Local classification and examples of an important class of paracontact metric manifolds

arXiv:1408.6784v1 [math.DG] 28 Aug 2014 LOCAL CLASSIFICATION AND EXAMPLES OF AN IMPORTANT CLASS OF PARACONTACT METRIC MANIFOLDS VER ´ ONICA MART´ IN-MOLINA Abstract. We study paracontact metric (κ, µ)-spaces with κ=−1, equivalent to h2= 0 but not h= 0. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric (−1,2)-spaces and (−1,0)-spaces of arbitrary dimension with tensor hof every possible constant rank. We will also show explicit examples of paracontact metric (−1, µ)- spaces with tensor hof non-constant rank, which were not known to exist until now. 1. Introduction Paracontact metric manifolds, the odd-dimensional analogue of paraHermitian manifolds, were first introduced in [10] and they have been the object of intense study recently, particularly since the publication of [14]. An important class among paracontact metric manifolds is that of the (κ, µ)-spaces, which satisfy the nullity condition [5] (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, where κand µare constants and h=1 2Lξϕ. This class includes the paraSasakian manifolds [10, 14], the paracontact metric manifolds satisfying R(X, Y )ξ= 0 for all X, Y [15], certain g-natural paracontact metric structures constructed on unit tangent sphere bundles [7], etc. The definition of a paracontact metric (κ, µ)-space was motivated by the relationship between contact metric and paracontact geometry. More precisely, it was proved in [4] that any non-Sasakian contact metric (κ, µ)-space accepts two paracontact metric (eκ, eµ)-structures with the same contact form. On the other hand, under certain natural conditions, every non-paraSasakian paracontact (eκ, eµ)-space admits a contact metric (κ, µ)-structure compatible with the same contact form ([5]). Paracontact metric (κ, µ)-spaces satisfy that h2= (κ+ 1)φ2but this condition does not give any type of restriction over the value of κ, unlike in contact metric geometry, because the metric of a paracontact metric manifold is not positive definite. However, it is useful to distinguish the cases κ > −1, κ < −1 and κ=−1. In the first two, equation (1) determines the curvature completely and either the tensor hor ϕh are diagonalisable [5]. The case κ=−1 is equivalent to h2= 0 but not to h= 0. Indeed, there are examples of paracontact metric (κ, µ)-spaces with h2= 0 but h6= 0, as was first shown in [2, 5, 8, 12]. However, only some particular examples were given of this type of space and no effort had been made to understand the general behaviour of the tensor hof a paracontact metric (−1, µ)-space until the author published [11], where a local classification depending on the rank of hwas given in Theorem 3.2. The author also provided explicit examples of all the possible constant values of the rank of hwhen µ= 2. She explained why the values µ= 0 and µ= 2 are special and studying them is enough. Finally, she showed some paracontact metric (−1,0)-spaces of any dimension with 2010 Mathematics Subject Classification. Primary 53C15, 53B30; Secondary 53C25, 53C50. Key words and phrases. paracontact metric manifold; (κ, µ)-spaces; paraSasakian; contact metric; nullity distribution. 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. 