Parallel surfaces in the real special linear gorup <i>SL</i> (2,ℝ)

Belkhelfa, Mohamed; Dillen, Franki; Inoguchi, Jun-ichi

doi:10.1017/s0004972700020220

Cited by 9 publications

(17 citation statements)

References 3 publications

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“…(cf. [4]) and the (1, 1)-tensor φ by the matrix (2, R), φ, ξ, η, g The Legendrian curve with constant curvature h in SL(2, R) is obtained by the horizontal lift of the curveγ = (x(s), y(s)) in H 2 with constant curvature h. If h > 2, the curveγ is a closed circle. More precisely, it is given by…”

Section: The Stability Of L-minimal Legendrian Curves In Sasakian Space Formsmentioning

confidence: 99%

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Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds

Kajigaya

2013

Tohoku Math. J. (2)

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In this paper, we investigate compact Legendrian submanifolds L in Sasakian manifolds M, which have extremal volume under Legendrian deformations. We call such a submanifold L-minimal Legendrian submanifold. We derive the second variational formula for the volume of L under Legendrian deformations in M. Applying this formula, we investigate the stability of L-minimal Legendrian curves in Sasakian space forms, and show the L-instability of L-minimal Legendrian submanifolds in S 2n+1 (1). Moreover, we give a construction of L-minimal Legendrian submanifolds in R 2n+1 (−3). 1. Introduction. In [10], [11], Y. G. Oh introduced the notion of Hamiltonianminimal (H-minimal) Lagrangian submanifolds in Kähler manifolds. Such a submanifold is a critical point of the volume functional under the Hamiltonian deformation. This is an extension of the notion of minimal submanifold, and has been studied by many authors (for example, [7], [8], [11], [14] and see references therein). An H-minimal Lagrangian submanifold is called Hamiltonian-stable (H-stable) if the second variation is non-negative for any Hamiltonian deformation. Oh studied H-stablity of some examples of H-minimal Lagrangian submanifold in a specific Kähler manifold ([10], [11]). For example, the real projective space RP n and the Clifford torus in CP n , and the standard tori in C n are H-stable. Besides these examples, Schoen and Wolfson studied the H-stablity of two-dimensional H-minimal Lagrangian cones ([14]), and Iriyeh studied the three-dimensional case ([8]). Furthermore, in [1], Amarzaya and Ohnita proved that all compact Lagrangian submanifolds with parallel second fundamental form in C n and CP n are H-stable.On the other hand, there is a notion of Sasakian manifolds, which is an odd-dimensional counterpart to Kähler manifolds. In Sasakian manifolds, we consider Legendrian-minimal (Lminimal) Legendrian submanifolds which correspond to H-minimal Lagrangian manifolds in Kähler manifolds. An L-minimal Legendrian submanifold is a critical point of the volume function under the Legendrian deformation (for more details, see Section 3). In [7], [8], the authors constructed examples of L-minimal Legendrian submanifolds in odd-dimensional unit spheres (in [7], such submanifolds are called C-minimal). Moreover, they show that a certain L-minimal Legendrian submanifold in a Sasakian manifold is related to an H-minimal Lagrangian submanifold in a Kähler manifold. For example, the cone over a Legendrian

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Section: The Stability Of L-minimal Legendrian Curves In Sasakian Space Formsmentioning

confidence: 99%

“…where r ∈ R + is a positive constant (cf. [4]), and the smooth function μ : R → R satisfies the equationμ…”

Section: Then (Slmentioning

confidence: 99%

Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds

Kajigaya

2013

Tohoku Math. J. (2)

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“…. , ϕ n (t)) be a parametrization of an (n − 1)-dimensional sphere S n−1 (1) in E n . Then M can be described by…”

Section: Rotation Hypersurfaces With G|cmentioning

confidence: 99%

Divisibility of the Second Fundamental Form of Hypersurfaces of Space Forms

Dillen

Verbouwe

2008

Result. Math.

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“…Choose strictly positive real constants λ 1 , λ 2 , λ 3 and define (c 1 , c 2 , c 3 ), defined by the condition that {e 1 , e 2 , e 3 } is an orthonormal basis, is In particular, if c 1 = c 2 = c 3 = c > 0, then the space is of constant curvature c 2 /4. In general, the Levi Civita connection and the curvature tensor of (SU (2), g(c 1 , c 2 , c 3 )) are given by Proposition 2.…”

Section: Proposition 3 [15] Any Left-invariant Metric On Sumentioning

confidence: 99%

“…Therefore, these submanifolds are usually one of the first families to study. See for example [1][2][3]6].…”

Section: Introductionmentioning

confidence: 99%

A complete classification of parallel surfaces in three-dimensional homogeneous spaces

Inoguchi

Veken²

2007

Geom Dedicata

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Parallel surfaces in the real special linear gorup SL (2,ℝ)

Abstract: PARALLEL SURFACES IN THE REAL SPECIAL LINEAR GROUP SL(2,R)MOHAMED BELKHELFA, FRANKI DlLLEN AND JUN-ICHI INOGUCHI Dedicated to Professor Koichi Ogiue on his sixtieth birthday We show that the only parallel surfaces in SL(2, M) are rotational surfaces with constant mean curvature.

Cited by 9 publications

References 3 publications

Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds

Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds

Divisibility of the Second Fundamental Form of Hypersurfaces of Space Forms

A complete classification of parallel surfaces in three-dimensional homogeneous spaces

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