J. Geom. 82 (2005) 146 – 155 0047–2468/05/020146 – 10 © Birkh¨auser Verlag, Basel, 2005 DOI 10.1007/s00022-005-1753-7
On vertical skew symmetric almost contact 3-structures Adela Mihai and Radu Rosca
Abstract. We consider a (2m + 3)-dimensional Riemannian manifold M(ξr , ηr , g) endowed with a vertical skew symmetric almost contact 3-structure. Such manifold is foliated by 3-dimensional submanifolds M ⊥ of constant curvature tangent to the vertical distribution and the square of the length of the vertical structure vector field is an isoparametric function. If, in addition, M(ξr , ηr , g) is endowed with an f -structure ϕ, M turns out to be a framed f-CR-manifold. The fundamental 2-form associated with ϕ is a presymplectic form. Locally, M is the Riemannian product M = M × M ⊥ of two totally geodesic submanifolds, where M is a 2m-dimensional Kaehlerian submanifold and M ⊥ is a 3-dimensional submanifold of constant curvature. If M is not compact, a class of local Hamiltonians of is obtained. Mathematics Subject Classification (2000): 53C15, 53D15, 53C25. Key words: Vertical skew symmetric almost contact 3-structure, f -structure, framed f -manifold, presymplectic form, local Hamiltonian, infinitesimal homothety.
0. Introduction In the present paper we consider a (2m+3)-dimensional Riemannian manifold M(ξr , ηr , g) carrying 3 Reeb vector fields ξr , r ∈ {2m + 1, 2m + 2, 2m + 3}. If the vertical distribution H ⊥ = {ξr } is structured by a skew symmetric connection [13] defined by a vertical vector field ξ ∈ H ⊥ , then, since the class of the Reeb covectors ηr is 1, we agree to say that M(ξr , ηr , g) is endowed with a vertical skew symmetric (abbr. VSS) almost contact 3-structure. It is proved that any such manifold is foliated by 3-dimensional submanifolds M ⊥ of constant curvature tangent to H ⊥ and the square ξ 2 of the length of ξ is an isoparametric function, in the sense of [19]. Next we assume that M(ξr , ηr , g) is, in addition, endowed with an f -structure ϕ [21], i.e., ϕ is a field of endomorphisms of its tangent spaces such that 3 ϕ + ϕ = 0, ϕ 2 = −I d + r ηr ⊗ ξr , ϕξr = 0, ηr ◦ ϕ = 0, (0.1) g(Z, Z ) = g(ϕZ, ϕZ ) + r ηr (Z)ηr (Z ), Z, Z ∈ TM (Id is the identity morphism on TM). Supported by a JSPS Postdoctoral Fellowship.
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Under these conditions, M turns out to be a framed f -manifold, endowed with a VSS almost contact 3-structure. It is shown that the fundamental 2-form associated with the f -structure is a presymplectic form having as symplectic vector space the horizontal distribution H of M(Hp = {Z ∈ Tp M; ηr (Z) = 0}). It is proved that such an M is a framed f − CR manifold [1] and is the local Riemannian product M = M × M ⊥ of two totally geodesic submanifolds such that i) M is a 2m-dimensional Kaehlerian submanifold tangent to the horizontal distribution; ii) M ⊥ is a 3-dimensional submanifold of constant curvature tangent to the vertical distribution. If M is not compact, then any vector field X such that ∇Z X = τ ϕZ, τ ∈ C ∞ M, is a local Hamiltonian of and, in this case, ϕX is an infinitesimal homothety of , i.e., LϕX = 2τ , τ = constant. 1. Preliminaries Let (M, g) be a Riemannian C ∞ -manifold and ∇ be the covariant differential operator with respect to the metric tensor g. We assume that M is oriented and ∇ is the Levi-Civita connection. Let TM be the set of sections of the tangent bundle and : TM → T ∗ M and = −1 the classical musical isomorphisms defined by g. Following [16], we denote by Aq (M, TM) = Hom( q TM, TM) the set of vector valued q-forms, q < dim M, and by d ∇ : Aq (M, TM) → Aq+1 (M, TM) the covariant derivative operator with respect to ∇ (in general d ∇ = d ∇ ◦ d ∇ = 0, unlike d 2 = d ◦ d = 0). The vector valued 1-form dp ∈ A1 (M, TM) is the identity vector valued 1-form, called the soldering form of M (see [4]). Since ∇ is symmetric, one has d ∇ (dp) = 0. 2
The operator d ω = d + e(ω)
(1.1)
acting on M is called the cohomology operator [5]. In (1.1), e(ω) means the exterior product by the closed 1-form ω, i.e., d ω u = du + ω ∧ u,
(1.2)
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for any u ∈ M. One has d ω ◦ d ω = 0.