1 2 V. MART´ IN-MOLINA rank(h) = 1 and of paracontact metric (−1,0)-spaces of dimension 5 and 7 for any possible constant rank of h. These were the first examples of this type with µ6= 2 and dimension greater than 3. In the present paper, after the preliminaries section, we will give an alternative proof of Theorem 3.2 of [11] that does not use [13] and we will complete the examples of all the possible cases of constant rank of hby presenting (2n+ 1)-dimensional paracontact metric (−1,0)-spaces with rank(h) = 2,...,n. Lastly, we will also show the first explicit examples ever known of paracontact metric (−1,2)-spaces and (−1,0)-spaces with hof non-constant rank. 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 [10]: (i) η(ξ) = 1, ϕ2=I−η⊗ξ, (ii) the eigendistributions D+and D−of ϕcorresponding to the eigenvalues 1 and −1, respectively, have equal dimension n. It follows that ϕξ = 0, η◦ϕ= 0 and rank(ϕ) = 2n. If an almost paracontact manifold admits 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. Then gis necessarily of signature (n+ 1, n) and satisfies η=g(·, ξ) and g(·, ϕ·) = −g(ϕ·,·). We can now define the fundamental 2-form of the almost paracontact metric manifold by Φ(X, Y ) = g(X, ϕY ). If dη = Φ, then ηbecomes a contact form (i.e. η∧(dη)n6= 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 symmetric with respect to g(i.e. g(hX, Y ) = g(X, hY ), for all X, Y ), anti-commutes with ϕand satisfies hξ = trh= 0 and the identity ∇ξ=−ϕ+ϕh ([14]). 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 [ϕ, ϕ]−2dη ⊗ξ= 0, where [ϕ, ϕ] is the Nijenhuis tensor of ϕ[14]: [ϕ, ϕ](X, Y ) = ϕ2[X, Y ] + [ϕX, ϕY ]−ϕ[ϕX, Y ]−ϕ[X, ϕY ]. A normal paracontact metric manifold is said to be a paraSasakian manifold and is in particular K-paracontact. The converse holds in dimension 3 ([6]) but not in general in higher dimensions. However, it was proved in Theorem 3.1 of [11] that it also holds for (−1, µ)-spaces. Every paraSasakian manifold satisfies (2) R(X, Y )ξ=−(η(Y)X−η(X)Y), for every X, Y on M. The converse is not true, since Examples 3.8–3.11 of [11] and Examples 4.1 and 4.5 of the present one show that there are examples of paracontact metric manifolds satisfying equation (2) but with h6= 0 (and therefore not K-paracontact or paraSasakian). Moreover, it is also clear in Example 4.5 that the rank of hdoes not need to be constant either, since hcan be zero at some points and non-zero in others. The main result of [11] is the following local classification of paracontact metric (−1, µ)-spaces: Theorem 2.1 ([11]).Let Mbe a (2n+ 1)-dimensional paracontact metric (−1, µ)-space. Then we have one of the following possibilities: (1) either h= 0 and Mis paraSasakian, (2) or h6= 0 and rank(hp)∈ {1,...,n}at every p∈Mwhere hp6= 0. Moreover, there exists a basis {ξp, X1, Y1,...,Xn, Yn}of Tp(M)such that the only non-vanishing components of g are gp(ξp, ξp) = 1, gp(Xi, Yi) = ±1, LOCAL CLASSIFICATION AND EXAMPLES 3 and hp|hXi,Yii=0 0 1 0or hp|hXi,Yii=0 0 0 0, where obviously there are exactly rank(hp)submatrices of the first type. If n= 1, such a basis {ξp, X1, Y1}also satisfies that ϕpX1=±X1, ϕpY1=∓Y1, and the tensor hcan be written as hp|hξp,X1,Y1i=  0 0 0 0 0 0 0 1 0 . Many examples of paraSasakian manifolds are known. For instance, hyperboloids H2n+1 n+1 (1) and the hyperbolic Heisenberg group H2n+1 =R2n×R, [9]. We can also obtain (η-Einstein) paraSasakian manifolds from contact (κ, µ)-spaces with |1−µ 2|<√1−κ. In particular, the