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Turk J Math (2015) 39: 467 – 476 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ c TÜBİTAK ⃝ doi:10.3906/mat-1412-24 Research Article Invariant distributions and holomorphic vector fields in paracontact geometry Mircea CRASMAREANU1 , Laurian-Ioan PIŞCORAN2,∗ Faculty of Mathematics, University “Al. I. Cuza”, Iaşi, Romania 2 North University Center of Baia Mare, Technical University of Cluj, Baia Mare, Romania 1 Received: 09.12.2014 • Accepted/Published Online: 13.05.2015 • Printed: 30.07.2015 Abstract: Having as a model the metric contact case of V. Brı̂nzănescu; R. Slobodeanu, we study two similar subjects in the paracontact (metric) geometry: a) distributions that are invariant with respect to the structure endomorphism φ ; b) the class of vector fields of holomorphic type. As examples we consider both the 3 -dimensional case and the general dimensional case through a Heisenberg-type structure inspired also by contact geometry. Key words: Paracontact metric manifold, invariant distribution, paracontact-holomorphic vector field 1. Introduction Paracontact geometry [7, 13] appears as a natural counterpart of the contact geometry in [9]. Compared with the huge literature in (metric) contact geometry, it seems that new studies are necessary in almost paracontact geometry; a very interesting paper connecting these fields is [5]. The present work is another step in this direction, more precisely from the point of view of some subjects of [4]. The first section deals with the distributions V , which are invariant with respect to the structure endomorphism φ, one trivial example being the canonical distribution D provided by the annihilator of the paracontact 1 -form η . As in the contact case, the characteristic vector field ξ must belong to V or V ⊥ . Two important tools in this study are the second fundamental form and the integrability tensor field, both satisfying important (skew)-commutation formulas in the paracontact metric and para-Sasakian geometries. Let us remark that another important class of paracontact geometries, namely the para-Kenmotsu case, was studied recently in [2] from the same points of view. The second subject of the present paper is the class of paracontact-holomorphic vector fields that form a Lie subalgebra on a normal almost paracontact manifold; recently this type of vector fields was studied as providing the potential vector field of Ricci solitons in (3 -dimensional) almost paracontact geometries in [1]. These vector fields vanish a ∂¯ -operator expressed in terms of Levi-Civita as well as the canonical paracontact connection from [14]. We also give a relationship between the paracontact-holomorphicity on the manifold M and the holomorphicity on the cone manifold C(M ). The last result gives a characterization of paracontact-holomorphic vector fields X in terms of para-Cauchy–Riemann equations for the components of X in a paracontact-holomorphic frame. Two types of examples are examined: firstly in dimension 3 and secondly in arbitrary dimension following the Heisenberg-type example of contact metric geometry from [3, p. 60–61]. For the former case we compute the ∗Correspondence: plaurian@gmail.com 2010 AMS Mathematics Subject Classification: 53C15; 53C25. 