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1
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A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors

100%
Wang Y.
Open Mathematics
|
2016
|
tom 14
|
nr 1
977-985
EN
Let M3 be a three-dimensional almost Kenmotsu manifold satisfying ▽ξh = 0. In this paper, we prove that the curvature tensor of M3 is harmonic if and only if M3 is locally isometric to either the hyperbolic space ℍ3(-1) or the Riemannian product ℍ2(−4) × ℝ. This generalizes a recent result obtained by [Wang Y., Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math., 2016, 116, 79-86] and [Cho J.T., Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J., 2016, 45, 435-442].
2
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Ricci solitons on almost Kenmotsu 3-manifolds

100%
Wang Y.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1236-1243
EN
Let (M3, g) be an almost Kenmotsu 3-manifold such that the Reeb vector field is an eigenvector field of the Ricci operator. In this paper, we prove that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either the hyperbolic space ℍ3(−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. In particular, when g represents a gradient Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either ℍ3(−1) or ℍ2(−4) × ℝ.
3
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Weighted minimal translation surfaces in the Galilean space with density

100%
Yoon D. W.
Open Mathematics
|
2017
|
tom 15
|
nr 1
459-466
EN
Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.
4
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On compact holomorphically pseudosymmetric Kählerian manifolds

100%
Olszak Z.
Open Mathematics
|
2009
|
tom 7
|
nr 3
442-451
EN
For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric Kählerian manifolds are discovered in [4]. In these examples, the structure functions change their signs on the manifold.
5
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The Cotton Tensor and the Ricci Flow

100%
Mantegazza C., Mongodi S., Rimoldi M.
Geometric Flows
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2017
|
tom 2
|
nr 1
EN
We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.
6
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Smooth metric measure spaces, quasi-Einstein metrics, and tractors

76%
Case J.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1733-1762
EN
We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.
7
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Toric extremal Kähler-Ricci solitons are Kähler-Einstein

76%
Calamai S., Petrecca D.
Complex Manifolds
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2017
|
tom 4
|
nr 1
179-182
EN
In this short note, we prove that a Calabi extremal Kähler-Ricci soliton on a compact toric Kähler manifold is Einstein. This settles for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.
8
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Einstein-Weyl structures on lightlike hypersurfaces

76%
Atindogbe C., Bérard-Bergery L., Ogouyandjou C.
Open Mathematics
|
2013
|
tom 11
|
nr 10
1850-1862
EN
We study Weyl structures on lightlike hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambient Lorentzian space ℝ1n+2 and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves.
9
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Erratum to: “Smooth metric measure spaces, quasi-Einstein metrics, and tractors”

76%
Case J.
Open Mathematics
|
2013
|
tom 11
|
nr 1
196-196
EN
The original version of the article was published in Central European Journal of Mathematics, 2012, 10(5), 1733–1762, DOI: 10.2478/s11533-012-0091-x. Unfortunately, it contains a typographical error in display of (27). Here we display the correct version of the equations.
10
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Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections

64%
Keller J., Tønnesen-Friedman C.
Open Mathematics
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2012
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tom 10
|
nr 5
1673-1687
EN
We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.
11
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On the characteristic connection of gwistor space

64%
Albuquerque R.
Open Mathematics
|
2013
|
tom 11
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nr 1
149-160
EN
We give a brief presentation of gwistor spaces, which is a new concept from G 2 geometry. Then we compute the characteristic torsion T c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T c is ∇c-parallel; this allows for the classification of the G 2 structure with torsion and the characteristic holonomy according to known references. The case of an Einstein base manifold is envisaged.
12
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A spectral estimate for the Dirac operator on Riemannian flows

64%
Ginoux N., Habib G.
Open Mathematics
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2010
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tom 8
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nr 5
950-965
EN
We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.
13
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Curvature properties of φ-null Osserman Lorentzian S-manifolds

64%
Brunetti L., Caldarella A.
Open Mathematics
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2014
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tom 12
|
nr 1
97-113
EN
We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian S-manifold and the Jacobi operators with respect to particular spacelike unit vectors. We study the number of the eigenvalues of such operators on Lorentzian S-manifolds satisfying the φ-null Osserman condition, under suitable assumptions on the dimension of the manifold. Then, we provide in full generality a new curvature characterization for Lorentzian S-manifolds and we use it to obtain an algebraic decomposition for the Riemannian curvature tensor of φ-null Osserman Lorentzian S-manifolds.
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