tangent sphere bundle T1Nof any space form N(c) with c < 0 admits a canonical η-Einstein paraSasakian structure, [3]. Finally, we can see how to construct explicitly a paraSasakian structure on a Lie group (see Example 3.4 of [11]) or on the unit tangent sphere bundle, [7]. On the other hand, until [11] only some types of non-paraSasakian paracontact metric (−1, µ)- spaces were known: •(2n+ 1)-dimensional paracontact metric (−1,2)-space with rank(h) = n, [5]. •3-dimensional paracontact metric (−1,2)-space with rank(h) = n= 1, [12]. •3-dimensional paracontact metric (−1,0)-space with rank(h) = n= 1. This example is not paraSasakian but it satisfies (2), [8]. The answer to why there seems to be only examples of paracontact metric (−1, µ)-spaces with µ= 2 or µ= 0 is a Dc-homothetic deformation, i.e. the following change of a paracontact metric structure (M, ϕ, ξ, η, g) [14]: ϕ′:= ϕ, ξ′:= 1 cξ, η′:= cη, g′:= cg +c(c−1)η⊗η, for some c6= 0. It is known that (ϕ′, ξ′, η′, g′) is again a paracontact metric structure on Mand that K-paracontact and paraSasakian structures are also preserved. However, curvature conditions like R(X, Y )ξ= 0 are destroyed, since paracontact metric (κ, µ)-spaces become other paracontact metric (κ′, µ′)-spaces with κ′=κ+ 1 −c2 c2, µ′=µ−2 + 2c c. In particular, if (M, ϕ, ξ, η, g) is a paracontact metric (−1, µ)-space, then the deformed manifold is another paracontact metric (−1, µ′)-space with µ′=µ−2+2c c. Therefore, given a (−1,2)-space, a Dc-homothetic deformation with arbitrary c6= 0 will give us another paracontact metric (−1,2)-space. Given a paracontact metric (−1,0)-space, if we Dchomothetically deform it with c=2 2−µ6= 0 for some µ6= 2, we will obtain a paracontact metric (−1, µ)-space with µ6= 2. A sort of converse is also possible: given a (−1, µ)-space with µ6= 2, a Dc-homothetic deformation with c= 1 −µ 26= 0 will give us a paracontact metric (−1,0)-space. The case µ= 0, h6= 0 is also special because the manifold satisfies (2) but it is not paraSasakian. Examples of non-paraSasakian paracontact metric (−1,2)-spaces of any possible dimension and constant rank of hwere presented in [11]: Example 2.2 ((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, Y1,...,Xn, Yn}such that the only 4 V. MART´ IN-MOLINA non-zero Lie brackets are: [ξ, Xi] = Yi, i = 1,...,m, [Xi, Yj] =          δij (2ξ+√2(1 + δim)Ym) +(1 −δij)√2(δimYj+δjmYi), i, j = 1,...,m, δij (2ξ+√2Yi), i, j =m+ 1,...,n, √2Yi, i = 1, . . . , m, j =m+ 1,...,n. 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, ϕXi=Xi, ϕYi=−Yi, i = 1,...,n, η(ξ) = 1, η(Xi) = η(Yi) = 0, i = 1,...,n. The only non-vanishing components of the metric are g(ξ, ξ) = g(Xi, Yi) = 1, i = 1,...,n. A straightforward computation gives that hXi=Yiif i= 1,...,m,hXi= 0 if i=m+ 1,...,n and hYj= 0 if j= 1,...,n, so h2= 0 and rank(h) = m. Furthermore, the manifold is a (−1,2)-space. Examples of non-paraSasakian paracontact metric (−1,0)-spaces of any possible dimension and rank(h) = 1 were also given in [11]: Example 2.3 ((2n+ 1)-dimensional paracontact metric (−1,0)-space with rank(h) = 1).Let gbe the (2n+ 1)-dimensional Lie algebra with basis {ξ, X1, Y1,...,Xn, Yn}such that the only non-zero Lie brackets are: [ξ, X1] = X1+Y1,[ξ, Y1] = −Y1,[X1, Y1] = 2ξ, [Xi, Yi] = 2(ξ+Yi),[X1, Yi] = X1+Y1,[Y1, Yi] = −Y1, i = 2,...,n. 