Classification results for three-dimensional (para)contact metric and almost (para)cosymplectic (κ,μ) -spaces
aa r X i v : . [ m a t h . DG ] S e p CLASSIFICATION RESULTS FOR THREE-DIMENSIONAL(PARA)CONTACT METRIC AND ALMOST(PARA)COSYMPLECTIC ( κ, µ ) -SPACES. PIOTR DACKO
Abstract.
It is provided an overview of existed results concerning classi-fication of contact metric, almost cosymplectic and almost Kenmotsu ( κ, µ ) -manifolds. In the case of dimension three it is described in full details structureof contact metric or almost cosymplectic ( κ, µ ) -spaces. The second part of thepaper addresses three-dimensional paracontact metric and almost paracosym-plectic ( κ, µ ) -spaces. There is obtained local classification of paracontact met-ric ( κ, µ ) -spaces, and almost paracosymplectic ( κ, µ ) -spaces, for every possiblevalue of κ .. Introduction
In this paper there are studied almost contact metric and almost paracontactmetric three-dimensional ( κ, µ ) -manifolds. In particular contact metric manifold oralmost cosymplectic ( κ, µ ) -space can be realized as three-dimensional unimodularLie group equipped with left-invariant almost contact metric structure. As thesemanifolds are already classified the paper is an overview of existing result in case ofdimension three. Novelty of our approach is, that there is created on R , smoothone-parameter family of almost contact metric structures, such that for particularvalue of parameter R turns into contact metric ( κ, µ ) -space of almost cosymplectic ( κ, µ ) -space. From other hand every such manifold is locally isometric to uniquesimply connected connected Lie group equipped with left-invariant almost contactmetric structure.The case of paracontact metric or almost paracosymplectic ( κ, µ ) -spaces is morechallenging. Still up to the author knowledge there is no full classification of suchmanifolds. For the literature on the subject see [8], [10], [11].The reason to provide detailed study of three-dimensional manifolds is connectedto possible decomposition theorem. Idea is that every higher dimensional ( κ, µ ) -spaces, is constructed from three dimensional ( κ, µ ) -space. Conjecturing there isreverse decomposition classification of three-dimensional ( κ, µ ) -spaces is enough toprovide local classification in all dimensions.As almost contact metric and almost paracontact metric manifolds share somesimilarities it seems to be interesting to mix these classes. In the sense to considerpseudo-Riemannian manifolds with φ -4 structure: φ = Id − η ⊗ ξ . Such manifoldis equipped with corresponding fundamental form and usual classes can be defined:contact metric with pseudo-metric, almost cosymplectic with pseudo-metric, etc.For example we can equip odd-dimensional Lorentzian manifold with structure ofcontact metric manifold with Lorentzian metric. Mathematics Subject Classification. Preliminaries
All manifolds considered in the paper are smooth, connected without boundary.Tensor fields on smooth manifold are assumed to be smooth. If not otherwise statedletters X , Y , Z , ... denote vector fields.2.1. Almost contact metric manifolds.
Let M be (2 n + 1) -dimensional man-ifold, n > . Almost contact metric structure is a quadruple of tensor fields ( φ, ξ, η, g ) , where φ is (1 , -tensor field, ξ is a characteristic (or Reeb) vector field, η a chracteristic 1-form and g - a Riemannian metric. By definition φ = − Id + η ⊗ ξ, η ( ξ ) = 1 , (2.1) g ( φX, φY ) = g ( X, Y ) − η ( X ) η ( Y ) . (2.2)Tensor field Φ( X, Y ) = g ( X, φY ) is skew-symmetric, Φ( X, Y ) + Φ(
Y, X ) = 0 . Itdetermines a 2-form on M . called fundamental form. From definition of Φ , thereis η ∧ Φ n = 0 , at every point of M . Manifold equipped with some fixed almostcontact metric structure is called almost contact metric manifold.Denote by N S Nijenhuis torsion of a (1 , -tensor field SN S ( X, Y ) = S [ X, Y ] + [
SX, SY ] − S ([ SX, Y ] + [