467 CRASMAREANU and PIŞCORAN/Turk J Math fundamental functions α, β occurring in the Levi-Civita differential of φ while for the latter we use an adapted frame of D . Let us remark that our Heisenberg-type example 2.11 is different from the hyperbolic Heisenberg group of [8, p. 85]. For the 3-dimensional example we point out the vanishing of the mixed sectional curvature of the pair (D, ξ) of invariant distributions in a short Appendix. 2. Invariant distributions on almost paracontact metric manifolds Let M be a (2n + 1)-dimensional smooth manifold, φ a (1, 1)-tensor field called the structure endomorphism, ξ a vector field called the characteristic vector field, η a 1 -form called the paracontact form, and g a pseudoRiemannian metric on M of signature (n + 1, n). In this case, we say that (φ, ξ, η, g) defines an almost paracontact metric structure on M if [14]: φ2 = I − η ⊗ ξ, η(ξ) = 1, g(φX, φY ) = −g(X, Y ) + η(X)η(Y ). (2.1) From the definition it follows φ(ξ) = 0, η ◦ φ = 0, η(X) = g(X, ξ) , g(ξ, ξ) = 1 and the fact that φ is g -skewsymmetric: g(φX, Y ) = −g(φY, X). The associated 2 -form ω(X, Y ) := g(X, φY ) is skew-symmetric and is called the fundamental form of the almost metric paracontact manifold (M, φ, ξ, η, g). The 2n-dimensional distribution D := ker η is called the canonical distribution associated to the almost paracontact metric structure (φ, ξ, η, g). The vector field ξ is g -orthogonal to D and we have the orthogonal splitting of the tangent bundle T M = D ⊕ span{ξ} ; let vξ and hξ be the corresponding projectors; thus vξ (X) = X − η(X)ξ . We assume given a distribution V on M . The main hypothesis for our framework is the existence of a g -orthogonal complementary distribution V ⊥ . Let Γ(V) be the C ∞ (M ) -module of its sections. We denote with v and h the orthogonal projectors with respect to the decomposition T M = V ⊕ V ⊥ . Inspired by [4] we introduce: Definition 2.1 The distribution V is called invariant if φ(V) ⊆ V , i.e. h ◦ φ ◦ v = 0 . The first result provides an example and a characterization: Proposition 2.2 On (M, φ, ξ, η, g) we have: i) D is an invariant distribution; ii) V is invariant if and only if V ⊥ is invariant. Hence the invariance means φ ◦ v = v ◦ φ respectively φ ◦ h = h ◦ φ. Proof i) From η ◦ φ = 0. ii) From the skew-symmetry of φ. 2 With the same proof as that of Lemma 2.1. from [4, p. 194] we have: Proposition 2.3 If V is an invariant distribution then ξ ∈ Γ(V) or ξ ∈ Γ(V ⊥ ). Moreover, if ξ ∈ Γ(V) then V⊥ ⊆ D . We consider a particular class of almost paracontact metric geometry after [14, p. 39]: Proposition 2.4 The almost paracontact metric manifold (M, φ, ξ, η, g) is a paracontact metric manifold if ω = dη where d is given by: 2dη(X, Y ) = X(η(Y )) − Y (η(X)) − η([X, Y ]) for all vector fields X, Y . 