Induced Riemannian structures on null hypersurfaces

Gutiérrez, Manuel; Olea, Benjamín

doi:10.1002/mana.201400355

Cited by 37 publications

(45 citation statements)

References 29 publications

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“…It is known that a totally umbilic null hypersurface with a closed normalization splits locally as a twisted product, the decomposition being global if M is simply connected and the rigged vector field complete, [20,Theorem 5.3]. We show here that if moreover it admits a closed conformal rigging in an ambient space form, the local twisted product structure of the rigged metric is in fact a warped product.…”

Section: Completeness Of (M G)mentioning

confidence: 82%

“…Proof. Using [20,Theorem 5.3], the only point we are going to show is the warped decomposition of (M, g).…”

Section: Completeness Of (M G)mentioning

confidence: 99%

“…Returning to the approach in [11], the basic question is how to reduce as much as possible the arbitrary choices and to have a reasonable coupling between the properties of the null hypersurface and the ambient space. In [20], the authors used the rigging technique to study null hypersurfaces. It is based on the arbitrary choice of a unique vector field in a neighborhood of the null hypersurface, called rigging vector field, from which is constructed both a null section defined on the null hypersurface, called rigged vector field, and a screen distribution.…”

Section: Introductionmentioning

confidence: 99%

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New Properties on Normalized Null Hypersurfaces

2018

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Rigging technique introduced in [20] is a convenient way to address the study of null hypersurfaces. It offers in addition the extra benefit of inducing a Riemannian structure on the null hypersurface which is used to study geometric and topological properties on it. In this paper we develop this technique showing new properties and applications. We first discuss the very existence of the rigging fields under prescribed geometric and topological constraints. We consider the completeness of the induced rigged Riemannian structure. This is potentially important because it allows to use most of the usual Riemannian techniques.Keywords. Lorentzian manifolds, null hypersurface, normalization, rigging vector field, rigged vector field, completeness. MSC(2010). 53C50, 53B30, 53B50.Our aim is to show that in a 4-dimensional Lorentzian manifold, compact null hypersurfaces with finite fundamental group can not admit such normalizations. We prove first the following:Proposition 3.7. Let ζ be a rigging for a compact null hypersurface M in a Lorentzian manifold (M , g) of constant curvature. If the screen S (ζ) is conformal and the first De Rham cohomology group H 1 (M, R) is trivial, then M is totally geodesic.Proof. Since M is screen conformal there exists a non vanishing function ρBut the left hand side of (3.2) vanishes since M has constant curvature, and ξ is orthogonal to both X and Y . Moreover,MANUEL GUTIÉRREZ, AND RAYMOND HOUNNONKPE since A N = ρ ⋆ A ξ . Using the fact that H 1 (M, R) is trivial, there exists a function (say) φ defined on M such that τ = dφ. Define a new rigging vector field by ζ = exp(−φ)ζ, so N = exp(−φ)N . Moreover, it follows (from [2], Lemma 2.1) that τ = τ + d(ln(exp(−φ)) = 0 as τ = dφ. Denote respectively by ξ, H, ⋆ A ξ the rigged vector field, the mean curvature function and the screen shape operator form of ζ, we have ([5], Remark 3)But Ric( ξ) = 0 since M has constant curvature and τ ( ξ) = 0, it fol-We conclude that M is totally geodesic.We can get now the following result. Proposition 3.8. Let (M 4 , g) be a 4-dimensional Lorentzian manifold of constant curvature and M a compact null hypersurface. If M has finite fundamental group then there is no normalization such that M is screen conformal.Proof. Let M be as above. Suppose there is a normalization such that M is screen conformal. Since M has finite fundamental group, the first De Rham cohomology group H 1 (M, R) is trivial. It follows from Proposition 3.7 that M is totally geodesic. Elsewhere, M being screen conformal, S (ζ) is integrable and induces a foliation on M . We show that the leaves of the screen distribution S (ζ) are totally geodesic in (M, g). For this, recall (from [20], Proposition 3.7)that for X and Y in S (ζ),In other words, the second fundamental form of each leaf of S (ζ) in (M, g) is B and then each of them is totally geodesic in (M, g) as M is totally geodesic in (M 4 , g). It follows that there exits a totally geodesic codimension one foliation on the compact 3-manifold M , hence M must have infinite fundamental g...

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Section: Completeness Of (M G)mentioning

confidence: 82%

“…Proof. Using [20,Theorem 5.3], the only point we are going to show is the warped decomposition of (M, g).…”

Section: Completeness Of (M G)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New Properties on Normalized Null Hypersurfaces

2018

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“…Using (42) one has ∇ ∂ ∂u a ξ F = −x 0 ∇ ∂ ∂u a n − ∂x 0 ∂u a n = −x 0 ∇ ∂ ∂u a n + F u a x 0 ξ F . This latter together with (8) and (50) give Aξ= x 0 ∇ · n and τ = η.…”

Section: A Specialmentioning

confidence: 86%

“…For this reason, we are going to relate Ricci tensor of ∇ α with the one of ∇ for sections of T M . In [8], such a relationship was found for α = 1 and by assuming that M is totally geodesic. We are going to relate this Ricci tensor for a function α constant on the leaves of the screen distribution and without total geodesibility condition.…”

Section: Now Using Equation (11) This Becomesmentioning

confidence: 89%

alpha-Associated metrics on rigged null hypersurfaces

Ngakeu¹,

Tetsing²

2019

Turk J Math

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Let x : M → M be the canonical injection of a Null Hypersurface (M, g) in a semi-Riemannian manifold (M ,ḡ). A rigging for M is a vector field L defined on some open set of M containing M such that Lp / ∈ TpM for each p ∈ M . Such a vector field induces a null rigging N . Letη be the 1-form which isḡ-metrically equivalent to N and η = x η its pull back on M . We introduce and study for a given non vanishing function α on M the so-called α-associated (semi-)Riemannian metric gα = g + αη ⊗ η. For a closed rigging N we give a constructive method to find an α-associated metric whose Levi-Civita connection coincides with the connection ∇ induced on M by the Levi-Civita connection ∇ of M and the null rigging N . We relate geometric objects of gα to those of g and g. As application, we show that given a null Monge hypersurface M in R n+1 q , there always exists a rigging and an α-associated metric whose Levi-Civita connection coincides with the induced connection on M .keyword: Perturbation of metric, Monge hypersurface, Null hypersurface, screen distribution, Rigging vector field, Associated Metric MSC[2010] 053C23, 53C25, 53C44, 53C50

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Null Hypersurfaces and the Rigged Metric

Gutiérrez

Olea

2022

Developments in Lorentzian Geometry

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Induced Riemannian structures on null hypersurfaces

Cited by 37 publications

References 29 publications

New Properties on Normalized Null Hypersurfaces

New Properties on Normalized Null Hypersurfaces

alpha-Associated metrics on rigged null hypersurfaces

Null Hypersurfaces and the Rigged Metric

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