Contact metric manifolds satisfying a nullity condition

doi:10.1007/BF02761646

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Contact metric manifolds satisfying a nullity condition

Journal Article•DOI•

Contact metric manifolds satisfying a nullity condition

David E. Blair¹, Themis Koufogiorgos², Basil J. Papantoniou³•Institutions (3)

Michigan State University¹, University of Ioannina², University of Patras³

01 Oct 1995-Israel Journal of Mathematics (Springer-Verlag)-Vol. 91, Iss: 1, pp 189-214

TL;DR: In this article, a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below, is presented.

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Abstract: This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric.

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Journal Article•DOI•

Generalized Sasakian-space-forms

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Pablo Alegre¹, David E. Blair², Alfonso Carriazo¹•Institutions (2)

University of Seville¹, Michigan State University²

01 Dec 2004-Israel Journal of Mathematics

TL;DR: In this paper, generalized Sasakian-space-forms are introduced and studied, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations.

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Abstract: Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New results on generalized complex-space-forms are also obtained.

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202 citations

Journal Article•DOI•

Certain Results on K-Contact and (k, μ)-Contact Manifolds

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Ramesh Sharma¹•Institutions (1)

University of New Haven¹

22 Sep 2008-Journal of Geometry

TL;DR: In this article, Boyer and Galicki showed that a complete K-contact gradient soliton is a Jacobi vector field along the geodesics of the Reeb vector field.

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Abstract: Inspired by a result of Boyer and Galicki, we prove that a complete K-contact gradient soliton is compact Einstein and Sasakian. For the non-gradient case we show that the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Next we show that among all complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field (HPCV). Finally we show that, if a (k, μ)-contact manifold admits a non-zero HPCV, then it is either Sasakian or locally isometric to E3 or En+1 × Sn (4).

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157 citations

Cites methods from "Contact metric manifolds satisfying..."

...(∇X h)Y = (1 − k)g(X, ϕY )+ g(X, hϕY ) ξ + η(Y )h(ϕX + ϕhX) − μη(X)ϕhY (13) which occur in [ 2 ]....
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...Finally, we consider a class of contact metric manifolds known as (k, μ)-contact manifolds (which were introduced by Blair, Koufogiorgos and Papantoniou [ 2 ], 142 R. Sharma J. Geom....
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Posted Content•

Contact geometry

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Hansjörg Geiges

17 Jul 2003

TL;DR: In this article, an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol 2, is presented, along with a detailed exposition of the original proof of the Lutz-Martinet theorem.

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Abstract: This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol 2 After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a detailed exposition of the original proof of the Lutz-Martinet theorem The text ends with a guide to the literature

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106 citations

Journal Article•DOI•

A survey on cosymplectic geometry

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Beniamino Cappelletti-Montano¹, Antonio De Nicola², Ivan Yudin²•Institutions (2)

University of Cagliari¹, University of Coimbra²

16 May 2013-arXiv: Differential Geometry

TL;DR: An up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds is given in this paper, where the authors also mention some of their applications to time-dependent mechanics.

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Abstract: We give an up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds. We also mention some of their applications to time-dependent mechanics.

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91 citations

Additional excerpts

...eal numbers κand µone can deﬁne a distribution N(κ,µ) on Mby Nx(κ,µ) := {Z∈TxM|R(X,Y)Z= κ(g(Y,Z)X−g(X,Z)Y) +µ(g(Y,Z)hX−g(X,Z)hY)}. The distribution N(κ,µ) is called the (κ,µ)-nullity distribution. In [17], the case when the Reeb vector ﬁeld of a contact metric manifold belongs to the (κ,µ)-nullity distribution was considered. A few years later, the almost coK¨ahler case was also considered ([69]). Thu...
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Journal Article•DOI•

Structures on generalized Sasakian-space-forms

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Pablo Alegre¹, Alfonso Carriazo¹•Institutions (1)

University of Seville¹

01 Dec 2008-Differential Geometry and Its Applications

TL;DR: In this paper, contact metric and trans-Sasakian generalized Sasakian-space-forms are deeply studied and general results for manifolds with dimension greater than or equal to 5 are presented.

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Abstract: In this paper, contact metric and trans-Sasakian generalized Sasakian-space-forms are deeply studied. We present some general results for manifolds with dimension greater than or equal to 5, and we also pay a special attention to the 3-dimensional cases.