468 (2.2) CRASMAREANU and PIŞCORAN/Turk J Math The same proof as that of Proposition 2.1 from [4, p. 195] yields: Proposition 2.5 Suppose that V is an invariant distribution in a paracontact metric manifold satisfying one of the following conditions: (i) dim(V) = 2k + 1 with k ≤ n , (ii) V is integrable. Then ξ ∈ Γ(V). In particular, an integrable invariant distribution must be odd-dimensional. Recall now two important tensor fields associated to a given distribution: Definition 2.6 If V is a distribution on the Riemannian manifold (M, g) then: i) its second fundamental form is B V : Γ(V) × Γ(V) → Γ(V ⊥ ) given by: B V (X, Y ) = 1 h(∇X Y + ∇Y X) 2 (2.3) where ∇ is the Levi-Civita connection of g ; ii) its integrability tensor is B V : Γ(V) × Γ(V) → Γ(V ⊥ ) given by: I V (X, Y ) = h([X, Y ]). (2.4) For the class of paracontact metric structures we determine a relationship between the second fundamental form and the integrability tensor for invariant distributions transversally to the characteristic vector field: Proposition 2.7 Let V be an invariant distribution on the paracontact metric manifold (M, φ, ξ, η, g) such that ξ ∈ Γ(V ⊥ ) . If X, Y ∈ Γ(V) then [ ] 2 B V (φX, Y ) − B V (X, φY ) = φ ◦ I V (φX, φY ) − φ ◦ I V (X, Y ). (2.5) In particular, for V = D we have the symmetry B D (φX, Y ) = B D (X, φY ), Proof B D (φX, φY ) = B D (X, Y ). (2.6) From Lemma 2.7 of [14, p. 42] we have for all vector fields X, Y : (∇φX φ)φY − (∇X φ)Y = 2g(X, Y )ξ − (X − hX + η(X)ξ)η(Y ) (2.7) where h = 21 Lξ φ . The Proposition 2.3 gives V ⊆ D and then the second term in the right hand-side is zero. Hence ∇φX Y − φ(∇φX φY ) − ∇X φY + φ(∇X Y ) = 2g(X, Y )ξ = ∇φY X − φ(∇φY φX) − ∇Y φX + φ(∇Y X) gives (∇φX Y + ∇Y φX) − (∇φY X + ∇X φY ) = φ([φX, φY ] − [X, Y ]) yielding ( ) 2 B V (φX, Y ) − B V (X, φY ) = h ◦ φ([φX, φY ] − [X, Y ]) (2.8) 469 CRASMAREANU and PIŞCORAN/Turk J Math which is (2.5) . For V = D we take the g -inner product of (2.8) with ξ and use the g -skew-symmetry of φ and φ(ξ) = 0 to obtain (2.61 ). With Y replaced by φY in (2.61 ) it results (2.62 ) . 2 Let us study now the complementary case when ξ ∈ Γ(V). We recall that a para-Sasakian manifold is a normal paracontact metric manifold; the normality means the integrability of the almost paracomplex structure J on the cone C(M ) = M × R: ( ) ( ) d d J X, f = φX + f ξ, η(X) . (2.9) dt dt A characterization of this case is given in [14, p. 42] (∇X φ)Y = −g(X, Y )ξ + η(Y )X (2.10) for all vector fields X, Y . In a para-Sasakian manifold we have ∇X ξ = −φX (2.11) ∇φX ξ = φ(∇X ξ) = −φ2 X. (2.12) which yields the commutation formula Proposition 2.8 Let V be an invariant distribution with ξ ∈ Γ(V ) in a para-Sasakian manifold. Then for all X, Y ∈ Γ(V) we have [ ] 2 B V (X, φY ) − φ ◦ B V (X, Y ) = −φ ◦ I V (X, φY ) − φ ◦ I V (X, Y ). (2.13) 2B V (X, ξ) = −I V (X, ξ) (2.14) B V (φX, Y ) = φ ◦ B V (X, Y ) = B V (X, φY ). (2.15) In particular, and if V is integrable then Proof By using the relation (2.10) the left-hand side of (2.13) is h(∇X φY + ∇φY X − φ(∇X Y ) − φ(∇Y X)) = h(∇φY X − φ(∇Y X)). Now, using the metric character of ∇, the last term is h(∇X φY − [X, φY ] − φ(∇X Y ) − φ([X, Y ])) and we get the conclusion (2.13). With Y = ξ in (2.13) we obtain (2.14) while (2.15) is a direct consequence of (2.13) . 