X, SY ]) . Almost contact metric manifold M is called normal if N φ + 2 dη ⊗ ξ = 0 . Normalityis equivalent to the existence of complex structure on the product f M = M × S ,with the circle.Almost contact metric manifold M is called contact metric if dη = Φ , almostcosymplectic (or almost coKähler) if dη = 0 , d Φ = 0 , almost Kenmotsu if dη =0 , d Φ = 2 η ∧ Φ . Assuming normality we obtain Sasakian (contact metric andnormal), cosymplectic (or coKähler) and Kenmotsu manifolds. Almost contactmetric manifold with maximal isometry group is locally isometric either to Sasakianmanifold of constant sectional curvature c = +1 , Kenmotsu manifold of constantsectional curvature c = − , or cosymplectic manifold of constant sectional curvature c = 0 . General literature on the subject can be found eg. in [4], [12], [16], [24], [33],[35], [36].Let ∇ denote Levi-Civita connection of the metric, R ( X, Y ) = [ ∇ X , ∇ Y ] Z −∇ [ X,Y ] Z , curvature of ∇ . Define h = L ξ φ . Let κ , µ , be real constants. Almostcontact metric manifold is called ( κ, µ ) -space if R ( X, Y ) ξ = κ ( η ( Y ) X − η ( X ) Y ) + µ ( η ( Y ) hX − η ( X ) hY ) . (2.3)Note if h = 0 it is not possible to determine µ . But condition is still formally validfor any possible value of µ . Ambiguity also arrives if(2.4) R ( X, Y ) ξ = κ ( η ( Y ) X − η ( X ) Y ) , for example in case µ = 0 .We denote D = { η = 0 } , kernel distribution of characteristic form. D -homothetyis deformation of almost contact metric structure, α ∈ R + φ φ ′ = φ, ξ ξ ′ = α − ξ, η η ′ = αη,g g ′ = αg + α ( α − η ⊗ η. lmost contact, paracontact metric structures ... 3 Theorem 1 (Blair, Koufogiorgos, Papantoniou, 1995, [5] ) . Let M be contactmetric ( κ, µ ) -manifold. Then κ . The following relations hold ( ∇ X φ ) Y = g ( X, Y + hY ) ξ − η ( Y )( X + hX ) , (2.5) ( ∇ X h ) Y = ((1 − κ ) g ( X, φY ) − g ( X, φhY )) ξ − (2.6) η ( Y )((1 − κ ) φX + φhX ) − µη ( X ) φhY. For contact metric non-Sasakian ( κ, µ ) -space M , let define(2.7) I M = (1 − µ/ / √ − κ,I M is called Boeckx invariant. Theorem 2 (Boeckx, 2000, [6]) . Let M i , i = 1 , , be two non-Sasakian ( κ i , µ i ) -spaces of the same dimension. Then I M = I M if and only if, up to D -homothetictransformation, the two spaces are locally isometric as contact metric spaces. Inparticular, if both spaces are simply connected and complete, they are globally iso-metric up to D -homothetic transformation. The cited result is base for classification of non-Sasakian contact metric ( κ, µ ) -manifolds. It is enough to provide an example of manifold M , for every allowablevalue I of Boeckx invariant, such that I M = I .For almost cosymplectic manifold distribution { η = 0 } is completely integrable.Therefore it determines canonical foliation of the manifold. Let F denote a leafpassing through some point ∈ M . Then F inherits structure of almost Kählermanifold. Assuming structure is Kähler for every leaf manifold is called almostcosymplectic with Kähler leaves. Theorem 3 (Olszak, 1987, [38]) . Define A = −∇ ξ . Almost cosymplectic manifoldhas Kählerian leaves if and only if (2.8) ( ∇ X φ ) Y = − g ( φAX, Y ) + η ( Y ) φAX. Theorem 4 (Dacko, Olszak, 2005, [22]) . Let M be non-cosymplectic almost cosym-plectic ( κ, µ ) -manifold. Then κ . If κ = 0 , M is locally isometric to product ofreal line and almost Kähler manifold. For κ < , M has Kähler leaves and eachleaf is locally flat Kähler manifold. There is following identity L ξ φ = 2 h, L ξ h = − κφ − µφh, L ξ ( φh ) = µh, (2.9)The theorem allows to classify, by analytic solution, almost cosymplectic ( κ, µ ) -manifolds in terms of so-called models. For every µ ∈ R there is almost cosymplectic ( − , µ ) -manifold - called model - and every other ( κ, µ ) -manifold is locally isometricup to D -homothety to particular model [23]. The value µ √− κ , κ < is D -homothetyinvariant. We set C M = − µ/ √− κ , κ < and call C M Dacko-Olszak invariant of almostcosymplectic ( κ, µ ) -manifold.For almost Kenmotsu manifolds there are following basic results. Theorem 5 (Dileo, Pastore, 2009) . Let M be almost Kenmotsu ( κ, µ ) -manifold.Then κ = − , h = 0 and M is locally warped product of an almost Kähler manifoldand open interval. If M is locally symmetric then M is locally isometric to thehyperbolic space H ( − of constant sectional curvature − . Piotr Dacko
Theorem 6 (Dileo, Pastore, 2009 ) . Let M be almost Kenmotsu manifold suchthat h = 0 and (2.10) R ( X, Y ) ξ = κ ( η ( Y ) X − η ( X ) Y ) + µ ( η ( Y ) hφX − η ( X ) hφY ) , then M is locally isomeric to warped products (2.11) H n +1 ( κ − λ ) × f R n , B n +1 ( κ + 2 λ ) × f ′ R n , where H n +1 ( κ − λ ) is the hyperbolic space of constant sectional curvature k − λ < − , B n +1 ( κ +2 λ ) is a space of constant sectional curvature κ +2 λ , f = ce (1 − λ ) t , f ′ = c ′ e (1+ λ ) t , λ = p | κ | . It is known that almost Kenmotsu manifold as Riemannian manifold is locallyconformal to almost cosymplectic manifold. For this point of view see [31], [38]. In[40] authors study generalized nullity distribution on almost Kenmotsu manifold.Curvature properties of more general class of almost cosymplectic and almostKenmotsu so-called ( κ, µ, ν ) -manifolds are studied in [13], [22].2.2. Almost paracontact metric manifolds.