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, ϕX1=X1, ϕY1=−Y1, ϕXi=−Xi, ϕYi=Yi, i = 2,...,n, η(ξ) = 1, η(Xi) = η(Yi) = 0, i = 1,...,n. The only non-vanishing components of the metric are g(ξ, ξ) = g(X1, Y1) = 1, g(Xi, Yi) = −1, i = 2,...,n. A straightforward computation gives that hX1=Y1,hY1= 0 and hXi=hYi= 0,i= 2,...,n, so h2= 0 and rank(h) = 1. Moreover, by basic paracontact metric properties and Koszul’s formula we obtain that ∇ξX1= 0,∇ξY1= 0,∇ξXi=Xi,∇ξYi=−Yi, i = 2,...,n, ∇XiY1=δi1ξ, ∇XiYj=δij(ξ+ 2Yi),∇Y1X1=−ξ, ∇YiXj=−δij ξ, i, j = 2,...,n, ∇X1Xj= 0,∇Y1Y1=∇Y1Yj= 0,∇YjY1=Y1, i = 2,...,n, and thus R(Xi, ξ)ξ=−Xi, i = 1,...,n, R(Yi, ξ)ξ=−Yi, i = 1,...,n, R(Xi, Xj)ξ=R(Xi, Yj)ξ=R(Yi, Yj)ξ= 0, i, j = 1,...,n. Therefore, the manifold is also a (−1,0)-space. LOCAL CLASSIFICATION AND EXAMPLES 5 To our knowledge, the previous example is the first paracontact metric (−1, µ)-space with h2= 0, h6= 0 and µ6= 2 that was constructed in dimensions greater than 3. For dimension 3, Example 4.6 of [8] was already known. In dimension 5, there also exist examples of paracontact metric (−1,0)-space with rank(h) = 2 and in dimension 7 of rank(h) = 2,3, as shown in [11]. Higher-dimensional examples of paracontact metric (−1,0)-spaces with rank(h)≥2 were not included, which will be remedied in Example 4.1. We will also see how to construct a 3-dimensional paracontact metric (−1,0)-space and (−1,2)-space where the rank of his not constant. 3. New proof of Theorem 2.1 We will now present a revised proof of Theorem 2.1 that does not use [13] when h6= 0 but constructs the basis explicitly. Proof. Since κ=−1, we know from [5] that h2= 0. If h= 0, then R(X, Y )ξ=−(η(Y)X−η(X)Y), for all X, Y on Mand ξis a Killing vector field, so Theorem 3.1 of [11] gives us that the manifold is paraSasakian. If h6= 0, then let us take a point p∈Msuch that hp6= 0. We know that ξis a global vector field such that g(ξ, ξ) = 1, that hξ = 0 and that his self-adjoint, so Kerηpis h-invariant and hp:Kerηp7→ Kerηpis a non-zero linear map such that h2 p= 0. We will now construct a basis {X1, Y1,...,Xn, Yn}of Kerηpthat satisfies all of our requirements. Take a non-zero vector v∈Kerηpsuch that hpv6= 0, which we know exists because hp6= 0. Then we write Kerηp=L1⊕L⊥ 1, where L1=hv, hpvi. Both L1and L⊥ 1are hp-invariant because hpis self-adjoint. Moreover, gp(v, hpv)6= 0 because gp(hpv, hpv) = 0 = gp(hpv, w) for all w∈L⊥ 1, hpv6= 0 and gis a non-degenerate metric.We now distinguish two cases: (1) If gp(v, v) = 0, then we can take Xi=1 √|gp(v,hpv)|vand Yi=1 √|gp(v,hpv)|hpv, which satisfy that gp(Xi, Xi) = 0 = gp(Yi, Yi), gp(Xi, Yi) = ±1 and hpXi=Yi. (2) If gp(v, v)6= 0, then v′=v−gp(v,v) gp(v,hpv)hpvsatisfies that gp(v′, v′) = 0, so we can take Xi=1 √|gp(v′,hpv)′|v′,Yi=1 √|gp(v′,hpv′)|hv′. We have again that gp(Xi, Xi) = 0 = gp(Yi, Yi), gp(Xi, Yi) = ±1 and hpXi=Yi. In both cases, L1=hXi, Yii, so we now take a non-zero vector v∈L⊥ 1and check if hpv6= 0. We know that we can take vsuch that hpv6= 0 in this step as many times as the rank of hp, which is at minimum 1 (since hp6= 0) and at most nbecause dim Kerηp= 2nand the spaces L1have dimension 2. If we denote by mthe rank of hp, then we can write Kerηpas the following direct sum of mutually orthogonal subspaces: Kerηp=L1⊕L2⊕···⊕Lm⊕V=hX1, Y1,...,Xm, Ymi⊕V, where hpv= 0 for all v∈V. Each Liis of signature (1,1) because {˜ Xi=1 √2(Xi+Yi),˜ Yi= 1 √2(Xi−Yi)}is a