An isoparametric function on almost k-contact manifolds

Transcription

1 An. Şt. Univ. Ovidius Constanţa Vol. 17(1), 2009, 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,

2 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 ).

3 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}].

4 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.

5 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.

6 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)

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 [2] Y. Y. Kuo, On almost contact 3-structure, Tohoku Math. J., 22 (1970), [3] A. Mihai, R. Roşca, On vertical skew symmetric almost contact 3-structures, J. Geom., 82(2005), [4] B. Cappelletti Montano, A. De Nicola, 3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem, J. Geom. Phys., 57(2007), [5] C. Udrişte, Structures presque coquaternioniennes, Bull. Math. R. S. Roumanie, 13(61)(1969), Adara M. Blaga Department of Mathematics and Computer Science West University of Timişoara Bld. V. Pârvan nr.4, Timişoara, România adara@math.uvt.ro

8 22 Adara M. Blaga