共查询到20条相似文献,搜索用时 31 毫秒
1.
Amalendu Ghosh 《Results in Mathematics》2014,65(1-2):81-94
We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S 2n+1. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to E n+1 × S n (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained. 相似文献
2.
For a Lagrangian submanifold M of S 6 with nearly Kaehler structure, we provide conditions for a canonically induced almost contact metric structure on M by a unit vector field, to be Sasakian. Assuming M contact metric, we show that it is Sasakian if and only if the second fundamental form annihilates the Reeb vector field ξ, furthermore, if the Sasakian submanifold M is parallel along ξ, then it is the totally geodesic 3-sphere. We conclude with a condition that reduces the normal canonical almost contact metric structure on M to Sasakian or cosymplectic structure. 相似文献
3.
Domenico Perrone 《Mathematische Zeitschrift》2009,263(1):125-147
It is well known that a Hopf vector field on the unit sphere S
2n+1 is the Reeb vector field of a natural Sasakian structure on S
2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector
field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.
Supported by funds of the University of Lecce and M.I.U.R.(PRIN). 相似文献
4.
Domenico Perrone 《Differential Geometry and its Applications》2012,30(1):49-58
The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R×N, where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map. 相似文献
5.
If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere. 相似文献
6.
Yaning Wang 《Czechoslovak Mathematical Journal》2017,67(1):73-86
A necessary and sufficient condition for the Reeb vector field of a three dimensional non-Kenmotsu almost Kenmotsu manifold to be minimal is obtained. Using this result, we obtain some classifications of some types of (k, μ, v)-almost Kenmotsu manifolds. Also, we give some characterizations of the minimality of the Reeb vector fields of (k, μ, v)-almost Kenmotsu manifolds. In addition, we prove that the Reeb vector field of an almost Kenmotsu manifold with conformal Reeb foliation is minimal. 相似文献
7.
The aim of this paper is to study slant curves of three-dimensional normal almost contact manifolds as natural generalization of Legendre curves. Such a curve is characterized by means of the scalar product between its normal vector field and the Reeb vector field of the ambient space. In the particular case of a helix we show that it has a proper (non-harmonic) mean curvature vector field. The general expressions of the curvature and torsion of these curves and the associated Lancret invariant are computed as well as the corresponding variants for some particular cases, namely β-Sasakian and cosymplectic. A class of examples is discussed for a normal not quasi-Sasakian 3-manifold. 相似文献
8.
Amalendu Ghosh 《Mediterranean Journal of Mathematics》2013,10(2):1051-1065
We study generalized Sasakian space form M(f 1, f 2, f 3) when (i) the Reeb vector field of the almost contact metric structure is Killing, (ii) the Ricci tensor satisfies Einstein-like conditions and (iii) the fundamental 2-form of the almost contact metric structure is a twistor form. 相似文献
9.
Gianluca Bande David E. Blair Amine Hadjar 《Mediterranean Journal of Mathematics》2013,10(2):989-1009
We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-Kähler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional. 相似文献
10.
An almost contact metric structure is parametrized by a section σ of an associated homogeneous fibre bundle, and conditions
for σ to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field ξ, and the
almost complex structure in the contact subbundle. Several examples are given where the harmonic section equations for σ reduce
to those for ξ, regarded as a section of the unit tangent bundle. These include trans-Sasakian structures. On the other hand,
there are examples where ξ is harmonic but σ is not a harmonic section. Many examples arise by considering hypersurfaces of
almost Hermitian manifolds, with the induced almost contact structure, and comparing the harmonic section equations for both
structures.
相似文献
11.
Bayram Sahin 《Proceedings Mathematical Sciences》2008,118(4):573-581
We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic
map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map
under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK
manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally,
we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds. 相似文献
12.
In this paper, contact metric manifolds whose characteristic vector field ξ is a harmonic vector field are called H-contact manifolds. We show that a (2n+1)-dimensional contact metric manifold is an H-contact manifold if and only if ξ is an eigenvector of the Ricci operator (J.C. González-Dávila and L. Vanhecke [J. Geom. 72 (2001) 65–76] proved this result for n=1). Consequently, the class of H-contact manifolds is very large. Moreover, we give some application about the topology of a compact H-contact manifold. 相似文献
13.
Amalendu Ghosh 《Annals of Global Analysis and Geometry》2009,35(4):431-441
In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete
K-contact manifold admitting both the Einstein-Weyl structures W
± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove
that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f
η) are presented.
相似文献
14.
Amalendu Ghosh 《Acta Mathematica Hungarica》2010,129(4):357-367
We study holomorphically planar conformal vector fields (HPCV) on contact metric manifolds under some curvature conditions.
In particular, we have studied HPCV fields on (i) contact metric manifolds with pointwise constant ξ-sectional curvature (under this condition M is either K-contact or V is homothetic), (ii) Einstein contact metric manifolds (in this case M becomes K contact), (iii) contact metric manifolds with parallel Ricci tensor (under this condition M is either K-contact Einstein or is locally isometric to E
n+1×S
n
(4)). 相似文献
15.
We construct a self-dual geometry of quasi-Sasakian 5-manifolds. Namely, we intrinsically define the notion of contact conformally semiflat (i.e., contact self-dual or contact antiself-dual) almost contact metric manifolds and also obtain a number of results concerning contact conformally semiflat quasi-Sasakian 5-manifolds. Themost important results concerning Sasakian and cosymplectic manifolds reveal interesting relationships between the characteristics of these manifolds such as contact self-duality and constancy of the Φ-holomorphic sectional curvature, contact anti-self-duality and Ricci flatness, etc. 相似文献
16.
Izu Vaisman 《Israel Journal of Mathematics》1976,24(3-4):338-351
In the first section of this note, we discuss locally conformal symplectic manifolds, which are differentiable manifoldsV 2n endowed with a nondegenerate 2-form Ω such thatdΩ=θ ∧ Ω for some closed form θ. Examples and several geometric properties are obtained, especially for the case whendΩ ≠ 0 at every point. In the second section, we discuss the case when Ω above is the fundamental form of an (almost) Hermitian manifold, i.e. the case of the locally conformal (almost) Kähler manifolds. Characterizations of such manifolds are given. Particularly, the locally conformal Kähler manifolds are almost Hermitian manifolds for which some canonically associated connection (called the Weyl connection) is almost complex. Examples of locally conformal (almost) Kähler manifolds which are not globally conformal (almost) Kähler are given. One such example is provided by the well-known Hopf manifolds. 相似文献
17.
Thomas Leistner 《Differential Geometry and its Applications》2006,24(5):458-478
The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves. 相似文献
18.
Cs. Vincze 《Differential Geometry and its Applications》2006,24(1):1-20
As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F−1(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian. 相似文献
19.
Giovanni Catino 《Mathematische Zeitschrift》2012,271(3-4):751-756
In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n ? 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons. 相似文献
20.
Mukut Mani Tripathi Jeong-Sik Kim Seon-Bu Kim 《Proceedings Mathematical Sciences》2002,112(3):415-423
For submanifolds tangent to the structure vector field in locally conformal almost cosymplectic manifolds of pointwise constantφ-sectional curvature, we establish a basic inequality between the main intrinsic invariants of the submanifold on one side,
namely its sectional curvature and its scalar curvature; and its main extrinsic invariant on the other side, namely its squared
mean curvature. Some applications including inequalities between the intrinsic invariantδ
M
and the squared mean curvature are given. The equality cases are also discussed. 相似文献