2021
DOI: 10.1007/s10231-020-01063-5
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Lagrangian submanifolds of the complex hyperbolic quadric

Abstract: We consider the complex hyperbolic quadric Q * n as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on Q * n , which are then used to define local angle functions on any Lagrangian submanifold of Q * n . We prove that a Lagrangian immersion into Q * n can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersu… Show more

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Cited by 2 publications
(6 citation statements)
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“…By using such approach, some canonical submanifolds with constant sectional curvature (even under more general conditions) in some canonical Riemannian manifold have been classified, cf. [1,5,6,14,21,22], etc. The main purpose of this paper is to classify the hypersurfaces of S 2 × S 2 with constant sectional curvature.…”
Section: Introductionmentioning
confidence: 99%
“…By using such approach, some canonical submanifolds with constant sectional curvature (even under more general conditions) in some canonical Riemannian manifold have been classified, cf. [1,5,6,14,21,22], etc. The main purpose of this paper is to classify the hypersurfaces of S 2 × S 2 with constant sectional curvature.…”
Section: Introductionmentioning
confidence: 99%
“…As mentioned in the introduction, the almost product structure A and the Kähler structure J of Q m * also satisfy (2.1) and (2.2) (see [13,25,27,28]), and it makes the tensors ϕ, T, μ, and η on M also satisfy the equations (2.7)-(2.11) in Lemma 2.2. Because the Riemannian curvature tensor of Q m * is exactly opposite to the Riemannian curvature tensor of Q m , it makes both the Gauss-Codazzi equations and the Ricci tensor of M a slightly different from (2.12)-(2.14): some terms change at most by negative signs.…”
Section: Proof Of Theorem 11 For Q M *mentioning
confidence: 90%
“…These two structures satisfy the same properties of those two in Q m . For more details about the complex hyperbolic quadric Q m * , we refer the readers to [13,19,25,27,28]. Now, we can state the main result of this paper as the following nonexistence theorem.…”
Section: Introductionmentioning
confidence: 90%
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