Paracontact metric manifolds without a contact metric counterpart

TAIWANESE JOURNAL OF MATHEMATICS Vol. 19, No. 1, pp. 175-191, February 2015 DOI: 10.11650/tjm.19.2015.4447 This paper is available online at http://journal.taiwanmathsoc.org.tw PARACONTACT METRIC MANIFOLDS WITHOUT A CONTACT METRIC COUNTERPART Ver ´ onica Mart´ın-Molina Abstract. We study non-paraSasakian paracontact metric (κ, μ)-spaces with κ= −1(equivalent to h2=0but h=0). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of h. We will also give explicit examples of every possible constant rank of h. 1. INTRODUCTION A remarkable class of paracontact metric manifolds (M, φ, ξ, η,g)is that of paracontact metric (κ, μ)-spaces, which satisfy the nullity condition (1.1) R(X, Y )ξ=κ(η(Y)X−η(X)Y)+μ(η(Y)hX −η(X)hY ), for all X, Y vector fields on M,whereκand μare constants and h=1 2Lξϕ. This definition, which may appear quite technical, arises from the deep and meaningful relationship between contact metric (κ, μ)-spaces and paracontact geometry. More precisely, it was proved in [7] that any non-Sasakian contact metric (κ, μ)- space accepts two paracontact metric (κ, μ)-structures with the same contact form. On the other hand, under certain natural conditions, every non-paraSasakian paracontact (κ, μ)-space carries a contact metric (κ, μ)-structure compatible with the same contact form ([8]). The class of paracontact metric (κ, μ)-spaces includes the paraSasakian ones (see [12] and [15]) and the ones satisfying R(X, Y )ξ=0for all X, Y (studied in [16]), among others. There are some notable differences between a contact metric (κ, μ)-space (M, φ, ξ, η,g)and a paracontact metric (κ, μ)-space ( M,  φ,  ξ, η, g). First of all, while they Received February 18, 2014, accepted May 15, 2014. Communicated by Bang-Yen Chen. 2010 Mathematics Subject Classification: 53C15, 53C25, 53C50. Key words and phrases: Paracontact metric manifold, ParaSasakian, Nullity distribution, (κ, μ)-spaces. The author is is partially supported by the PAI group FQM-327 (Junta de Andaluc´ıa, Spain), the group Geometr´ıa E15 (Gobierno de Arag´ on, Spain), the MINECO grant MTM2011-22621 and the “Centro Universitario de la Defensa de Zaragoza” grant ID2013-15. 175 176 Ver ´ onica Mart´ın-Molina satisfy h2=(κ−1)ϕ2and  h2=(κ+1)ϕ2, respectively, the first condition means that κ≤1but the second one does not give any type of restriction for κbecause the metric of a paracontact metric manifold is not positive definite (see [2] for the contact metric case and [8] for the paracontact metric one). Another difference is that, in the contact metric case, κ=1(i.e. h2=0)is also equivalent to the manifold being Sasakian (and thus h=0). However, there are paracontact metric (κ, μ)-spaces with κ=−1(and thus  h2=0) but  h=0. The first example of paracontact metric (−1,2)-space (M2n+1, φ,  ξ, η,g)with  h=0was given in [5] for n=2. Later, an example with arbitrary nappeared in [8] (constructed by deforming the contact metric structure defined on the unit tangent sphere bundle) and a numerical example with n=1was shown in [13]. It is worth mentioning that all these spaces have μ=2and rank( h)=n. Lastly, an example of 3-dimensional paracontact metric (−1,0)-space with  h=0appeared in [10]. To our knowledge, no effort has been made to better understand the general behaviour of the tensor hof a paracontact metric (κ, μ)-space when h2=0but h=0, which we will address in Theorem 3.2. We will study the form of the tensor hin this remarkable situation and we will later construct explicit examples that illustrate all the possible constant values of the rank of h(from 1to n)whenμ=2. Finally, we will discuss the situation when μ=2and show some paracontact metric (−1,0)-spaces with h=0in Examples 3.8-3.11. These are the first examples of this type with μ=2 and dimension greater than 3. 