Almost paracontact metric struc-ture on (2 n + 1) -dimensional manifold M , n > , is a quadruple of tensor fields ( φ, ξ, η, g ) , where φ is an affinor, ξ characteristic vector field , 1-form η and pseudo-Riemannian metric g of signature ( n + 1 , n ) . It is assumed that(2.12) φ = Id − η ⊗ ξ, η ( ξ ) = 1 ,g ( φX, φY ) = − g ( X, Y ) + η ( X ) η ( Y ) . The immediate consequences of the definition are that tensor field φ - viewedas linear map on tangent space - has three eigenvalues (0 , − , of multiplicities (1 , n, n ) . Eigendistributions p
7→ V − p , p
7→ V +1 p are totally isotropic, g ( V − , V − ) =0 , g ( V + , V + ) = 0 . Tensor field Φ( X, Y ) = g ( X, φY ) is fundamental 2-form, η ∧ Φ n =0 everywhere. Manifold M equipped with fixed almost paracontact metric structureis called almost paracontact metric manifold.Let M be an almost paracontact metric manifold. M is said to be:(1) normal if N φ − dη ⊗ ξ = 0; (2) paracontact metric if dη = Φ; (3) almost paracosymplectic if dη = 0 , d Φ = 0; (4) almost para-Kenmotsu if dη = 0 , d Φ = 2 η ∧ Φ . Let define h = L ξ φ . Applying derivative L ξ to both sides of identity φ = Id − η ⊗ ξ , we obtain(2.13) hφ + φh = −
12 ( ξ x dη ) ⊗ ξ, We will use also term Reeb vector field lmost contact, paracontact metric structures ... 5 therefore h and φ anti-commute if and only if ξ x dη = 0 . For all mentioned aboveclasses of manifolds - normal etc., the condition ξ x dη = 0 is satisfied. Therefore forall above classes we have hφ + φh = 0 . Let denote by ∇ the Levi-Civita connection of M , R XY Z = [ ∇ X , ∇ Y ] Z −∇ [ X,Y ] Z , curvature operator of ∇ . Almost paracontact metric manifold is called ( κ, µ ) -space if its curvature satisfies R XY Z = κ ( η ( Y ) X − η ( X ) Y ) + µ ( η ( Y ) hX − η ( X ) hY ) , κ, µ ∈ R . In similar as for almost contact metric manifold there is introduced notion of D -homothety of almost paracontact metric manifold.Let M be (2 n + 1) -dimensional almost paracontact metric manifold. A localframe of vector fields ξ , E i , E i + n , i = 1 , . . . n , is called Artin frame if φE i = E i , φE i + n = − E i + n , i = 1 , . . . n (2.14) g ( ξ, E i ) = g ( ξ, E i + n ) = 0 , g ( E i , E i + n ) = 1 , i = 1 , . . . n. (2.15)Gauge of local Artin frame is deformation of Artin into another Artin frame definedby E i E ′ i = f i E i , E i E ′ i + n = f − i E i + n , i = 1 , . . . n, (2.16)where f , . . . f n is a family of locally defined function, non-zero everywhere ondomain of their definition.2.3. Three dimensional connected simply connected Lie groups.
Curva-ture of arbitrary left-invariant Riemannian metric on 3-dimensional Lie group wasdescribed in simple and intuitive way by John Milnor in his paper [34]. In particularfor unimodular groups there is
Theorem 7 (J. Milnor [34]) . Let G be 3-dimensional unimodular Lie group, equippedwith left-invariant Riemannian metric. There is orthonormal frame ( e , e , e ) ofleft-invariant vector fields and constants λ , λ , λ , such that (2.17) [ e , e ] = λ e , [ e , e ] = λ e , [ e , e ] = λ e . Signs of λ i up to the order determine G uniquely if G is connected and simplyconnected. Define µ , µ , µ (2.18) µ i = 12 ( λ + λ + λ ) − λ i . The orthonormal base ( e , e , e ) diagonalizes Ricci quadratic form, the principalRicci curvatures r i = r ( e i ) , i = 1 , , , being given by (2.19) r = 2 µ µ , r = 2 µ µ , r = 2 µ µ . Let v × w be vector product determined by e × e = e , e × e = e , e × e = e .On Lie algebra connection maps are given by x
7→ ∇ e i x = µ i ( e i × x ) . The base change e i
7→ − e i follows λ i
7→ − λ i . Piotr Dacko Three-dimensional contact metric and almost cosymplectic ( κ, µ ) -manifolds Let ( φ, ξ, η, g ) be three-dimensional left-invariant almost paracontact metric struc-ture on unimodular Lie group G . We assume there is an orthonormal frame ofleft-invariant vector fields ( ξ, E , E ) , such that φE = E and commutators aregiven by(3.1) [ E , E ] = 2 kξ, [ E , ξ ] = − ( λ + c ) E , [ ξ, E ] = ( λ − c ) E ,k , λ , c are some constants.We define × -matrix A = ad ξ | { E ,E } . Proposition 1.