(1.3)
A form u such that d ω u = 0 is said to be a d ω -closed and if ω is an exact form, then u is called d ω -exact. A vector field X ∈ TM such that ∇ 2 X = π ∧ dp ∈ A2 (M, TM),
(1.4)
for some 1-form π, is said to be an exterior concurrent vector field [18], [15]. The 1-form π , which is called the concurrence form, is defined by π = λ(X), λ ∈ C ∞ M.
(1.5)
If R is the Ricci tensor of ∇, then, by (1.4), one has R(X, Z) = −(n − 1)λg(X, Z), Z ∈ TM,
(1.6)
where n = dim M. A function f : Rn → R is isoparametric [19] if ∇f 2 and div(∇f ) are functions of f (∇f = grad f ). A vector field X such that ∇X = λdp + ω ⊗ X
(i.e.,∇Z X = λZ + ω(Z)X),
(1.7)
is called a torse forming [20]. Let O = {eA ; A = 1, . . . , n} be a local field of adapted vectorial frames over M and let O ∗ = {ωA } be its associated coframe. Then the soldering form dp of M is expressed by dp = α A ⊗ ξA and E. Cartan’s structure equations written in indexless manner are ∇e = θ ⊗ e,
(1.8)
dω = −θ ∧ ω,
(1.9)
dθ = −θ ∧ θ + .
(1.10)
In the above equations, θ (resp. ) are the local connection forms in the tangent bundle TM (resp. the curvature forms on M). 2. Manifolds with ξ as vertical structure vector field and η as vertical structure Pfaffian Let (M, g) be a (2m + 3)-dimensional oriented Riemannian manifold carrying 3 Reeb vector fields ξr , r, s ∈ {2m + 1, 2m + 2, 2m + 3}, that is ηr (ξs ) = δrs , where ηr = (ξr ) denotes the Reeb covectors.
(2.1)
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One may decompose the tangent space Tp M at ∀p ∈ M as Tp M = Hp ⊕ Hp⊥ ,
(2.2)
where H ⊥ is a 3-distribution determined by {ξr }, called the vertical distribution, and its orthogonal complement H (Hp = {Z ∈ Tp M; ηr (Z) = 0}), called the horizontal (or the almost contact) distribution. In consequence of this decomposition, any vector field Z ∈ TM may be split as Z = Z + Z⊥,
(2.3)
where Z ⊥ ∈ H ⊥ (respectively Z ∈ H ) is the vertical component of Z (respectively the horizontal component of Z). Similarly, the soldering form dp may be split as dp = dp + dp ⊥ ,
(2.4)
where dp and dp ⊥ are the vertical and the horizontal components of dp, respectively. On the other hand, by reference to [17], the connections forms θsr in the vertical subbundle O ⊥ (M)(r, s = 2m + 1, 2m + 2, 2m + 3) are called the vertical connection forms. Let ξ=
λ r ξr ∈ H ⊥ ,
λr ∈ C ∞ M,
(2.5)
r
be a vertical vector field. Assume that ξ defines a skew symmetric connection in the sense of [13]. Thus, the following relations θsr = ξ, ξs ∧ ξr
(2.6)
hold good. Since ξs ∧ ξr are expressed by ξs ∧ ξ r = η r ⊗ ξ s − η s ⊗ ξ r ,
(2.7)
θsr = λs ηr − λr ηs .
(2.8)
it follows that
It should be noticed that one has θsr (ξ ) = 0, which shows that the vertical connection forms are relations of integral invariance for the vertical vector field ξ (in the sense of [10]). If relations (2.6) hold good, we agree to say that the (2m + 3)-dimensional Riemannian manifold M(ξr , ηr , g) is endowed with a vertical skew symmetric (abbr. VSS) almost contact 3-structure.
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In the following we call ξ the vertical structure (abbr. VS) vector field and its dual η = (ξ ) the VS 1-form. Making use of the structure equation (1.9), one infers on behalf of (2.8) dηr = η ∧ ηr
(2.9)
and it follows that the VS 1-form η is closed. Therefore, in terms of d ω -cohomology, one may express (2.9) as d −η ηr = 0,
(2.10)
i.e., the Reeb covectors are d −η -closed. In addition, by the structure equation (1.8) and by (2.8), one gets ∇ξr = λr dp ⊥ − ηr ⊗ ξ,
(2.11)
where the vertical component dp ⊥ of dp is expressed by dp ⊥ = ηr ⊗ ξr .
