Hypersurfaces in the Lorentz-Minkowski space satisfying L k ψ = Aψ + b

Lucas, Pascual; Ramírez-Ospina, H. Fabián

doi:10.1007/s10711-010-9562-z

Cited by 22 publications

(18 citation statements)

References 15 publications

(21 reference statements)

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“…where D k = c k /2. From now on, we will follow a similar reasoning to that given in [14,Lemma 9]. The proof continues according to the type of the shape operator S.…”

Section: The Case Where a Is Self-adjointmentioning

confidence: 77%

“…Following the ideas contained in [4], we have completely extended to the Lorentz-Minkowski space the previous classification theorem obtained by Alías and Gürbüz. In particular, the following classification result was given in [14,Theorem 1].…”

mentioning

confidence: 99%

“…Theorem A. ( [14]) Let ψ : M → L n+1 be an orientable hypersurface immersed into the Lorentz-Minkowski space L n+1 , and let L k be the linearized operator of the (k + 1)th mean curvature of M , for some fixed k = 0, 1, . .…”

mentioning

confidence: 99%

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HYPERSURFACES IN NON-FLAT LORENTZIAN SPACE FORMS SATISFYING Lkψ = Aψ + b

Lucas¹,

Ramírez-Ospina²

2012

Taiwanese J. Math.

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We study hypersurfaces either in the De Sitter space S n+1 1 ⊂ R n+2 1 or in the anti De Sitter space H n+1 1 ⊂ R n+2 2 whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k+1)-th mean curvature of the hypersurface, for a fixed k = 0, . . . , n − 1, A is an (n + 2) × (n + 2) constant matrix and b is a constant vector in the corresponding pseudo-Euclidean space. For every k, we prove that when A is selfadjoint and b = 0, the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)-th mean curvature and constant k-th mean curvature, open pieces of standard pseudo-Riemannian products in S n+1When H k is constant and b is a non-zero constant vector, we show that the hypersurface is totally umbilical, and then we also obtain a classification result (see Theorem 2). (2010): 53C50, 53B25, 53B30 and hectorfabian.ramirez@um.es c ⊂ R n+2 q satisfies the condition L k ψ = Aψ + b, for some self-adjoint constant matrix A ∈ R (n+2)×(n+2) and some non-zero constant vector b. Since H k is assumed to be constant on M , from Lemma 9 we know that H k+1 is also constant on M . The case H k+1 = 0 cannot occur, because in that case we have b = 0 (see Example 1). Mathematics Subject ClassificationsLet us assume that H k+1 is a non-zero constant. From (35) we obtain that b ⊤ = 0 and that the function b, ψ is constant on M . Now we use (34) to deduce that b, N = cH k H k+1 b, ψ = constant. Since b = ε b, N N + c b, ψ ψ, taking covariant derivative in this equation we have −ε b, N SX + c b, ψ X = 0, for any tangent vector field X. If b, N = 0, then M is totally umbilical (but not totally geodesic). Otherwise, b = c b, ψ ψ and then b, ψ = 0, but this implies b = 0. That concludes the proof of Theorem 2.

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“…where D k = c k /2. From now on, we will follow a similar reasoning to that given in [14,Lemma 9]. The proof continues according to the type of the shape operator S.…”

Section: The Case Where a Is Self-adjointmentioning

confidence: 77%

mentioning

confidence: 99%

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HYPERSURFACES IN NON-FLAT LORENTZIAN SPACE FORMS SATISFYING Lkψ = Aψ + b

Lucas¹,

Ramírez-Ospina²

2012

Taiwanese J. Math.

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“…Most recently, the complete classication of biconservative surfaces in 4-dimensional Lorentzian space forms is obtained in [11] Let M be a hypersurface in E n+1 s , s = 0, 1 with the shape operator S, mean curvature H and x : M → E m an isometric immersion. M is said to be biharmonic if the equation ∆ 2 x = 0 is satised or, equivalently, the system of dierential equations (BC) S(∇H) + ε nH 2 ∇H = 0, (BH1) ∆H + HtrS 2 = 0 is satised, where N is the unit normal vector eld (see [6,13]) and ε = N, N .…”

Section: Introductionmentioning

confidence: 99%

“…. , kn, where the eigenvector e1 of S is light-like, (see also [13,16]). With the abuse of terminology, we call these vector elds e1, e2, .…”

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confidence: 99%

Numerical solution of fourth order parabolic partial differential equation using parametric septic splines

Khan¹

2016

HJMS

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In this paper we study hypersurfaces with the mean curvature function H satisfying ∇H, ∇H = 0 in a Minkowski space of arbitrary dimension. First, we obtain some conditions satised by connection forms of biconservative hypersurfaces with the mean curvature function whose gradient is light-like. Then, we use these results to get a classication of biharmonic hypersurfaces. In particular, we prove that if a hypersurface is biharmonic, then it must have at least 6 distinct principal curvatures under the hypothesis of having mean curvature function satisfying the condition above.

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Some Classifications of Biharmonic Lorentzian Hypersurfaces in Minkowski 5-Space

Turgay

2014

Mediterr. J. Math.

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Hypersurfaces in the Lorentz-Minkowski space satisfying L k ψ = Aψ + b

Cited by 22 publications

References 15 publications

HYPERSURFACES IN NON-FLAT LORENTZIAN SPACE FORMS SATISFYING Lkψ = Aψ + b

HYPERSURFACES IN NON-FLAT LORENTZIAN SPACE FORMS SATISFYING Lkψ = Aψ + b

Numerical solution of fourth order parabolic partial differential equation using parametric septic splines

Some Classifications of Biharmonic Lorentzian Hypersurfaces in Minkowski 5-Space

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