Let matrix A be non-nilpotent. The almost contact metric manifold M is ( k − λ , k + c )) -space, ie. (3.2) R XY ξ = ( k − λ )( η ( Y ) X − η ( X ) Y ) + 2( k + c )( η ( Y ) hX − η ( X ) hY ) . Proof.
We employ Milnor’s method to find Levi-Cvita connection coefficients andthen directly compute: R E ξ ξ = ( k − λ + 2 λ ( k + c )) E , R E ξ ξ = ( k − λ − λ ( k + c )) E , (3.3) R E E ξ = 0 , (3.4)from (3.4) it follows that(3.5) R XY ξ = η ( Y ) R Xξ ξ − η ( X ) R Y ξ ξ, Jacobi operator J ξ X = R Xξ ξ by the first set of identities has decomposition(3.6) J ξ X = ( k − λ )( Id − η ( X ) ξ ) + 2( k + c ) h, therefore R XY ξ = η ( Y ) R Xξ ξ − η ( X ) R Y ξ ξ = ( k − λ )( η ( Y ) X − η ( X ) Y ) +2( k + c )( η ( Y ) hX − η ( X ) hY ) . (cid:3) Following Milnor’s paper we obtain
Proposition 2.
Pincipal Ricci curvatures of M are given by (3.7) r = Ric ( ξ, ξ ) = 2( k − λ ) , r = Ric ( E , E ) = − k + c )( k − λ ) ,r = Ric ( E , E ) = − k + c )( k + λ ) , scalar curvature (3.8) s = X r i = − k + λ ) − kc. Remark 1 (Classification - contact metric case, | I M | 6 = 1 ) . Let k = 1 . Then M is contact metric ( κ, µ ) -manifold with κ = 1 − λ , µ = 2(1 + c ) . Boeckx invariant | I M | 6 = 1 . Resolving these equations λ = √ − κ , µ = c + 1 , we obtain explicitform (3.9) [ E , E ] = 2 ξ, [ E , ξ ] = − ( √ − κ + µ/ − E , [ ξ, E ] = ( √ − κ − µ/ E , , Mentioned earlier local isometry with Lie group justifies use of Milnor’s theorem. | I M | = 1 implies λ − c = 0 which contradicts assumptions of the Proposition ?? . lmost contact, paracontact metric structures ... 7 In terms of Boeckx invariant for non-Sasakian manifolds resp. coefficients can beexpressed as (3.10) ( I M − √ − κ, ( I M + 1) √ − κ. By 3.7, for non-Sasakian manifold we have (3.11) r = Ric ( ξ, ξ ) = 2 κ, r = Ric ( E , E ) = − µ (1 − √ − κ ) ,r = Ric ( E , E ) = − µ (1 + √ − κ ) , scalar curvature (3.12) s = r + r + r = 2( κ − µ ) . For example
Ric > if and only if < κ < , µ < . In this case manifold iscompact with compact covering. By list below universal cover is S . Remark 2 (Classification - almost cosymplectic case, | C M | 6 = 1 ) . Let k = 0 . Themanifold M is almost cosymplectic with | C M | 6 = 1 . We find λ = √− κ , µ = 2 c ,explicit form (3.13) [ E , E ] = 0 , [ E , ξ ] = − ( √− κ + µ/ E , [ ξ, E ] = ( √− κ − µ/ E , In terms of Dacko-Olszak invariant C M corresponding coefficients are given by (3.14) ( C M − √− κ, ( C M + 1) √− κ, By (3.7) principal curvatures of Ricci tensor being given by (3.15) r = Ric ( ξ, ξ ) = 2 κ, r = Ric ( E , E ) = µ √− κ,r = Ric ( E , E ) = − µ √− κ, scalar curvature (3.16) s = r + r + r = 2 κ. For example we easily see that there are only two possibilities for signature of Riccitensor in case manifold is non-cosymplectic: ( − , , or ( − , − , +1) . Case A nilpotent is more complex, as there is no symmetry between conditions λ − c = 0 and λ + c = 0 . Proposition 3.