(2.12)
If Z1⊥ , Z2⊥ ∈ H ⊥ are any vertical vector fields, from (2.11) and (2.12) it follows that ∇Z ⊥ Z1⊥ ∈ H ⊥ , 2
(2.13)
which shows that the vertical distribution H ⊥ defines an autoparallel foliation (see [8]). Next, since dη = 0, we assume that ∇λr = tξr ⇐⇒ dλr = tηr , t ∈ C ∞ M,
(2.14)
and by (2.9) it follows at once dt + tη = 0.
(2.15)
dλ = tη ⇒ t + λ = c = constant,
(2.16)
Setting 2λ = ξ 2 , one gets
and taking the covariant differential of ξ , one derives by (2.11) and (2.14) ∇ξ = (2λ + t)dp⊥ − η ⊗ ξ.
(2.17)
We notice that if Z is any vector field, it follows from (2.17) ∇Z ξ = (2λ + t)Z ⊥ − η(Z)ξ, and on behalf of (1.7) one may say that ξ is a vertical torse forming.
(2.18)
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On the other hand, since by (2.9) one has ηr ∧ dηr = 0, following a known definition (see also [9]), it is seen that the Reeb covectors ηr are of class 1. This, as is known, is in accordance with the almost contact 3-structure character defined by {ξr }. Operating now on (2.12) by the covariant differential operator d ∇ , one derives by (2.9) and (2.11) d ∇ (dp ⊥ ) = 0,
(2.19)
which shows that dp ⊥ is the soldering form of the leaf M ⊥ of H . Further, operating on ∇ξr by d ∇ , one derives by (2.11) and (2.19) d ∇ (∇ξr ) = ∇ 2 ξr = 2cηr ∧ dp ⊥ .
(2.20)
From (2.20) it follows that the Reeb vector fields ξr are exterior concurrent vector fields with 2c as conformal scalar. Since the property of exterior concurrency is invariant by linearity, it follows that, if Z ⊥ is any vertical vector field, one has ∇ 2 Z ⊥ = 2c(Z ⊥ ) ∧ dp ⊥ .
(2.21)
Therefore, on behalf of [15], we conclude that the 3-dimensional manifold M ⊥ is of constant curvature −2c. By (2.16), one also has ∇ξ 2 = gradξ 2 = 2tξ and consequently ∇ξ 2 = (2c − ξ 2 )ξ.
(2.22)
On the other hand, since in general div Z = trace ∇Z, one derives from (2.17) div ξ =
1 ξ 2 + 3c, 2
(2.23)
and by (2.21) one finds 1 2 div(∇ξ ) = (2c − ξ ) 3c − ξ . 2 2
2
(2.24)
By (2.22) and (2.21) it follows that ∇ξ 2 2 and div(∇ξ 2 ) are both functions of ξ 2 , i.e., ξ 2 is an isoparametric function. THEOREM 1. Let M(ξr , ηr , g) be a (2m+3)-dimensional Riemannian manifold endowed with a VSS almost contact 3-structure with ξ as vertical structure vector field and (ξ ) = η as vertical structure Pfaffian. Any such manifold is foliated by 3-dimensional submanifolds M ⊥ of constant curvature −2c, tangent to the vertical distribution H ⊥ = {ξr } and ξ 2 is an isoparametric function.
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3. Manifolds with an f -structure We assume in this section that the manifold M(ξr , ηr , g) under discussion is equipped with an f -structure ϕ [21], i.e., ϕ is a field of endomorphisms of its tangent spaces of rank 2m, which satisfies ϕ 3 + ϕ = 0, ϕ 2 = −I d + r ηr ⊗ ξr , ϕξr = 0, ηr ◦ ϕ = 0, (3.1) g(Z, Z ) = g(ϕZ, ϕZ ) + r ηr (Z)ηr (Z ), Z, Z ∈ TM where Id is the identity morphism on TM and r ∈ {2m + 1, 2m + 2, 2m + 3}. In addition, the fundamental 2-form associated with the f -structure satisfies (Z, Z ) = g(ϕZ, Z ),
m ∧ η2m+1 ∧ η2m+2 ∧ η2m+3 = 0.
(3.2)
We agree to say that in this case M(ϕ, , ξr , ηr , g) is a framed f -manifold with a VSS almost contact 3-structure. With respect to the cobasis O ∗ = {ωA , ηr } of O = {eA , ξr }, A ∈ {1, . . . , 2m}, the 2-form is expressed by ∗ = ωa ∧ ωa , a ∈ {1, . . . , m}, a ∗ = a + m, (3.3) and by (2.11) and (3.1) the connection forms θAr (the transversal connection forms [17]) and the connection forms θAB (the horizontal connection forms) satisfy θAr = 0,
(3.4)
and the well known Kaehlerian relations ∗
∗
∗
θba = θba∗ , θba = θab .
