…”

Section: Legendrian Submanifolds Of the Sasakian Space Form ñ 2n+1 (C)mentioning

confidence: 99%

New characterizations of the Whitney spheres and the contact Whitney spheres

Hu¹,

Xing²

2021

Preprint

Self Cite

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show abstract

“…1 The method used in the proof of Lemma 3.1 is elementary and is called the Tsinghua principle. This remarkable technique has been applied in many different situations since its first successful attempt in [8], see [1,6,7,16,29] for details. Essentially, it establishes a bridge between the Codazzi equation and the Ricci identity by calculating the cyclic sum of the second covariant derivative of the second fundamental form.…”

Section: I(w X Y Z) (34)mentioning

confidence: 99%