2 Corollary 2.9 Let N be an invariant submanifold of the para-Sasakian manifold (M, φ, ξ, η, g) containing ξ and B its second fundamental form. Then for all X, Y ∈ Γ(N ) we have: B(X, ξ) = 0, B(φX, Y ) = φ ◦ B(X, Y ) = B(X, φY ). We finish this section with some examples other than those of [8]: Example 2.10 Suppose that n = 1. After [11, p. 379] we have 470 (2.16) CRASMAREANU and PIŞCORAN/Turk J Math (∇X φ)Y = g(φ(∇Xξ), Y )ξ − η(Y )φ(∇X ξ) (2.17) and (M, φ, ξ, η, g) is normal if and only if there exist smooth functions α, β on M such that (∇X φ)Y = β (g(X, Y )ξ − η(Y )X) + α (g(φX, Y )ξ − η(Y )φ(X)) , ∇X ξ = α(X − η(X)ξ) + βφ(X). (2.18) Hence, the para-Sasakian case is provided by α = 0 and β = −1. (M, φ, ξ, η, g) admits locally a frame {ξ, E, φE} with g(E, E) = 1 = −g(φE, φE), which means that ξ and E are space-like vector fields while φE is a time-like vector field. We have I D (E, φE) = η([E, φE])ξ . In order to handle a concrete example let N be an open connected subset of R2 , (a, b) an open interval in R, and let us consider the manifold M = N × (a, b). Let (x, y) be the coordinates on N induced from the Cartesian coordinates on R2 and let z be the coordinate on (a, b) induced from the Cartesian coordinate on R. Thus (x, y, z) are the coordinates on M . Now we choose the functions σ, f : M → R∗+ , ω1 , ω2 : N → R, (2.19) and following the idea from [10] we define  ω 2 + σe2f 1 1 ω1 ω2 g= 4 ω1  ω1 ( ) 1 1 ω2  = σe2f dx2 − dy 2 + η ⊗ η, η = (dz + ω1 dx + ω2 dy), 4 2 1 ω1 ω2 ω22 − σe2f ω2  ∂ ξ=2 , ∂z 0 φ= 1 −ω2  0 0 . 0 1 0 −ω1 (2.20) (2.21) It follows an almost paracontact metric manifold with 2e−f E= √ σ From   [E, ξ] =  [E, φE] ( ∂ ∂ − ω1 ∂x ∂z ) 2e−f φE = √ σ , ( ∂ ∂ − ω2 ∂y ∂z 2fz σ+σz z E, [φE, ξ] = 2fz σ+σ φE σ σ [ ] √ −f −f −2f = e−fσ E( e√σ )φE − φE( e√σ )E + 2e σ ( ∂ω1 ∂y − ) . ∂ω2 ∂x (2.22) ) (2.23) ξ it follows that D is integrable if and only if the 1-form ω1 dx + ω2 dy is closed; hence η is closed. We have the Levi-Civita connection √  −f z ∇E E = − e−fσ E( e√σ )φE + 2fz σ+σ  σ (ξ  )  √ −f −2f 4 σ ∂ω1 2 ξ ∇E φE = − e−f φE( e√σ )E + e σ − ∂ω (2.24) ∂y ∂x ( )    ∇ ξ = 2fz σ+σz E + e−2f ∂ω1 − ∂ω2 φE E σ σ ∂y ∂x ( ) √  −f 4 σ ∂ω1 ∂ω2 e−2f e√  ∇ E = − ξ )φE − − E(  σ ∂y ∂x e−f σ  φE √ −f 4 σ 2fz σ+σz e√ ∇φE φE = − e−f φE( σ )E + ξ σ ( )   −2f  ∇ ξ=e ∂ω1 − ∂ω2 E + 2fz σ+σz φE φE e−2f ∇ξ E = σ ( ∂ω1 ∂ω2 − ∂y ∂x σ ∂y ) φE, ∂x e−2f ∇ξ φE = σ (2.25) σ ( ∂ω1 ∂ω2 − ∂y ∂x ) E, ∇ξ ξ = 0 (2.26) 471 CRASMAREANU and PIŞCORAN/Turk J Math and then e−2f β= σ σz α = 2fz + , σ ( ∂ω1 ∂ω2 − ∂y ∂x ) . (2.27) ∂ω1 ∂ω2 − = −σe2f . ∂y ∂x (2.28) Hence, (M, φ, ξ, η, g) is a para-Sasakian manifold if and only if σe2f = σe2f (x, y), The first relation expresses the normality of the paracontact structure while the second condition means the metrical condition of the Definition 2.4 and yields the nonintegrability of D since I D (E, φE) = −2ξ . Some cases when both equations hold are: i) ω1 = −y , ω2 = 0 = f , σ = 1 ; ii) ω1 = −y , ω2 = x, σ = 2 , f = 0 . Other examples of 3-dimensional (almost) paracontact manifolds appear in [6, 11, 12]. Example 2.11 On M = R2n+1 with the splitting Rn × Rn × R we consider a Heisenberg-type structure inspired by the contact metric example from [3, p. 60-61]:     n δij + y i y j 0 −y i 0 δij 0 ∑ ∂ 1 1 0 −δij 0  , φ =  δij 0 0  , ξ = 2 , η = (dz − y i dxi ). (2.29) g=  4 ∂z 2 j j i=1 −y 0 1 0 y 0 It follows that (R2n+1 , φ, ξ, η, g) is a paracontact metric manifold with { } ∂ ∂ i ∂ D = span Ai = + y , B = ; 1 ≤ i ≤ n . i ∂xi ∂z ∂y i (2.30) Two classes of invariant distributions are indexed by k ∈ {1, ..., n − 1} : Vkeven = span {Aα , Bα ; 1 ≤ α ≤ k} , Vkodd = Vkeven ∪ {ξ}. (2.31) Let us remark that for n = 1 we recover the previous Example with: ω1 = −y , ω2 = 0 = f , σ = 1 . It is a para-Sasakian manifold with nonintegrable D : [E, ξ] = [φE, ξ] = 0, [E, φE] = −2ξ . The sectional curvature of the plane spanned by E and φE is pK = K M (E, φE) = g(R(E, φE)φE, E) = g(∇φE ξ + 2∇ξ φE, E) = g(−E − 2E, E) = −3 (2.32) similar to the metric contact case. 3. Infinitesimal paracontact-holomorphicity Definition 3.1 The vector field X ∈ Γ(T M ) is called paracontact-holomorphic if vξ ◦ LX φ = 0. Let phol(M ) be the set of all paracontact-holomorphic vector fields. (3.1) The distribution V is paracontact- holomorphic if its sections are elements of phol(M ). The condition (3.1) says that for all vector fields Y we have that (LX φ)Y is collinear with ξ ; let us denote αX (Y ) the collinearity factor. We have αX (Y ) = g([X, φY ] − φ([X, Y ]), ξ) = η([X, φY ]). (3.2) The next result shows the invariance of the above defined holomorphicity and its proof is exactly as in [4]: 472 CRASMAREANU and PIŞCORAN/Turk J Math Proposition 3.2 Let X be a paracontact-holomorphic vector field on the normal almost paracontact metric manifold (M, φ, ξ, η, g). Then φX is also a paracontact-holomorphic vector field. Remarks 3.3 i) Fix X a paracontact-holomorphic vector field. Then computing αX (ξ) with (3.2) we get αX (ξ) = 0 (3.3) which means that [X, ξ] is collinear with ξ , i.e. vξ ([X, ξ]) = 0 . ii) The vanishing of the tensor field N (3) = Lξ φ means that ξ is a paracontact-holomorphic vector field with αξ = 0 . iii) The paracontact-holomorphicity of a fixed X implies for every vector field Y LX Y = η(LX Y )ξ + φ(LX φY ), LX φY = αX (Y )ξ + φ([X, Y ]). (3.4) In both relations, the first term in the right-hand side belongs to spanξ while the second belongs to D . By using these remarks we get: Proposition 3.4 If (M, φ, ξ, η, g) is a normal almost paracontact manifold then phol(M ) is a Lie subalgebra in the Lie algebra of vector fields of M . Proof Let X and Y be paracontact-holomorphic vector fields and Z an arbitrary vector field. Then (L[X,Y ] φ)Z = [X, (LY φ)Z] − (LY )([X, Z]) − [Y, (LX φ)Z] + (LX φ)([Y, Z]). (3.5) From the property of X , Y we have that the second and fourth terms are collinear with ξ . Also [X, (LY φ)Z] = X(αY (Z))ξ − αY (Z)[X, ξ] and the first relation (3.4) gives that this expression is collinear with ξ . The same fact holds for the third term 2 of (3.5). As in the contact case we can express the paracontact-holomorphicity by the vanishing of some ∂¯ -operator. More precisely, we define the map ∂¯ : Γ(T M ) → End(T M ) given by ¯ ∂(X)(Y ) = φ (∇Y X − φ(∇φY X) + φ(∇X φ)Y ) . (3.6) ¯ Thus, X is a paracontact-holomorphic vector field if and only if ∂(X) = 0 . For a general vector field X , if (M, φ, ξ, η, g) is a para-Sasakian manifold then ¯ ∂(X)(ξ) = φ([ξ, X]) (3.7) ¯ ∂(X)(Y ) = φ (∇Y X − φ(∇φY X)) . (3.8) and for Y ∈ D we have If n = 1 then the expression (3.6) reduces to ¯ ∂(X)(Y ) = φ (∇Y X − φ(∇φY X) − η(Y )(αX + βφX)) . (3.9) 473 CRASMAREANU and PIŞCORAN/Turk J Math ˜ of [14, p. 49] we have For the general n and using the canonical paracontact connection ∇ ( ) ¯ ˜ Y X − φ(∇ ˜ φY X) + φ(∇ ˜ X φ)Y + 2η(X)(φN (3) Y − φ2 N (3) φY ) − η(Y )φN (3) X . ∂(X)(Y )=φ ∇ (3.6can) Recall now that on the cone C(M ) we have d d [(X, f ), (Y, g )] = dt dt ( ) dg df d [X, Y ], (X(g) − Y (f ) + f −g ) dt dt dt (3.10) which yields: d Proposition 3.5 Fix X ∈ Γ(T M ) and f ∈ C ∞ (M × R). Then (X, f dt ) is a paraholomorphic vector field on the cone C(M ) if and only if the following three conditions hold: i) (LX φ)Y = −Y (f )ξ , ii) (LX η)(Y ) = φY (f ) + η(Y ) df dt , iii) LX ξ = − df dt ξ , d where Y ∈ Γ(T M ) is arbitrary. Consequently, if (X, f dt ) is a paraholomorphic vector field on C(M ) then X is paracontact-holomorphic vector field on M and f is a first integral if ξ . Proof By using (3.10) we get with respect to J of (2.9) ( (L(X,f d dt ) J)(Y, 0) = df d (LX φ)Y + Y (f )ξ, (X(η(Y )) − φY (g) − η(Y ) − η([X, Y ])) dt dt d (L(X,f d ) J)(0, ) = dt dt ( df d [X, ξ] + , −ξ(f ) dt dt ) (3.11) ) . (3.12) d The paraholomorphicity of (X, f dt ) means the vanishing of the above left-hand sides and this is equivalent with f being first integral of ξ and the relations i)-iii). However, with Y = ξ in i) and using iii) it follows that ξ(f ) = 0 . The equation i) means that X is a paracontact-holomorphic vector field. 2 Corollary 3.6 The paracontact-holomorphic vector fields on M , which come about by projection of the paraholomorphic fields on C(M ), form a Lie subalgebra of phol(M ), denoted by pholpr (M ). They are paracontactholomorphic fields X with two additional properties: a) The 1 -form αX is exact: there exists a smooth function f on M such that αX = d(−f ), b) η([X, ξ]) is a (locally) constant, i.e. constant on any connected component of M . Proof a) it results by applying η to i); more precisely Y (−f ) = η([X, φY ]) for all vector fields Y . By applying η to iii) we get df dt = η([ξ, X]) and then around a point p0 ∈ M we have the following expression of f: f (p, t) = η([ξ, X])(p)t − F (p). Plugging this expression in a) we get: Y (F ) + Y (η([X, ξ]))t = η([X, φY ]) and it results in b). 474 (3.13) 2 CRASMAREANU and PIŞCORAN/Turk J Math Corollary 3.7 On a normal almost paracontact metric manifold (M, φ, ξ, η, g) we have: iv) aξ is a contact-holomorphic vector field, for any function a ∈ M ; so aξ ∈ phol(M ) but it is not necessarily the case that aξ ∈ pholpr (M ) , d v) (ξ, c dt ) is a holomorphic vector field on C(M ) if and only if c is a constant. Proof The first part is a direct consequence of (Laξ φ)Y = a(Lξ φ)Y − φY (a)ξ. (3.14) Let us remark that the normality implies that αaξ (Y ) = −φY (a). For the second part, from iii) of Proposition 3.5 it results that dc dt 2 = 0 while i) gives that Y (c) = 0 for all vector fields Y . Proposition 3.8 Let (M, φ, ξ, η, g) be a paracontact metric manifold. Then any two of the following conditions imply the third one: (i) (LX g)(Y, Z) = 0 for all Y, Z ∈ Γ(D), (ii) iX dη is a closed form, (iii) X is a paracontact-holomorphic vector field. Proof It is a direct consequence of the formula (LX g)(Y, φZ) = (LX dη)(Y, Z) − g(Y, (LX φ)Z) (3.15) 2 for all vector fields Y, Z . Example 3.9 Returning to Example 2.11, let n ∑ ∂ ∂ i ∂ X = α Ai + β Bi + γξ = α +β + (2γ + ( y j αj )) . ∂xi ∂y i ∂z j=1 i i i (3.16) Then X ∈ phol(M ) if and only if the coefficients α and β satisfy the para-Cauchy–Riemann equations with respect to the variables (x, y) and are constant with respect to z : { ∂αi ∂xj ∂αi ∂z = = ∂β i ∂αi ∂y j , ∂y j i ∂β = 0. ∂z = ∂β i ∂xj (3.17) The following analogy with the contact case shows that these computations have a general nature: Proposition 3.10 On a normal almost paracontact metric manifold there always exist (local) adapted frames (Ei , φEi , ξ) consisting of contact-holomorphic vector fields. If the vector field X has the expression X = αi Ei + β i φEi + γξ then X is a paracontact-holomorphic vector field if and only if the coefficients α, β satisfy the generalized para-Cauchy–Riemann equations: Ej (αi ) = φEj (β i ), φEj (αi ) = Ej (β i ) (3.18) and are first integrals of ξ . 475 CRASMAREANU and PIŞCORAN/Turk J Math 4. Appendix: The mixed sectional curvature The main result of [4] is the Bochner-type Theorem 5.1 stated on page 206. The technical ingredient of this result is the mixed sectional curvature: ∑ smix (V, V ⊥ ) = K M (ei ∧ fα ) (a.1) where {ei } respectively {fα } are local orthonormal frames for the given distribution. The cited Bochner-type result deals with an invariant distribution V of dimension 2p + 1 in the Sasakian case and concerns the case smix ≥ 2(n − p). The aim of this short Appendix is to compute this quantity for our example 2.10: smix (D, ξ) = K M (E ∧ ξ) + K M (φE ∧ ξ) = g(R(E, ξ)ξ, E) + g(R(φE, ξ)ξ, φE) (a.2) Since E is a space-like vector field while φE is a time-like one, a direct computation yields the vanishing: smix (D, ξ) = 0 . Acknowledgments The authors are grateful to both referees for their several constructive remarks. References [1] Bejan CL, Crasmareanu M. Second order parallel tensors and Ricci solitons in 3 -dimensional normal paracontact geometry. Ann Global Anal Geom 2014; 46: 117–127. [2] Blaga AM. Invariant and holomorphic distributions on para-Kenmotsu manifolds. Annali dell’Universita’ di Ferrara (in press) DOI 10.1007/s11565-014-0220-5. [3] Blair DE. Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, Vol. 203. 2nd ed. Boston, MA, USA: Birkhäuser, 2010. [4] Brı̂nzănescu V, Slobodeanu R. Holomorphicity and Walczak formula on Sasakian manifolds. J Geom Phys 2006; 57: 193–207. [5] Cappelletti Montano B. Bi-paracontact structures and Legendre foliations. Kodai Math J 2010; 33: 473–512. [6] Dacko P, Olszak Z. On weakly para-cosymplectic manifolds of dimension 3 . J Geom Phys 2007; 57: 561–570. [7] Gündüzalp Y, Şahin B. Paracontact semi-Riemannian submersions. Turk J Math 2013; 37: 114–128. [8] Ivanov S, Vassilev D, Zamkovoy S. Conformal paracontact curvature and the local flatness theorem. Geom Dedicata 2010; 144: 79–100. [9] Kaneyuki S, Williams FL. Almost paracontact and parahodge structures on manifolds. Nagoya Math J 1985; 99: 173–187. [10] Welyczko J. On Legendre curves in 3 -dimensional normal almost contact metric manifolds. Soochow J Math 2007; 33: 929–937. [11] Welyczko J. On Legendre curves in 3 -dimensional normal almost paracontact metric manifolds. Results Math 2009; 54: 377–387. [12] Welyczko J. Slant curves in 3 -dimensional normal almost paracontact metric manifolds. Mediterr J Math 2014; 11: 965–978. [13] Yüksel Perktas S, Tripathi MM, Kiliç K, Keles S. ξ ⊥ -submanifolds of para-Sasakian manifolds. Turk J Math 2014; 38: 905–919. [14] Zamkovoy S. Canonical connections on paracontact manifolds. Ann Global Anal Geom 2008; 36: 37–60. 476