pseudo-orthonormal basis such that gp(˜ Xi,˜ Xi) = −gp(˜ Yi,˜ Yi) = gp(Xi, Yi) = ±1, gp(˜ Xi,˜ Yi) = 0. Therefore, hX1, Y1,...,Xm, Ymiis of signature (m, m) and, since Kerηpis of signature (n, n), we can take a pseudo-orthonormal basis {v1,...,vn−m, w1,...,wn−m}of Vsuch that gp(vi, vj) = δij and gp(wi, wj) = −δij. Therefore, it suffices to define Xm+i=1 √2(vi+ wi), Ym+i=1 √2(vi−wi) to have gp(Xi, Xi) = 0 = gp(Yi, Yi), gp(Xi, Yi) = 1 and hpXi=hpYi= 0, i=m+ 1,...,n. If n= 1, then ϕpX1=±X1and ϕpY1=∓Y1follow directly from paracontact metric properties and the definition of the basis {X1, Y1,...,Xn, Yn}. 6 V. MART´ IN-MOLINA It is worth mentioning that Theorem 2.1 is true only pointwise, i.e. rank(hp) does not need to be the same for every p∈M. Indeed, we will see in Examples 4.3 and 4.5 that we can construct paracontact metric (−1, µ)-spaces such that his zero in some points and non-zero in others. 4. New examples We will first present an example of (2n+ 1)-dimensional paracontact metric (−1,0)-space with rank of hgreater than 1. This means that, together with Examples 2.2 and 2.3, we have examples of paracontact metric (−1, µ)-spaces of every possible dimension and constant rank of hwhen µ= 0 and µ= 2. Example 4.1 ((2n+1)-dimensional paracontact metric (−1,0)-space with rank(h) = m∈ {2,...,n}). Let gbe the (2n+ 1)-dimensional Lie algebra with basis {ξ, X1, Y1,...,Xn, Yn}such that the only non-zero Lie brackets are: [ξ, X1] = X1+X2+Y1,[ξ, Y1] = −Y1+Y2, [ξ, X2] = X1+X2+Y2,[ξ, Y2] = Y1−Y2, [ξ, Xi] = Xi+Yi, i = 3,...,m, [ξ, Yi] = −Yi, i = 3,...,m, [Xi, Xj] =      √2X1,if i= 1, j = 2, −√2Xjif i= 2, j = 3 . . . , m, √2[ξ, Xi],if i= 1, . . . , m, j =m+ 1,...,n, [Yi, Yj] = (√2(−Y1+Y2),if i= 1, j = 2, √2Yj,if i= 1,2, j = 3,...,m, [Xi, Yi] =          2ξ+√2(X2+Y2)if i= 1, −2ξ+√2X1,if i= 2, −2ξ+√2(X1−X2−Y2),if i= 3,...,m, −2ξ−√2Xi,if i=m+ 1,...,n, [Xi, Yj] i6=j =                √2(Y1+X2)if i= 1, j = 2, √2X1,if i= 2, j = 1, √2Xj,if i= 1,2, j = 3,...,m, √2Yi,if i= 3, . . . , m, j = 2, −√2[ξ, Yj],if i=m+ 1, . . . , n, j = 1,...,m. 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, ϕXi=Xi, ϕYi=−Yi, i = 1,...,n, η(ξ) = 1, η(Xi) = η(Yi) = 0, i = 1,...,n. The only non-vanishing components of the metric are g(ξ, ξ) = g(X1, Y1) = 1, g(Xi, Yi) = −1, i = 2,...,n. A straightforward computation gives that hXi=Yi,i= 1,...,m,hXi= 0,i=m+ 1,...,n and hYi= 0,i= 1,...,n, so h2= 0 and rank(h) = m. Moreover, very long but direct computations give that R(Xi, ξ)ξ=−Xi, i = 1,...,n, R(Yi, ξ)ξ=−Yi, i = 1,...,n, R(Xi, Xj)ξ=R(Xi, Yj)ξ=R(Yi, Yj)ξ= 0, i, j = 1,...,n. LOCAL CLASSIFICATION AND EXAMPLES 7 Therefore, the manifold is also a (−1,0)-space. Remark 4.2. Note that the previous example is only possible when n≥2. If n= 1, then we can only construct examples of rank(h) = 1, as in Example 2.3. In the definition of the Lie algebra of the previous example, some values of iand jare not possible for m= 2 or m=n. In that case, removing the affected Lie brackets from the definition will give us valid examples nonetheless. We will present now an example of 3-dimensional paracontact metric (−1,2)-space and one of 3-dimensional paracontact metric (−1,0)-space, such that rank(hp) = 0 or 1 depending on the point pof the manifold. These are the first examples of paracontact metric (κ, µ)-spaces with hof non-constant rank that are known. Example 4.3 (3-dimensional