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86 citations

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References

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Journal Article•DOI•

Curvatures of left invariant metrics on lie groups

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John Milnor¹•Institutions (1)

Institute for Advanced Study¹

01 Sep 1976-Advances in Mathematics

TL;DR: In this paper, the author outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a metric invariant under left translation.

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1,403 citations

Book•

Contact manifolds in Riemannian geometry

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David E. Blair

01 Jan 1976

TL;DR: In this paper, the tangent sphere bundle is shown to be a contact manifold, and the contact condition is interpreted in terms of contact condition and k-contact and sasakian structures.

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Abstract: Contact manifolds.- Almost contact manifolds.- Geometric interpretation of the contact condition.- K-contact and sasakian structures.- Sasakian space forms.- Non-existence of flat contact metric structures.- The tangent sphere bundle.

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1,259 citations

"Contact metric manifolds satisfying..." refers background in this paper

...tangent sphere bundle of a flat Riemannian manifold admits such a structure [ 2 ]....
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...of a manifold M is described in Chapter VII of [ 2 ] and in [5]....
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...,n be a local orthonormal Q-basis (see [ 2 ], p.22)....
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...For more details concerning contact manifolds and related topics we refer the reader to [ 2 ]....
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Journal Article•DOI•

The topology of contact Riemannian manifolds

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Shûkichi Tanno¹•Institutions (1)

Tohoku University¹

01 Dec 1968-Illinois Journal of Mathematics

217 citations

"Contact metric manifolds satisfying..." refers background or methods in this paper

...Applying a D-homothetic deformation [11] to a contact metric manifold with R(X, Y)~ = 0 we obtain a contact metric manifold satisfying ( ,) R (X , Y)~ = ~(~(Y)X - ,](X)Y) + # (~ (Y )hX - ~ (X)hY) where ~, p are constants and 2h is the Lie derivative of ~ in the direction ~....
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...By a D a - h o m o t h e t i c d e f o r m a t i o n [11] we mean a change of structure tensors of the form (3....
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...[11] S....
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Journal Article•DOI•

Curvature of the Induced Riemannian Metric on the Tangent Bundle of a Riemannian Manifold.

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Oldrich Kowalski

01 Sep 1971-Crelle's Journal

TL;DR: In this paper, the curvature tensor of the Riemannian space (TM, Tg) corresponding to (If, g) is computed, and it is shown that the space is not Symmetrie unless (M, g, tg) is locally euclidean.

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Abstract: If M is an ra-dimensional differentiable manifold (n £ N) with the Riemannian metric g, then the tangent b ndle TM of M admits a canonical Riemannian metric Tg (see [1], [2]). In other words, a metric connection v on M induces, in a canonical way, a metric connection v on TM. Further, A. J. Ledger and K. Yano ([3], [4]) found a different construction joining to any linear connection v on M a linear connection v on TM. The basic result of [4] says that the space (M, v) is locally Symmetrie if and only if the space (T M, v) is locally Symmetrie, In the present paper we compute the curvature tensor of the Riemannian space (TM, Tg) corresponding to (If, g). We deduce, in contrast to the Yano-Ledger's theory, the following result: The space (TM, Tg) is never locally Symmetrie unless (M, g) is locally euclidean.

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167 citations

Journal Article•DOI•

Ricci curvatures of contact riemannian manifolds

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Shukichi Tanno¹•Institutions (1)

Tokyo Institute of Technology¹

01 Jan 1988-Tohoku Mathematical Journal

TL;DR: In this paper, a variete de Riemann de contact de courbure φ-sectionnelle constante H.Riemann et al. satisfait Ric(X,X)+Ric(φX, φX)≤3n−1+(n+1)H pour chaque vecteur unite X∈T x M x∈M, tels que η(X)=0.

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Abstract: Soit (M, η,g) une variete de Riemann de contact de courbure φ-sectionnelle constante H. Alors les courbures de Ricci satisfont Ric(X,X)+Ric(φX, φX)≤3n−1+(n+1)H pour chaque vecteur unite X∈T x M x∈M, tels que η(X)=0. L'egalite est vraie pour tout x∈M et pour un vecteur unite X∈T x M tel que η(X)=0, si et seulement si (M, η, g) est sasakienne

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156 citations

"Contact metric manifolds satisfying..." refers background or result in this paper

...Proof: The proof of this lemma is similar to that of Proposition 5.1 of Tanno's paper [ 13 ] and hence we omit it....
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...is also well known ([10] or [ 13 ]) that a contact metric manifold with R(X, Y)~ = 0 satisfies...
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