Let A be nilpotent. If λ − c = 0 the commutators of ξ , E , E aregiven by (3.17) [ E , E ] = 2 kξ, [ E , ξ ] = − λE , [ ξ, E ] = 0 . If λ + c = 0 they are given by (3.18) [ E , E ] = 2 kξ, [ E , ξ ] = 0 , [ ξ, E ] = 2 λE . Proposition 4.
Let A be nilpotent. In case λ − c = 0 , the manifold M is almostcontact metric ( k − λ , k + λ )) -space. In case λ + c = 0 , M is ( k − λ , k − λ )) -space.Proof. Proof goes on in the same way as the proof of
Proposition 1 . (cid:3) Corollary 1 (Contact metric structure, I M = − ) . In the Proposition above, for k = 1 , λ − c = 0 , M is contact metric manifold with Boeckx invariant I M = − . Liealgebra of vector fields ( ξ, E , E ) is isomorphic to the Lie algebra of left-invariantvector fields on the Lie groups of rigid motions of hyperbolic plane E (1 , . Plane with indefinite flat pseudo-metric
Piotr Dacko
Corollary 2 (Contact metric structure, I M = 1 ) . In the case k = 1 , λ + c = 0 , M is contact metric manifold with Boeckx invariant I M = 1 . Lie algebra of vectorfields ( ξ, E , E ) is isomorphic to the Lie algebra of left-invariant vector fields onthe Lie group of rigid motions of Euclidean plane E (2) . Almost cosymplectic case is less troublesome, setting k = 0 in (3.17) or in (3.18)we obtain Lie algebra of 3-dimensional Heisenberg Lie group. The respective almostcosymplectic manifolds are ( κ, µ ) -manifolds with Dacko-Olszak invariant C M = − in the case (3.18) or C M = 1 for (3.18).4. Three-dimensional almost paracontact metric ( κ, µ ) -manifolds Family of examples of almost paracontact metric ( κ, µ ) -spaces. Westudy particular class of left-invariant almost paracontact metric structures on Liegroups. Let G be connected 3-dimensional Lie group and ( φ, ξ, η, g ) left-invariantstructure on G . Let ( ξ, E , E ) be an Artin frame of left-invariant vector fields on G . Lie algebra of G is given by [ E , E ] = p E + p E + 2 uξ, (4.1) [ ξ, E ] = aE + bE , [ ξ, E ] = cE + dE . (4.2)Set A = ( a cb d ) , p = ( p p ) , I - identity × -matrix. Jacobi identity requires(4.3) u ( a + d ) = 0 , ( A − ( a + d ) I ) p = 0 In what will follow if not otherwise explicitly stated we assume p = p = 0 , a + d = 0 . Proposition 5.
Levi-Civita connection coefficients are given by ∇ ξ ξ = 0 , ∇ ξ E = l E , ∇ ξ E = − l E , (4.4) ∇ E ξ = − k E − k E , ∇ E E = k ξ, ∇ E E = k ξ, (4.5) ∇ E ξ = − m E − m E , ∇ E E = m ξ, ∇ E E = m ξ. (4.6) Proof.
For example L ( X, Y ) = g ( ∇ ξ X, Y ) is skew-symetric tensor, therefore matrixof coefficients is just common skew-symmetric × -matrix(4.7) [ L ] = − l − l l − l l l . Notice that matrix of coefficients of ∇ ξ , is obtained by switching two last rows in [ L ] . Due to constraints coming from Lie algebra structure there must be l = l = 0 .And we apply the same procedure to ∇ E i , i = 1 , . (cid:3) As connection is torsion-less we have following relations between connection co-efficients and Lie algebra structure constants(4.8) a = l + k = l − m , b = k , c = m , k = − m u = k = − m . Jacobi operator J ξ associated to the vector field ξ is defined by X J ξ X = R Xξ ξ , R stands for curvature operator R XY Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z . Proposition 6.
The curvature R XY ξ , is determined by Jacobi operator J ξ (4.9) R XY ξ = η ( Y ) J ξ X − η ( X ) J ξ Y. lmost contact, paracontact metric structures ... 9 Proof.