paracontact metric (−1,2)-space with rank(hp) not constant).We consider the manifold M=R3with the usual cartesian coordinates (x, y, z). The vector fields e1=∂ ∂x +xz ∂ ∂y −2y∂ ∂z , e2=∂ ∂y , ξ =∂ ∂z are linearly independent at each point of M. We can compute [e1, e2] = 2 ξ, [e1, ξ] = −x e2,[e2, ξ] = 0. We define the semi-Riemannian metric gas the non-degenerate one whose only non-vanishing components are g(e1, e2) = g(ξ, ξ) = 1, and the 1-form ηas η= 2ydx +dz, which satisfies η(e1) = η(e2) = 0,η(ξ) = 1. Let ϕbe the (1,1)-tensor field defined by ϕe1=e1, ϕe2=−e2, ϕξ = 0. Then dη(e1, e2) = 1 2(e1(η(e2)) −e2(η(e1)) −η([e1, e2])) = −1 = −g(e1, e2) = g(e1, ϕe2), dη(e1, ξ) = 1 2(e1(η(ξ)) −ξ(η(e1)) −η([e1, ξ]) = 0 = g(e1, ϕξ), dη(e2, ξ) = 1 2(e2(η(ξ)) −ξ(η(e2)) −η([e2, ξ]) = 0 = g(e2, ϕξ). Therefore, (ϕ, ξ, η, g)is a paracontact metric structure on M. Moreover, hξ = 0,he1=xe2,he2= 0. Hence, h2= 0 and, given p= (x, y, z)∈R3, rank(hp) = 0 if x= 0 and rank(hp) = 1 if x6= 0. Let ∇be the Levi-Civita connection. Using the properties of a paracontact metric structure and Koszul’s formula (3) 2g(∇XY, Z) = X(g(Y, Z))+Y(g(Z, X))−Z(g(X, Y ))−g(X, [Y, Z])−g(Y, [X, Z])+g(Z, [X, Y ]), we can compute ∇ξξ= 0,∇e1ξ=−e1−xe2,∇e2ξ=e2,∇ξe1=−e1,∇ξe2=e2, ∇e1e1=xξ, ∇e2e2= 0,∇e1e2=ξ, ∇e2e1=−ξ. Using the following definition of the Riemannian curvature (4) R(X, Y )Z=∇X∇YZ−∇Y∇XZ−∇[X,Y ]Z, we obtain R(e1, ξ)ξ=−e1+ 2he1, R(e2, ξ)ξ=−e2+ 2he2, R(e1, e2)ξ= 0, so the paracontact metric manifold Mis also a (−1,2)-space. Remark 4.4. The previous example does not contradict Theorem 2.1, as we will see by constructing explicitly the basis of the theorem on each point pwhere hp6= 0, i.e., on every point p= (x, y, z) such that x6= 0. Indeed, let us take a point p= (x, y, z)∈R3. If x6= 0, then we define X1=e1p √|x|,Y1=hpe1p √|x|. We obtain that {ξp, X1, Y1}is a basis of Tp(R3)that satisfies that: 8 V. MART´ IN-MOLINA •the only non-vanishing components of gare gp(ξp, ξp) = 1, gp(X1, Y1) = sign(x), •the tensor hcan be written as hp|hξp,X1,Y1i=  0 0 0 0 0 0 0 1 0 , •ϕpξ= 0, ϕpX1=X1, ϕpY1=−Y1. Example 4.5 (3-dimensional paracontact metric (−1,0)-space with rank(hp) not constant).We consider the manifold M=R3with the usual cartesian coordinates (x, y, z). The vector fields e1=∂ ∂x +xe−2z∂ ∂y −2y∂ ∂z , e2=∂ ∂y , ξ =∂ ∂z are linearly independent at each point of M. We can compute [e1, e2] = 2 ξ, [e1, ξ] = 2xe−2ze2,[e2, ξ] = 0. We define the semi-Riemannian metric gas the non-degenerate one whose only non-vanishing components are g(e1, e2) = g(ξ, ξ) = 1, and the 1-form ηas η= 2ydx +dz, which satisfies η(e1) = η(e2) = 0,η(ξ) = 1. Let ϕbe the (1,1)-tensor field defined by ϕe1=e1, ϕe2=−e2, ϕξ = 0. Then dη(e1, e2) = 1 2(e1(η(e2)) −e2(η(e1)) −η([e1, e2])) = −1 = −g(e1, e2) = g(e1, ϕe2), dη(e1, ξ) = 1 2(e1(η(ξ)) −ξ(η(e1)) −η([e1, ξ]) = 0 = g(e1, ϕξ), dη(e2, ξ) = 1 2(e2(η(ξ)) −ξ(η(e2)) −η([e2, ξ]) = 0 = g(e2, ϕξ). Therefore, (ϕ, ξ, η, g)is a paracontact metric structure on M. Moreover, hξ = 0,he1=−2xe−2ze2,he2= 0. Hence, h2= 0 and, given p= (x, y, z)∈R3, rank(hp) = 0 if x= 0 and rank(hp) = 1 if x6= 0. Let ∇be the Levi-Civita connection. Using the properties of a paracontact metric structure and Koszul’s formula (3), we can compute ∇ξξ= 0,∇e1ξ=−e1+ 2xe−2ze2,∇e2ξ=e2,∇ξe1=−e1,∇ξe2=e2, ∇e1e1=−2xe−2zξ, ∇e2e2= 0,∇e1e2=ξ, ∇e2e1=−ξ. 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