Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces

Anciaux, Henri

doi:10.1090/s0002-9947-2013-05972-7

Cited by 27 publications

(77 citation statements)

References 16 publications

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“…similar construction for the Hyperbolic 3-space S 3 (−1) was established by Georgiou and Guilfoyle in [4]. Then, in [1], Anciaux used the fact that L(S n+1 (c)) is identified with the Grassmannian of oriented two-planes of R n+2 to extend this geometric construction for all non-flat real space forms. In particular, he showed that L(S n+1 (c)) admits a Kähler or a para-Kähler structure, where the metric (which will be denoted here by G e ) is Einstein and invariant under the isometry group of S n+1 (c).…”

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confidence: 88%

“…The invariance of the constructed (para-) Kähler metric in L(S n+1 (c)) under the action of the isometric group of S n+1 (c) allows one to study geometric problems in the base manifold S n+1 (c) by studying its space of oriented geodesics. For example, the set of all oriented geodesics orthogonal (called as the Gauss map) to a hypersurface in S n+1 (c) corresponds to a Lagrangian submanifold in L(S n+1 (c)), with respect to the canonical symplectic structure Ω (see [1]). In particular, G 0 -flat Lagrangian surfaces in L(S 3 (c)) are the Gauss map of Weingarten surfaces in S 3 (c)), i.e.…”

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confidence: 99%

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On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms

Georgiou¹,

Lobos²

2016

Arch. Math.

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In this article, we construct a new para-Kähler structure (G, J , Ω) in the space of oriented geodesics L(M ) in a non-flat, real space form M . We first show that the para-Kähler metric G is scalar flat and when M is a 3-dimensional real space form, G is locally conformally flat. Furthermore, we prove that the space of oriented geodesics in hyperbolic n-space, equipped with the constructed metric G, is minimally isometric embedded in the tangent bundle of the hyperbolic n-space. We then study the submanifold theory, and we show that G-geodesics correspond to minimal ruled surfaces in the real space form. Lagrangian submanifolds (with respect to the canonical symplectic structure Ω) play an important role in the geometry of the space of oriented geodesics as they are the Gauss map of hypersurfaces in the corresponding space form. We demonstrate that the Gauss map of a non-flat hypersurface of constant Gauss curvature is a minimal Lagrangian submanifold. Finally, we show that a Hamiltonian minimal submanifold is locally the Gauss map of a hypersurface Σ that is a critical point of the functional F (Σ) = Σ |K| dV , where K denotes the Gaussian curvature of Σ.

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confidence: 88%

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confidence: 99%

On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms

Georgiou¹,

Lobos²

2016

Arch. Math.

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“…The space L − (AdS 3 ) of oriented timelike geodesics in the anti-De Sitter 3-space AdS 3 is diffeomorphic to the product dS 2 × dS 2 . The para-Kähler metric G ǫ is invariant under the natural action of the isometry group Iso(AdS 3 , g) (see [1] and [3]). It is clear by Corollary 2 that every G − -minimal Lagrangian surface immersed in L − (AdS 3 ) is locally the product of two geodesics in dS 2 .…”

Section: Corollary 2 Let (σ G) Be a Non-flat Lorentzian Two Manifolmentioning

confidence: 99%

“…We know from [3] that every G − -minimal Lagrangian immersion in AdS 3 is the Gauss map of a equidistant tube along a geodesic γ in AdS 3 and following the example 2 it must be locally parametrised as the product of geodesics in dS 2 . In this example, we are going to see exactly how can we obtain this product of geodesics.…”

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Lagrangian immersions in the product of Lorentzian two manifolds

Georgiou

2014

Geom Dedicata

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Abstract. For Lorentzian 2-manifolds (Σ 1 , g 1 ) and (Σ 2 , g 2 ) we consider the two product para-Kähler structures (G ǫ , J, Ω ǫ ) defined on the product four manifold Σ 1 × Σ 2 , with ǫ = ±1. We show that the metric G ǫ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures κ 1 , κ 2 of g 1 , g 2 , respectively, are both constants satisfying κ 1 = −ǫκ 2 (resp. κ 1 = ǫκ 2 ). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product γ 1 × γ 2 ⊂ Σ 1 ×Σ 2 , where γ 1 and γ 2 are curves of constant curvature. We study Lagrangian surfaces in the product dS 2 × dS 2 with non null parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and H-minimal surfaces.

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Hopf hypersurfaces in spaces of oriented geodesics

Georgiou

Guilfoyle

2017

J. Geom.

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Abstract. A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector.We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the oriented geodesic space admits a canonical KaehlerEinstein and para-Kaehler-Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists.The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in the underlying space form. We prove that a tangent hypersurface is Hopf in the space of oriented geodesics with respect to this canonical (para-)Kaehler structure iff the underlying convex hypersurface is totally umbilic and non-flat.In the case of 3 dimensional space forms, however, there exists a second canonical complex structure which can also be used to define Hopf hypersurfaces. We prove that in this dimension, the tangent hypersurface of a convex hypersurface in the space form is always Hopf with respect to this second complex structure.

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Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces

Cited by 27 publications

References 16 publications

On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms

On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms

Lagrangian immersions in the product of Lorentzian two manifolds

Hopf hypersurfaces in spaces of oriented geodesics

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