It is enough to show that R E E ξ = 0 . So(4.10) R E E ξ = ∇ E ∇ E ξ − ∇ E ∇ E ξ − ∇ [ E ,E ] ξ = − m ∇ E E − m ∇ E E + k ∇ E E + k ∇ E E = − ( m k + m k ) ξ + ( m k + m k ) ξ = 0 , we have used ∇ [ E ,E ] ξ = u ∇ ξ ξ = 0 . (cid:3) If Jacobi operator can be decomposed into J ξ = κId + µh , then G equipped withsuch structure became almost paracontact metric ( κ, µ ) -space. Jacobi operator issymmetric in the sense that g ( J ξ X, Y ) = g ( X, J ξ Y ) . Possible non-zero coefficientsare a ij = g ( J ξ E i , E j ) , i, j = 1 , , a ij = a ji . Therefore(4.11) ( g ( J ξ E i , E j )) = (cid:18) a a a a (cid:19) ,a = a = κ and(4.12) J ξ E = κE + a E , J ξ E = a E + κE , For local components of h = 12 L ξ φ , we have(4.13) hξ = 0 , hE = k E , hE = − m E . Manifold is ( κ, µ ) -manifold, iff a = µk and a = − µm , for some constant µ .We verify a = g ( J ξ E , E ) = R ( E , ξ, ξ, E ) = − l k , (4.14) a = g ( J ξ E , E ) = R ( E , ξ, ξ, E ) = 2 l m . (4.15)So at this point we see that every manifold so far being considered is ( κ, µ ) -manifold with κ = a = R ( E , ξ, ξ, E ) = k m − k m = − ( u + bc ) , (4.16) µ = − l = 2( u − a ) , (4.17)All considered examples satisfy(4.18) dη = u Φ , d Φ = 0 . For particular values u = 1 , u = 0 we obtain paracontact metric ( κ, µ ) -spaces, resp.almost paracosymplectic ( κ, µ ) -spaces.Before proceeding further at first let focus on trying to find out D -homothetyinvariants similar to Boeckx or Dacko-Olszak invariants. Proposition 7.
Quantity ( u − µ/ / ( u + κ ) , is D -homothetically invariant (4.19) ( u − µ/ u + κ = ( u − µ ′ / u + κ ′ . Proof.
For Artin frame ( ξ, E , E ) the frame ( ξ ′ , E ′ , E ′ ) , ξ ′ = ξ/α , E ′ i = E i / √ α , i = 1 , is Artin for structure deformed by D -homothety. By (4.1), there is(4.20) [ E ′ , E ′ ] = 2 uξ ′ , [ ξ ′ , E ′ ] = a ′ E ′ + b ′ E ′ , [ ξ ′ , E ′ ] = c ′ E ′ − a ′ E ′ . Corresponding Lia algebras constants are related by(4.21) a ′ = αa, b ′ = αb, c ′ = αc. Moreover κ , κ ′ , µ , µ ′ are determined in terms of these constants(4.22) κ = − u − bc, κ ′ = − u − b ′ c ′ ,µ = 2( u − a ) , µ ′ = 2( u − a ′ ) . Therefore by above relations D -homothety coefficient(4.23) α = u + κu + κ ′ = (2 u − µ ) (2 u − µ ′ ) . (cid:3) For paracontact metric, u = 1 , ( κ, µ ) -space, κ = − we set E M = (1 − µ/ / (1 + κ ) . (4.24)For almost paracosymplectic, u = 0 , ( κ, µ ) -space, κ = 0 we set F M = ( − µ/ /κ. (4.25)By above remarks both E M , and F M are D -homothety invariant.Set N (1) = [ φ, φ ] − dη ⊗ ξ , N (2) ( X, Y ) = L φX η ( Y ) − L φY η ( X ) , Proposition 8.
For every almost paracontact metric manifold there is g (( ∇ X φ ) Y, Z ) = − d Φ( X, φY, φZ ) − d Φ( X, Y, Z ) − g ( N (1) ( Y, Z ) , φX )+ (4.26) N (2) ( Y, Z ) η ( X ) + 2 dη ( φY, X ) η ( Z ) − dη ( φZ, X ) η ( Y ) , Proof.
Proof is similar to the proof of analogous formula in case of almost contactmetric manifolds. The latter can be found in [4] (cid:3)
Corollary 3.
Three-dimensional almost paracontact metric manifold satisifies g (( ∇ X φ ) Y, Z ) = ( f Φ( Y, X ) + dη ( φY, X ) + g ( hY, X )) η ( Z ) − (4.27) ( f Φ( Z, X ) + dη ( φZ, X ) + g ( hZ, X )) η ( Y ) ,d Φ = 2 f η ∧ Φ (4.28) Proof.
On three-dimensional manifold − d Φ( X, φY, φZ ) − d Φ( X, Y, Z ) = 2 f η ( Y )Φ( X, Z ) − f η ( Z )Φ( X, Y ) . Moreover tensor fields N (1) and N (2) are zero on vector fields Y , Z , such that η ( Y ) = η ( Z ) = 0 . Having this in mind for vector fields Y , Z we set ¯ Y = Y − η ( Y ) ξ , ¯ Z = Z − η ( Z ) ξ and in virtue of (4.26), we have g (( ∇ X φ ) ¯ Y , ¯ Z ) = 0 . Hence g (( ∇ X φ ) Y, Z ) = η ( Y ) g ( ∇ X ξ, φZ ) − η ( Z ) g ( ∇ X ξ, φY ) , and g ( ∇ X ξ, φZ ) = f X, Z ) + dη ( X, φZ ) − g ( X, hZ ) ,h = L ξ φ . From these last two identities we obtain our claim. (cid:3) lmost contact, paracontact metric structures ... 11 Paracontact metric three-dimensional manifolds.
