An. Şt. Univ. Ovidius Constanţa Vol. 17(1), 2009, 15 22 An isoparametric function on almost k-contact manifolds Adara M. BLAGA Abstract The aim of this paper is to point out an isoparametric function on an almost k-contact manifold. 1 Introduction Almost 3-contact manifolds were introduced by Kuo [2] and independently, by Udrişte [5]. To their class belong also 3-Sasakian and 3-cosymplectic manifolds studied by Boyer and Galicki [1], whose properties were also analyzed by Montano and De Nicola [4]. In this paper, starting with a proposal for the notion of almost k-contact structure, we shall point out an isoparametric function which can be associated in this framework, by generalizing a similar construction initiated by Mihai and Rosca [3]. 2 Almost k-contact manifolds Recall that an almost contact manifold is an odd-dimensional manifold (M,Φ,ξ,η), where 1. Φ is a field of endomorphisms of the tangent space; 2. ξ is a vector field (called the Reeb vector field); 3. η is a 1-form, such that Key Words: almost k-contact manifold; isoparametric function; Riemannian metric. Mathematics Subject Classification: 53C15. Received: December, 2008 Accepted: March, 2009 15
16 Adara M. Blaga Φ 2 = I Γ(TM) + η ξ η(ξ) = 1, where I Γ(TM) denotes the identity on the Lie algebra of vector fields. Proposition 1 Any almost contact manifold (M, Φ, ξ, η) admits a Riemannian metric g (called compatible metric) with the properties: g(φ(x),φ(y )) = g(x,y ) η(x)η(y ), for any X,Y Γ(TM). g(ξ,x) = η(x), We call (M,Φ,ξ,η,g) almost contact metric manifold. In this case, the Reeb vector field ξ is orthonormal with respect to g [g(ξ,ξ) = η(ξ) = 1]. A natural generalization of almost 3-contact manifold [4] is given by the following definition: Definition 1 An almost k-contact manifold is an (n + k + nk)-dimensional manifold M with k almost contact structures (Φ 1,ξ 1,η 1 ),...,(Φ k,ξ k,η k ) such that: Φ i Φ j = δ ij I Γ(TM) + η j ξ i + k ε ijlφ l η i (ξ j ) = δ ij, for any i,j,l {1,...,k}, where ε ijl is the totally antisymmetric symbol. It follows that Φ i (ξ j ) = k ε ijlξ l and η i Φ j = k ε ijlη l, for any i,j {1,...,k}. A similar computation like in the almost contact case leads us to Φ i (ξ i ) = 0 and η i Φ i = 0, for any i {1,...,k}. Consider now the case i j. Then Φ i Φ j = η j ξ i + ε ijl Φ l and computing this relation on ξ i and respectively ξ j, we obtain ε ijl Φ l (ξ i ) = Φ i (Φ j (ξ i )) = ξ j, for any i j. Multiplying ξ l = Φ j (Φ l (ξ j )) with ε ijl and summing over l, we get ε ijl ξ l = ε ijl Φ j (Φ l (ξ j )) = Φ j ( ε ijl Φ l (ξ j )) = Φ j (ξ i ).
An isoparametric function 17 Then Φ i (ξ j ) = ε jil ξ l = ε ijl ξ l. Computing Φ 2 i = I Γ(TM)+η i ξ i for Φ j (X), with arbitrary X Γ(TM), we obtain It follows that Φ j (X) + η i (Φ j (X))ξ i = Φ 2 i(φ j (X)) = Φ i [(Φ i Φ j )(X)] = Φ i [η j (X)ξ i + ε ijl Φ l (X)] (η i Φ j )(X)ξ i = Φ j (X) + = Φ j (X) + = Φ j (X) + = ε ijl (Φ i Φ l )(X). ε ijl (Φ i Φ l )(X) ε ijl [ δ il X + η l (X)ξ i + ε ijl η l (X)ξ i Φ j (X) = for any X Γ(TM). Applying η i, we find ε ilp Φ p (X)] p=1 ε ijl η l (X)ξ i, for any X Γ(TM). (η i Φ j )(X) = ε ijl η l (X), Proposition 2 Any almost k-contact manifold (M,Φ i,ξ i,η i ) admits a Riemannian metric g compatible with each of the k almost contact structures: g(φ i (X),Φ i (Y )) = g(x,y ) η i (X)η i (Y ), for any X,Y Γ(TM),i {1,...,k}. g(ξ i,x) = η i (X), (1) We call (M,Φ i,ξ i,η i,g) almost k-contact metric manifold. In this case, the Reeb vector fields ξ 1,...,ξ k are orthonormal with respect to g [g(ξ i,ξ j ) = η i (ξ j ) = δ ij, for any i,j {1,...,k}].