2. PRELIMINARIES An almost paracontact structure on a (2n+1)-dimensional smooth manifold Mis given by a (1,1)-tensor field ϕ, a vector field ξand a 1-form ηsatisfying the following conditions [12]: (i) η(ξ)=1,ϕ2=I−η⊗ξ, (ii) the eigendistributions D+and D−of ϕcorresponding to the eigenvalues 1and −1, respectively, have equal dimension n. As an immediate consequence, ϕξ =0,η◦ϕ=0and the tensor ϕhas constant rank 2n. If an almost paracontact manifold is endowed with a semi-Riemannian metric gsuch that g(ϕX, ϕY )=−g(X, Y)+η(X)η(Y), for all X, Y on M,then(M, ϕ, ξ, η, g)is called an almost paracontact metric manifold. Note that such a semi-Riemannian metric is necessarily of signature (n+1,n)and the above condition (ii) of the definition of almost paracontact structures is automatically satisfied. Moreover, it follows easily that η=g(·,ξ)and g(·,ϕ·)=−g(ϕ·,·).We can now define the fundamental 2-form of the almost paracontact metric manifold by Paracontact Metric Manifolds 177 Φ(X, Y )=g(X, ϕY ).Ifdη =Φ,thenηbecomes a contact form (i.e. η∧(dη)n=0) and (M, ϕ, ξ, η, g)is said to be a paracontact metric manifold. We can also define on a paracontact metric manifold the tensor field h:= 1 2Lξϕ, which is a symmetric operator with respect to g, anti-commutes with ϕand satisfies hξ =trh=0and the identity ∇ξ=−ϕ+ϕh ([15]). Moreover, it vanishes identically if and only if ξis a Killing vector field, in which case (M, ϕ, ξ, η, g)is called a K-paracontact manifold. An almost paracontact structure is said to be normal if and only if the tensor Nϕ−2dη⊗ξvanishes identically, where Nϕis the Nijenhuis tensor of ϕ:Nϕ(X, Y )= [ϕ, ϕ](X,Y)=ϕ2[X, Y ]+[ϕX, ϕY ]−ϕ[ϕX, Y ]−ϕ[X, ϕY ]([15]). A normal paracontact metric manifold is said to be a paraSasakian manifold and satisfies (2.1) R(X, Y )ξ=−(η(Y)X−η(X)Y), for every X, Y on M. Unlike in the contact metric case, the condition (2.1) does not imply that the manifold is paraSasakian, as will be seen in Examples 3.8-3.11. It was also proved in [15] that an almost paracontact manifold is paraSasakian if and only if (2.2) (∇Xϕ)Y=−g(X, Y)ξ+η(Y)X, so, in particular, every paraSasakian manifold is K-paracontact. The converse holds in dimension 3([9]) and for (−1,μ)-spaces (which will be proved in Theorem 3.1) but we can construct explicit examples that show that these two concepts are not equivalent in general. Example 2.1. Let gbe the 5-dimensional Lie algebra with basis {ξ,X1,Y 1,X 2,Y 2} such that the only non-vanishing Lie brackets are [X1,Y 1]=2ξ, [X2,Y 2]=2ξ, [X1,X 2]=Y1. If we denote by Gthe Lie group whose Lie algebra is g, we can define a left-invariant paracontact metric structure on Gthe following way: ϕξ =0,ϕX i=Xi,ϕY i=−Yi,η(ξ)=1,η(Xi)=η(Yi)=0,i=1,2, g(ξ,ξ)=g(X1,Y 1)=g(X2,Y 2)=1, g(ξ,Xi)=g(ξ, Yi)=g(Xi,X j)=g(Yi,Y j)=0,i=1,2. A simple computation gives that h=0, so the manifold is K-paracontact. However, although R(X, ξ)ξ=−Xfor all vector field Xorthogonal to ξ, straightforward computations give that R(X1,X 2)ξ=−2Y1=0, so the manifold does not satisfy (2.1). In particular, it is not paraSasakian. 