(3.5)
Making use of (3.4) and (3.5), a standard calculation gives d = 0.
(3.6)
Hence one may say that the structure 2-form is a presymplectic (or a quasi-symplectic [9]) form of rank 2m. Next, by the same above relations, it is seen that H is also an autoparallel foliation. Thus, in the case under consideration, both distributions H ⊥ and H are parallel with respect to the Levi-Civita connection ∇. Therefore, if we will denote by M the 2m-dimensional leaf of H , then clearly M is a totally geodesic Kaehlerian submanifold of M. Obviously, by (3.1) and (2.11), one has ϕ∇ξr = ∇ϕξr = 0. Also, by (3.5) and (3.1), it is seen that ∇ϕZ = ϕ∇Z, for any horizontal vector field Z. Thus, ∇ and ϕ commute, or, in other words, the ϕ-covariant derivative of Z, (∇ϕ)Z, vanishes.
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We recall that the torsion tensor field S of an f -structure is the vector valued 2-form defined by S = Nϕ + 2 (3.7) dηr ⊗ ξr , where Nϕ (Z, Z ) = [ϕZ, ϕZ ] + ϕ 2 [Z, Z ] − ϕ[Z, ϕZ ] − ϕ[ϕZ, Z ]
(3.8)
is the Nijenhuis tensor field of ϕ. The f -structure (ϕ, ηr , ξr ) is said to be H -normal if S vanishes on H . Recall now the following. PROPOSITION [1]. A framed f -manifold endowed with a H -normal f -structure is a framed f − CR-manifold. Comming back to the case under discussion, it is seen by (2.9), (3.1) and (3.9) that S vanishes on H. Hence, it is proved that a framed f -manifold endowed with a VSS almost contact 3-structure is a framed f − CR-manifold. THEOREM 2. Any framed f -manifold M(ϕ, , ξr , ηr , g) endowed with a vertical skew symmetric almost contact 3-structure is a CR-manifold and may be viewed as the local Riemannian product M = M × M⊥ such that i) M is a Kaehlerian submanifold tangent to the horizontal distribution H of M and is totally geodesic immersed in M; ii) M ⊥ is a 3-dimensional submanifold of constant curvature tangent to the vertical distribution H ⊥ and totally geodesic immersed in M. 4. Properties of horizontal vector fields on a non-compact manifold Since |M is the structure symplectic form of M , one has the musical symplectic isomorphisms : Hp → Hp∗ , : Hp∗ → Hp [9]. Let X ∈ H be any horizontal vector field and assume that its covariant derivative ∇X satisfies ∇Z X = τ ϕZ, Z ∈ TM, τ ∈ C ∞ M.
(4.1)
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Since, in general, Z → −iZ = (Z) = Z, then, setting ∗
X = X a e a + X a ea ∗ , one finds
X = −iX = −
∗
(4.2)
∗
(Xa ωa − X a ωa ).
(4.3)
Now, in consequence of (4.1) and making use of the structure equation (1.9), one derives d((ϕX)) = 0,
(4.4)
which shows that X is a local Hamiltonian of the presymplectic form . Next, since (∇ϕ)Z = 0, one gets by (4.1) ∇ϕX = τ dp ⇔ ∇Z ϕX = τ Z ,
(4.5)
and by a short calculation one finds (ϕX) = −iϕX = (X).
Further, similarly as for X, we infer by a standard calculation LϕX = 2τ , which by dL = Ld implies τ = constant. Hence, the above equation shows that ϕX defines an infinitesimal homothety of the presymplectic form . We notice that in general case, since d |M = 0, the space E(M) = {U ∈ TM; LU = 0} of local Hamiltonian vector fields of has dim ≥ 3. THEOREM 3. If the manifold M(ϕ, , ξr , ηr , g) is not compact, any horizontal vector field X such that ∇Z X = τ ϕZ is a local Hamiltonian of the presymplectic form and, in this case, ϕX defines an infinitesimal homothety of , i.e., LϕX = 2τ , τ = const. Aknowledgements The first author would like to express her hearty thanks to Prof. Dr. Koji Matsumoto for usefull discussions and valuable advices and for the hospitality received during her visit to Yamagata University.
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[email protected]
Received 7 November 2003.
Radu Rosca 59 Avenue Emile Zola 75015 Paris France