18 Adara M. Blaga 3 Isoparametric function Let (M,Φ i,ξ i,η i,g) be an almost k-contact metric manifold and define H := k ker η i the horizontal distribution. Then the tangent bundle splits into the orthogonal sum of the horizontal and vertical distributions, TM = H < ξ 1,...,ξ k >. Consider the vector field ξ := k λ iξ i, λ i C (M) and define the 1-form η := i ξ g. Then η(x) = i ξ g(x) = g(ξ,x) = g( λ i ξ i,x) = for any X Γ(TM) and in particular for X = ξ, η(ξ) = λ i η i (ξ) = λ i g(ξ i,x) = λ i η i (X), λ i g(ξ i,ξ) = g(ξ,ξ) = ξ 2. (2) Let be the Levi-Civita connection associated to g. From Cartan s structure equations, for {e i } an orthonormal frame and θ the local connection form, we have e = θ e, with θ j i = λ iη j λ j η i, i,j {1,...,k}. If we assume that ξ defines a skew symmetric connection, then θ j i (ξ) = 0 and dη i = η η i, i {1,...,k}. It follows so 0 = d 2 η i = d(η η i ) = dη η i η dη i = dη η i η (η η i ) = dη η i, 0 = (dη η i )(X,Y ) = dη(x)η i (Y ) dη(y )η i (X), for any X,Y Γ(TM). In particular, for X = ξ i, Y = ξ j, i j, 0 = dη(ξ i )η i (ξ j ) dη(ξ j )η i (ξ i ), we find dη(ξ j ) = 0, for any j {1,...,k}. Now, for Y = ξ i, 0 = dη(x)η i (ξ i ) dη(ξ i )η i (X) = dη(x), for any X Γ(TM) and so dη = 0. Following the ideas of Mihai and Rosca [3], we shall prove that on an almost k-contact manifold, ξ 2 is an isoparametric function. Let (X) := i X g and := 1 be the musical isomorphisms.
An isoparametric function 19 Assume that λ i = fξ i, f C (M). Then (dλ i ) = fξ i 1 (dλ i ) = fξ i dλ i = (fξ i ) = i fξi g = fi ξi g = fη i and 0 = d 2 λ i = d(fη i ) = df η i + fdη i = df η i + fη η i = (df + fη) η i implies df + fη = 0. Set 2λ = ξ 2 [= g(ξ,ξ)]. Then dλ = d( g(ξ,ξ) 2 = 1 2 d[ 1 i,j k = 1 2 ( = f ) = 1 2 d[g( λ i ξ i, λ j ξ j )] = 1 2 d[ j=1 λ i λ j η i (ξ j )] = 1 2 d[ 2λ i dλ i ) = λ i η i = fη 1 i,j k λ i dλ i = 1 i,j k λ i λ j δ ij ] = 1 2 d[ λ i fη i λ i λ j g(ξ i,ξ j )] λ 2 i] and d(f + λ) = df + dλ = df + fη = 0 implies f + λ = c(constant). From the structure s equations follows that Z ξ i = λ i η j (Z)ξ j η i (Z)ξ, (3) j=1 for any Z Γ(TM), i {1,...,k} [3]. Therefore, Lemma 1 For any Z Γ(TM), Z ξ = (2λ + f) k j=1 η j(z)ξ j η(z)ξ.
20 Adara M. Blaga Proof. Indeed, Z ξ = = for any Z Γ(TM). [λ i Z ξ i + Z(λ i )ξ i ] [λ i (λ i η j (Z)ξ j η i (Z)ξ) + Z(λ i )ξ i ] j=1 = [ λ 2 i][ η j (Z)ξ j ] η(z)ξ + dλ i (Z)ξ i j=1 = ξ 2 [ η j (Z)ξ j ] η(z)ξ + fη i (Z)ξ i j=1 = (2λ + f)[ η i (Z)ξ i ] η(z)ξ, Theorem 1 Let on an almost k-contact metric manifold M a number of k- smooth functions λ i such that for all i, the gradient vector field λ i is parallel with ξ i with the same factor f C (M). Then, for the vector field ξ := k λ iξ i, its norm is an isoparametric function on M. Proof. Since {ξ i } is an orthonormal set for g we have: 2λ = λ 2 i and then: Therefore, Then, but λ = f( λ i ξ i ) = (c λ)( λ i ξ i ) = (c λ)ξ. (4) λ 2 = (c λ) 2 2λ. (5) div( λ) = (c λ)divξ ξ(λ), (6) ξ(λ) = 1 2 ξ(g(ξ,ξ)) = g( ξξ,ξ). (7)
An isoparametric function 21 From: it results: ξ ξ = (λ + c)ξ η(ξ)ξ = (c λ)ξ, (8) div( λ) = (c λ)[kc + k 2 2λ 2λ] = (c λ)[kc + (k 4)λ], (9) 2 which, for k = 3 gives the relation (2.24) of Rosca-Mihai. Acknowledgments I wish to express my thanks to the referees for their useful remarks. References [1] C. Boyer, K. Galicki, 3-Sasakian manifolds. Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., VI, Int. Press, Boston, MA 1999, pp. 123 184. [2] Y. Y. Kuo, On almost contact 3-structure, Tohoku Math. J., 22 (1970), 235 332. [3] A. Mihai, R. Roşca, On vertical skew symmetric almost contact 3-structures, J. Geom., 82(2005), 146 155. [4] B. Cappelletti Montano, A. De Nicola, 3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem, J. Geom. Phys., 57(2007), 2509 2520. [5] C. Udrişte, Structures presque coquaternioniennes, Bull. Math. R. S. Roumanie, 13(61)(1969), 487 507. Adara M. Blaga Department of Mathematics and Computer Science West University of Timişoara Bld. V. Pârvan nr.4, 300223 Timişoara, România adara@math.uvt.ro
22 Adara M. Blaga