For paracontact met-ric manifold f = 0 , dη = Φ . Corollary 3 follows ( ∇ X φ ) Y = − g ( X, Y − hY ) ξ + η ( Y )( X − hX ) , (4.29) ∇ X ξ = − φX + φhX. (4.30) Proposition 9.
Let M be three-dimensional non para-Sasakian paracontact metric ( κ, µ ) -space. Let p be a point of M . Assume there is local vector field X , onneighborhood of p , that φX = X , hX = 0 . Then on neighborhood of p there is alocal Artin frame ξ , E , E , function b , ǫ = ± , that ∇ ξ ξ = 0 , ∇ ξ E = − µ E , ∇ ξ E = µ E , (4.31) ∇ E ξ = − E − ǫE , ∇ E E = bE + ǫξ, ∇ E E = − bE + ξ, (4.32) ∇ E ξ = ǫ ( κ + 1) E + E , ∇ E E = − ξ, ∇ E E = − ǫ ( κ + 1) ξ, (4.33) if κ = − function b vanishes.Proof. Essentially proof is finished if we can show that there is local Artin so co-efficients of tensor field h are constants. By assumption manifold is ( κ, µ ) -space.Therefore Jacobi operator X J ξ = R Xξ ξ , is given by J ξ = κ ( Id − η ⊗ ξ ) + µh .From other hand J ξ X = ∇ X,ξ ξ − ∇ ξ,X ξ . With help of (4.30) we find ∇ X,ξ − ∇ ξ,X ξ = h X − X + η ( X ) ξ − φ ( ∇ ξ h ) X. (4.34)So we obtain equation ( κ + 1)( Id − η ⊗ ξ ) + µh = h − φ ( ∇ ξ h ) . (4.35)Due to symmetries of h the above equation splits into pair of separate equations h = ( κ + 1)( Id − η ⊗ ξ ) , (4.36) µh = − φ ( ∇ ξ h ) . (4.37)Note h = a a ( Id − η ⊗ ξ ) , where hE = a E , hE = a E . Hence a a = κ + 1 .If necessary we gauge frame to assure a = ǫ = ± , then a = ǫ ( κ + 1) .We are now going back to the proof. By ∇ ξ φ = 0 , (4.37), (4.30), and (4.29) ∇ ξ ξ = 0 , ∇ ξ E = f E , ∇ ξ E = − f E , (4.38) µE = µhE = − φ ( ∇ ξ h ) E = − ǫf E , (4.39) ∇ E ξ = − φE + φhE = − E − ǫE , ∇ E ξ = ǫ ( κ + 1) E + E , (4.40) ∇ E E = bE + ǫξ, ∇ E E = − bE + ξ, (4.41) ∇ E E = − cE − ξ, ∇ E E = cE − ǫ ( κ + 1) ξ, (4.42)with help of all these above formulas we obtain R E E ξ = ∇ E ∇ E ξ − ∇ E ∇ E ξ −∇ [ E ,E ] ξ = − b ( κ + 1) E − cE . Therefore c = 0 and b = 0 if κ = − . (cid:3) Corollary 4.
Let M be three-dimensional non-para-Sasakian paracontact metric ( κ, µ ) -space. Then around each point there is a local Artin frame ξ , E , E , suchthat [ E , E ] = − bE + 2 ξ, (4.43) [ ξ, E ] = (1 − µ E + ǫE , [ ξ, E ] = − ǫ ( κ + 1) E − (1 − µ E , (4.44) b = 0 if κ = − . Conversely having vector fields as in the above corollary define almost paracon-tact metric structure taking ξ , E , E as Artin frame. Such defined structure isalmost paracosymplectic and directly we verify that manifold equipped with thisstructure is paracontact metric ( κ, µ ) -space, where the case b = 0 corresponds to ( − , µ ) -spaces.By above corollary every three-dimensional paracontact metric ( κ, µ ) -space, κ = − , is locally isometric as almost paracontact metric manifold to some three-dimensional connected simply connected Lie group equipped with left-invariantalmost paracontact metric structure. For full list of such groups cf. Table 2. Example 1 (Family of non-isometric paracontact metric ( − , µ ) -spaces) . Underthe assumptions of the
Proposition 9 sectional curvature K D of contact distribu-tion of ( − , µ ) -space is given by K D : D → g ( R E E E , E ) = E b − (1 + µ ) . (4.45) Let function b be that E b = c , c ∈ R . Solutions leading to different values c , c determine non-isometric three-dimensional paracontact ( − , µ ) -space M , M as K D = const. , K D = const. , and K D = K D . Only what remains is to show thatsuch solutions exist. By Corollary 4 , around every point there is local coordinatesystem ( t, x, y ) , that ξ = ∂ t , E = e − (1 − µ ) t ∂ x , (4.46) we set b = b ( t, x, y ) = ( K D + (1 + µ )) e (1 − µ ) t x . Paracosymplectic ( κ, µ ) -spaces. For paracosymplectic manifolds ( ∇ X φ ) Y = g ( X, hY ) ξ − η ( Y ) hX, (4.47) ∇ X ξ = φhX. (4.48) Proposition 10.