178 Ver ´ onica Mart´ın-Molina Finally, we recall that a notion of Dc-homothetic deformation can be introduced in paracontact metric geometry ([15]). Indeed, given a non-zero constant c,aDchomothetic deformation on a paracontact metric manifold (M, ϕ, ξ, η, g)is the following change of the structure tensors: (2.3) ϕ:= ϕ, ξ:= 1 cξ, η:= cη, g:= cg +c(c−1)η⊗η. Then (ϕ,ξ,η,g)is again a paracontact metric structure on M. Moreover, Kparacontact and paraSasakian structures are also preserved under Dc-homothetic deformations. Although Dc-homothetic deformations destroy curvature conditions like R(X,Y )ξ= 0,aDc-homothetically deformed paracontact metric (κ, μ)-space is another paracontact metric (κ,μ )-space with (2.4) κ=κ+1−c2 c2,μ =μ−2+2c c. 3. PARACONTACT METRIC (−1,μ)-SPACES We now present our main results. Theorem 3.1. Let Mbe a paracontact metric manifold such that R(X, Y )ξ= −(η(Y)X−η(X)Y), for all X, Y on M.ThenMis paraSasakian if and only if ξis a Killing vector field. Proof. If Mis paraSasakian, it is in particular a K-paracontact manifold, hence ξis a Killing vector field. Conversely, if ξis a Killing vector field, then h=0and ∇Xξ=−ϕX holds for every Xon M([15]). We also know from [14, p.259] that R(ξ,X)Y=−∇X∇Yξ+∇∇XYξ, thus R(ξ,X)Y=(∇Xϕ)Y,forallX, Y on M. Therefore, it follows from the previous formula and from R(X, Y )ξ=−(η(Y)X− η(X)Y)that g((∇Xϕ)Y,Z)=g(R(ξ, X)Y,Z)=g(R(Y,Z)ξ, X)=g(−g(X, Y)ξ+η(Y)X, Z), hence equation (2.2) holds and the manifold is paraSasakian. Theorem 3.2. Let Mbe a (2n+1)-dimensional paracontact metric (−1,μ)-space. Then we have one of the following possibilities: (1) h=0and Mis paraSasakian Paracontact Metric Manifolds 179 (2) h=0and rank(hp)∈{1,...,n}at every p∈Mwhere hp=0. Moreover, there exists a basis {ξ,X1,Y 1,...,X n,Y n}of Tp(M)such that •The only non-vanishing components of gare gp(ξ,ξ)=1,g p(Xi,Y i)=±1, •and h|Xi,Yi=00 10 or h|Xi,Yi=00 00 , where obviously there are exactly rank(hp)submatrices of the first type. If n=1, such a basis {ξ,X1,Y 1}also satisfies that ϕX1=±X1,ϕY 1=∓Y1, and the tensor hcan be written as (3.1) h|ξ,Xi,Yi=⎛ ⎝ 000 000 010 ⎞ ⎠. Proof. We know from Lemma 3.2 of [8] that h2=0.Wehavenowtwo possibilities. If h=0,thenR(X, Y )ξ=−(η(Y)X−η(X)Y),forallX, Y on Mand ξis a Killing vector field. Therefore, it follows from Theorem 3.1 that the manifold is paraSasakian. Let us now suppose that h=0.Sincehis self-adjoint and Ker(η)is h-invariant, we have from [14, p.260] that, at each point p∈M,Ker(ηp)=V1⊕···⊕Vl(for some 1≤l≤2n), where Vkare mutually orthogonal subspaces that are h-invariant and on which h|Vkhas a matrix of either type: (3.2) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ λ 1λ0 1λ ...... 01λ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ relative to a basis X1,...,X rof Vk,r≥1, such that the only non-zero products are 180 Ver ´ onica Mart´ın-Molina gp(Xi,X j)=±1if i+j=r+1,oroftype (3.3) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ab −ba 0 10ab 01−ba 10ab 01−ba ...... 0 10ab 01−ba ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (b=0) relative to a basis X1,Y 1,...,X m,Y mof Vk, such that the only non-zero products are gp(Xi,X j)=1=−gp(Yi,Y j)if i+j=m+1. We will first see that the second case is not possible. Indeed, if there existed a subspace Vksuch that h|Vkhad a matrix of type (3.3), then h2 pX1=(a2−b2)X1− 2abY1+2aX2−2bY2+X3=0, which cannot happen for any value of mbecause b=0. If there exists a subspace Vksuch that h|Vkhas a matrix of type (3.2), then h2 pX1= λ2X1+2λX2+X3=0, which is only possible if λ=0and dim Vk=r≤2.Letus distinguish between both subcases: (1) If dim Vi=2,thenVi=Xi,Y i, the only non-zero product is gp(Xi,Y i)=±1 and h|Xi,Yi=00 10 . (2) If dim Vi=1,thenVi=Xiand hpXi=0, with Xia vector satisfying gp(Xi,X i)=±1. In fact, since Ker(ηp)is of signature (n, n)and the subspaces Viof the subcase (1) are of dimension two