On the classification of Killing submersions and their isometries

Manzano, José M.

doi:10.2140/pjm.2014.270.367

Cited by 36 publications

(75 citation statements)

References 18 publications

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“…A Riemannian submersion π : E → M of a 3-dimensional Riemannian manifold E over a surface M is called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field ξ (for more details, see [9,11]). Since ξ has constant norm, the fibers are geodesics in E and they form a foliation called the vertical foliation, denoted by F .…”

Section: Killing Submersionsmentioning

confidence: 99%

“…From now on, a Killing submersion will be denoted by E(G, r). The existence of a Killing submersion over a simply connected surface M, with a prescribed bundle curvature, r ∈ C ∞ (M), has been proved in [11]. Uniqueness, up to isomorphisms, is guaranteed under the assumption of simply connectedness for the total space also.…”

Section: Killing Submersionsmentioning

confidence: 99%

“…A Riemannian submersion π : E → M of a 3-dimensional Riemannian manifold E over a surface M will be called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field ξ. Killing submersions are determined by two functions on M, its Gaussian curvature G and the so called bundle curvature r (for more details, see [9,11] and Section 2) and, for this reason, we denote them by E(G, r). A remarkable family of Killing submersions are the homogeneous 3-spaces.…”

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Biharmonic constant mean curvature surfaces in Killing submersions

Montaldo

Onnis²,

Passamani³

2018

Journal of Geometry and Physics

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A 3-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part, we also classify the proper biharmonic Hopf cylinders in a Killing submersion.2000 Mathematics Subject Classification. 53C42, 58E20.1 is the curvature operator on (N, h). Equation (4) is a fourth order semi-linear elliptic system of differential equations. We also note that any harmonic map is an absolute minimum of the bienergy and, so, it is trivially biharmonic. Therefore, a general working plan is to study the existence of proper biharmonic maps, i.e. biharmonic maps which are not harmonic. We refer to [4,12] for existence results and general properties of biharmonic maps.In this paper, we study biharmonic surfaces in the total space of a Killing submersion. A Riemannian submersion π : E → M of a 3-dimensional Riemannian manifold E over a surface M will be called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field ξ. Killing submersions are determined by two functions on M, its Gaussian curvature G and the so called bundle curvature r (for more details, see [9,11] and Section 2) and, for this reason, we denote them by E(G, r). A remarkable family of Killing submersions are the homogeneous 3-spaces. In fact, with the exception of the hyperbolic 3-space H 3 (−1), simply-connected homogeneous Riemannian 3-manifolds with isometry group of dimension 4 or 6 can be represented by a 2-parameter family E(c, b), where c, b ∈ R. These E(c, b)-spaces are 3-manifolds admitting a global unit Killing vector field whose integral curves are the fibers of a certain Riemannian submersion over the simply-connected constant Gaussian curvature surface M(c) and, therefore, they determine Killing submersions π : E(c, b) → M(c) (for more details, see [6]). A local description of these examples can be given by using the so called Bianchi-Cartan-Vranceanu spaces (see Section 2). As it turns out, the E(c, b)-spaces are the only simply-connected homogeneous 3-manifolds admitting the structure of a Killing submersion (see [11]).

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Section: Killing Submersionsmentioning

confidence: 99%

Section: Killing Submersionsmentioning

confidence: 99%

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confidence: 99%

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Biharmonic constant mean curvature surfaces in Killing submersions

Montaldo

Onnis²,

Passamani³

2018

Journal of Geometry and Physics

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“…is a Killing submersion in the sense of [10,12]. Throughout this paper, we will denote by ∇ and ∇ the Levi-Civita connections on M 3 and B 2 (c), respectively.…”

Section: The Total Curvature Actionmentioning

confidence: 99%

Extremal curves of the total curvature in homogeneous 3-spaces

Barros

Ferrández

Garay

2015

Journal of Mathematical Analysis and Applications

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The space of curves which are extremal for the total curvature energy is well understood in isotropic homogeneous 3-spaces, said otherwise, spaces of constant curvature. In this paper we obtain that space of extremals in homogeneous 3-spaces whose isometry group has dimension four, that is, rotationally symmetric homogeneous 3-spaces. Most of the geometry in these spaces is governed by the existence of a unit Killing vector field, ξ, sometimes called the Reeb vector field, which turns the homogeneous 3-space into the source of a Riemannian submersion whose target space is a surface with constant curvature. Here, we show that a curve is an extremal of the total curvature energy if and only if ξ lies into either the rectifying plane or the osculating plane along that curve. Then, we prove that every rotationally symmetric homogeneous 3-space, except H 2 × R, admits a real one-parameter class of extremals with horizontal normal (Lancret helices). The whole family of extremals is completed with a second class made up of those curves with horizontal binormal. In contrast with the first class, it appears in any rotationally symmetric space, with no exception, and it can be modulated in the space of real valued functions. We also work out geometric algorithms to solve the so called solving natural equations for extremals problem, allowing us to determine them explicitly in many cases. In addition, we solve the closed curve problem by showing the existence of two families of closed extremals. Namely, a rational one-parameter class of closed Lancret helices that appears at any rotationally symmetric homogeneous 3-space, except in H 2 × S 1 , and a second class of extremals with horizontal binormal, which can be identified with the class of convex closed curves in the Euclidean plane. We also present a quantization principle, à la Dirac, for extremal values of the total curvature energy acting on closed curves in any rotationally symmetric homogeneous 3-space.

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“…The spaces E(κ, τ) are characterized by admitting a Riemannian submersion π : E(κ, τ) → M 2 (κ) with constant bundle curvature τ, where M 2 (κ) stands for the simply-connected Riemannian surface with constant curvature κ, such that the fibers of π are the integral curves of a unit Killing vector field in E(κ, τ) (see [Ma12]). In what follows, we will refer to this field as the vertical Killing vector field and it will be denoted by ξ.…”

Section: Introductionmentioning

confidence: 99%

On Complete Constant Mean Curvature Vertical Multigraphs in $\mathbb{E}(\kappa,\tau)$

Manzano

Rodríguez

2013

J Geom Anal

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Abstract. We prove that any complete surface with constant mean curvature in a homogeneous space E(κ, τ) which is transversal to the vertical Killing vector field is, in fact, a vertical graph. As a consequence we get that any orientable, parabolic, complete, immersed surface with constant mean curvature H in E(κ, τ) (different from a horizontal slice in S 2 × R) is either a vertical cylinder or a vertical graph (in both cases, it must be 4H 2 + κ ≤ 0).

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On the classification of Killing submersions and their isometries

Cited by 36 publications

References 18 publications

Biharmonic constant mean curvature surfaces in Killing submersions

Biharmonic constant mean curvature surfaces in Killing submersions

Extremal curves of the total curvature in homogeneous 3-spaces

On Complete Constant Mean Curvature Vertical Multigraphs in $\mathbb{E}(\kappa,\tau)$

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