Let M be three-dimensional almost paracosymplectic non paraco-symplectic ( κ, µ ) -space. Let p be a point of M . Assume there is local vector field X , on neighborhood of p , that φX = X , hX = 0 . Then on neighborhood of p thereis a local Artin frame ξ , E , E , function b , ǫ = ± , that ∇ ξ ξ = 0 , ∇ ξ E = − µ E , ∇ ξ E = µ E , (4.49) ∇ E ξ = − ǫE , ∇ E E = bE + ǫξ, ∇ E E = − bE , (4.50) ∇ E ξ = ǫκE , ∇ E E = 0 , ∇ E E = − ǫκξ, (4.51) if κ = 0 , function b vanishes b = 0 .Proof. Proof goes on the same way as proof of
Proposition 9 . The first we obtainfor almost paracosymplectic ( κ, µ ) -space h = κ ( Id − η ⊗ ξ ) , (4.52) µh = − φ ( ∇ ξ h ) , (4.53)so there is Artin frame hE = ǫE , hE = ǫκE , ǫ = ± . Now we use (4.47),(4.48) to obtain connection coefficients, finally we apply integrability condition R E E ξ = 0 . (cid:3) lmost contact, paracontact metric structures ... 13 Corollary 5.
Let M be three-dimensional almost paracosymplectic ( κ, µ ) -space.Then around each point there is a local Artin frame ξ , E , E , such that [ E , E ] = − bE , (4.54) [ ξ, E ] = − µ E + ǫE , [ ξ, E ] = − ǫκE + µ E , (4.55) if κ = 0 , b = 0 . Conversely having vector fields as in the above corollary define almost paracon-tact metric structure taking ξ , E , E as Artin frame. Such defined structure isalmost paracosymplectic and directly we verify that manifold equipped with thisstructure is almost paracosymplectic ( κ, µ ) -space, where the case b = 0 correspondsto (0 , µ ) -spaces.Corollary above allow us to create full list of three-dimensional Lie groups ad-mitting structure of paracosymplectic ( κ, µ ) -spaces, κ = 0 , up to D -homothety, cf.Table 4. Example 2 (Family of non-isometric (0 , µ ) -spaces) . For three-dimensional almostparacosymplectic (0 , µ ) -space sectional curvature K D is given by K D : D → g ( R E E E , E ) = E b. (4.56) If M , M , are two almost paracosymplectic (0 , µ ) -spaces and K D = const. , K D = const. , K D = K D , such spaces are non-isometric as almost paracon-tact metric manifolds. For every isometry preserving Reeb fields must preservesectional curvature of almost contact distributions. For every constant c ∈ R , thereis manifold with K D = c . We just need to solve equations ξb = − µ , E b = c, (4.57) by Corollary 5 , there is local coordinate system, such that ξ = ∂ t , E = e µ t ∂ x , soone of possible solution is b = K D e − µ t x . References [1] K. M. T. Abbassi and G. Calvaruso, g-natural contact metrics on unit tangent sphere bundles,
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Three-dimensional Lie groups with left-invariant, non-Sasakiancontact metric ( κ, µ ) -structures. Lie group Description E M = (1 − µ/ κ SO (3) or SU (2) simple, compact, S or S / {± } < E M < - - E M = 0 and κ > − SL (2 , R ) or O (1 , simple, R , compact quotients, E M < , E M > - - E M = 0 , κ < − E (2) or E (1 , solvable, R , compact quotients, E M = 1 Table 2.
Three-dimensional Lie groups with left-invariant, non para-Sasakian paracontact metric ( κ, µ ) -structures, κ = − . Lie group Description C M = − µ/ √− κ E (2) solvable, R , compact quotients | C M | > E (1 , solvable, R , compact quotients | C M | < Heisenberg Lie group H nilpotent, R , compact quotients | C M | = 1 Table 3.
Three-dimensional Lie groups with left-invariant, non-cosymplectic almost cosymplectic ( κ, µ ) -structures, κ < . Lie group Description F M = ( − µ/ κ E (2) solvable, R , compact quotients < F M < - - F M = 0 , κ > E (1 , solvable, R , compact quotients | F M | > - - F M = 0 , κ < Heisenberg Lie group H nilpotent, R , compact quotients F M = 1 Table 4.
Three-dimensional Lie groups with left-invariant, non paracosym-plectic almost paracosymplectic ( κ, µ ) -structures, κ = 0= 0