and signature (1,1), then there is an even number of subspaces of this type, half satisfying gp(Xi,X i)=1and half satisfying gp(Xi,X i)=−1. Taking one of each type, for example V1=X1 and V2=X2with gp(X1,X 1)=1=−gp(X2,X 2), a simple change of basis like  X1=1 √2(X1+X2)and  Y1=1 √2(X1−X2)would give us a basis such that the only non-vanishing component of the metric is gp( X1, Y1)=1. Finally, since h|Vi=0in the second subcase, the rank of hpdepends on the number of submatrixes of the first type, which have rank 1. Therefore, rank(hp)∈{1,...,n} at every point where hp=0. Paracontact Metric Manifolds 181 In dimension 3, the conditions over ϕare the only ones that remain to be proved. First of all, ϕξ =0on every paracontact metric structure. Hence ϕX1=aX1+bY1, for some constants a, b. It follows from gp(X1,ϕX 1)=dηp(X1,X 1)=0that b=0,soϕX1=aX1.On the other hand, ϕY1=ϕhX1=−hϕX1=−ahX1=−aY1, hence ϕY1=−aY1.We can then compute: gp(ϕX1,ϕY 1)=−gp(X1,Y 1)=∓1, gp(ϕX1,ϕY 1)=gp(aX1,−aY1)=−a2gp(X1,Y 1)=∓a2, so a2=1, and thus ϕX1=±X1,ϕY 1=∓Y1, which ends the proof. Remark 3.3. A paracontact metric manifold satisfying (3.4) R(X, Y )ξ=−(η(Y)X−η(X)Y), for all X, Y on M, can be either a (−1,μ)-space with h=0(thus paraSasakian by the previous Theorem and μis undetermined) or a paracontact metric (−1,0)-space with h=0(not K-paracontact or paraSasakian). This last case is possible because (3.4) does not imply h=0, as can be seen in Examples 3.8–3.11. Let us now see examples of all the possible constant ranks of hthat appear in Theorem 3.2. First of all, if h=0, the standard examples of paraSasakian manifolds are the hyperboloids H2n+1 n+1 (1) = (x0,y 0,...,x n,y n)∈R2n+2 |x2 0+...+x2 n−y2 0−...−y2 n=1  and the hyperbolic Heisenberg group H2n+1 =R2n×Rwith the structures defined in [11]. Other examples of (η-Einstein) paraSasakian manifolds can be obtained from contact (κ, μ)-spaces with |1−μ 2|<√1−κ, as seen in Theorem 3.4 of [6]. In particular, it was shown that the tangent sphere bundle T1Nof any space form N(c) with c<0admits a canonical η-Einstein paraSasakian structure. We can also construct paraSasakian examples by defining a paracontact metric structure on a Lie group: Example 3.4. (Canonical paraSasakian structure on the Heisenberg algebra). Let gbe the (2n+1)-dimensional Lie algebra with basis {ξ, X1,...,X 2n}such that the only non-vanishing Lie brackets are [X2i−1,X 2i]=2ξ, i =1,...,n. 182 Ver ´ onica Mart´ın-Molina If we denote by Gthe Lie group whose Lie algebra is g, we can define a left-invariant paracontact metric structure on Gthe following way: ϕξ =0,ϕX 2i−1=X2i,ϕX 2i=X2i−1,i=1,...,n, η(ξ)=1,η(Xi)=0,i=1,...,2n, the only non-vanishing components of the metric are g(ξ,ξ)=g(X2i,X 2i)=1,g(X2i−1,X 2i−1)=−1,i=1,...,n. A straightforward computation gives that h=0. Moreover, using properties of paracontact metric manifolds and Koszul’s formula, we obtain that ∇X2iξ=−X2i−1,∇X2i−1ξ=−X2i,i=1,...,n, ∇X2iX2i=∇X2i−1X2i−1=0,∇X2i−1X2j=−∇X2iX2j−1=δijξ, i =1,...,n. Therefore, R(Xi,ξ)ξ=−Xi,i=1,...,2n, R(Xi,X j)ξ=0,i,j=1,...,2n. We conclude that R(X, Y )ξ=−(η(Y)X−η(X)Y)for all X, Y on M, thus the manifold is paraSasakian because of Theorem 3.1. Alternatively, we could check the normality condition by proving that [ϕ, ϕ](X, Y)−2dη(X, Y )ξ=0for all vector fields X, Y on M. If we apply a Dc-homothetic deformation to any of the previous paraSasakian examples, we obtain again paraSasakian manifolds ([15]). Let us now construct some examples of paracontact metric (−1,μ)-spaces with h=0. We will begin by adapting some paracontact metric (−1,2)-spaces with h=0 that appear in the literature and will afterwards construct explicitly (2n+1)-dimensional paracontact metric (−1,2)-spaces with h=0such that rank of hattains all values in {1,...,n}. Lastly, the case μ=2will be discussed. Example 3.5. ((2n+1)-dimensional paracontact metric (−1,2)-space, rank(h)= n). We will begin with the examples that were presented in [8]. Let us take a flat Riemannian manifold Mand construct on it the tangent sphere bundle T1Mwith its standard contact metric structure (ϕ, ξ, η, g), which satisfies R(X, Y )ξ=0for every X, Y on T1M(see [2]). Then we can define a new structure by taking ϕ2=h, g2=dη(·,h·)+η⊗η, Paracontact Metric Manifolds 183 which is a paracontact metric (−1,2)-space (Theorem 3.4 of [8]). We will now see the form that h2has on these examples. Let us first take a ϕ-basis {ξ, X1,ϕX 1,...,X n,ϕX n}of the contact metric (κ, μ)- space such that hXi=Xiand hϕXi=−ϕXi(which exists because of [2]). Then {ξ,  X1, Y1,...,  Xn, Yn},where  Xi:= 1 √2Xi, Yi:= √2ϕXi,isabasisforwhichthe only non-vanishing components of the metric g2are g2(ξ,ξ)=1,g 2( Xi, Yi)=1, the tensor ϕ2satisfies ϕ2 Xi= Xi,ϕ 2 Yi=− Yi, and h2| Xi, Yi=00 10 , for all i=1,...,n. Therefore, h2= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 00 10 ... (n) 00 10 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , so rank(h2)=n. In dimension 3, we also have Example 6.2 from [13], which is a 3-dimensional paracontact metric (−1,2)-space with h=0satisfying (3.1). On the other hand, we can give examples using left-invariant paracontact metric structures on Lie groups of (2n+1)-dimensional paracontact metric (−1,2)-spaces with h=0and rank(h)=m∈{1,...,n}. Example 3.6. ((2n+1)-dimensional paracontact metric (−1,2)-space with rank(h) =m∈{1,...,n}). Let gbe the (2n+1)-dimensional Lie algebra with basis {ξ,X1,Y 1,...,X n,Y n}such that the only non-vanishing components are [ξ,Xi]=Yi,[Xi,Y j] =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ δij (2ξ+√2(1 + δim)Ym)+(1−δij )√2(δimYj+δjmYi),i,j=1,...,m, δij (2ξ+√2Yi),i,j=m+1,...,n, √2Yii=1,...,m, j =m+1,...,n. 190 Ver ´ onica Mart´ın-Molina Example 4.1. Let us take the 5-dimensional Lie algebra gof basis {ξ, X1,Y 1,X 2, Y2}such that the only non-vanishing Lie brackets are [ξ,X1]=Y1,[X1,Y 1]=2ξ, [X2,Y 2]=2(ξ+X2),[X1,X 2]=Y1. If we denote by Gthe Lie group whose Lie algebra is g, we can define on it a leftinvariant paracontact metric structure: ϕξ =0,ϕX i=Xi,ϕY i=−Yi,η(ξ)=1,η(Xi)=η(Yi)=0,i=1,2. The only non-vanishing components of the metric are g(ξ,ξ)=g(X1,Y 1)=g(X2,Y 2)=1. Therefore, hX1=Y1and hY1=hX2=hY2=0,soh2=0but h=0. Although R(X, ξ)ξ=−X+2hX for all vector field Xorthogonal to ξ, we can check that R(X1,X 2)ξ=−2Y1=0, so the manifold is not a (−1,2)-space. Remark 4.2. Note that the previous Lie algebra coincides with the one of Example 3.7 for n=2(hence the form of hcoincides) but that the construction of the paracontact metric structure is not the same, since both ϕand gare defined differently. ACKNOWLEDGMENTS The author would like to thank Prof. Mart´ın Avenda˜ no for his valuable suggestions and insights and the referee for his great knowledge and expertise on the field of paracontact geometry. REFERENCES 1. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Second Edition, Progress in Mathematics 203,Birkh ¨ auser, Boston, 2010. 2. D. E. Blair, T. Koufogiorgos and B. J. 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Ver ´ onica Mart´ın-Molina Centro Universitario de la Defensa Academia General Militar Ctra. de Huesca s/n, 50090 Zaragoza and I.U.M.A, Universidad de Zaragoza Spain E-mail: [email protected]