Harmonic Subtangent Structures

Inspired by the paper of Jianming [1], we introduce the notion of harmonic almost subtangent structure and underline the connection between harmonic subtangent structures and harmonic maps. It is well known that harmonic maps play an important role in many areas of mathematics. They often appear in nonlinear theories because of the nonlinear nature of the corresponding partial differential equations. In theoretical physics, harmonic maps are also known as sigma models. Remark also that harmonic maps between manifolds endowed with different geometrical structures have been studied in many contexts: Ianus and Pastore treated the case of contact metric manifolds [2], Bejan and Benyounes the almost para-Hermitian manifolds [3], Sahin the locally conformal Kähler manifolds [4], Ianus et al. the quaternionic Kähler manifolds [5], Jaiswal the Sasakian manifolds [6], Fetcu the complex Sasakian manifolds [7], Li the Finsler manifolds [8], and so forth. Fotiadis studied the noncompact case, describing the problem of finding a harmonic map between noncompact manifolds [9]. Let M be a smooth, m-dimensional real manifold for which we denote by C∞(M) the real algebra of smooth real functions onM, by Γ(TM) the Lie algebra of vector fields on M, and byT s (M) theC(M)-module of tensor fields of (r, s)type on M. An element of T1 1 (M) is usually called vector 1form or affinor. Recall the concept of almost tangent geometry. Definition 1 (see [10]). J ∈ T1 1 (M) is called almost tangent structure onM if it has a constant rank and


Introduction
Inspired by the paper of Jianming [1], we introduce the notion of harmonic almost subtangent structure and underline the connection between harmonic subtangent structures and harmonic maps.It is well known that harmonic maps play an important role in many areas of mathematics.They often appear in nonlinear theories because of the nonlinear nature of the corresponding partial differential equations.In theoretical physics, harmonic maps are also known as sigma models.Remark also that harmonic maps between manifolds endowed with different geometrical structures have been studied in many contexts: Ianus and Pastore treated the case of contact metric manifolds [2], Bejan and Benyounes the almost para-Hermitian manifolds [3], Sahin the locally conformal Kähler manifolds [4], Ianus et al. the quaternionic Kähler manifolds [5], Jaiswal the Sasakian manifolds [6], Fetcu the complex Sasakian manifolds [7], Li the Finsler manifolds [8], and so forth.Fotiadis studied the noncompact case, describing the problem of finding a harmonic map between noncompact manifolds [9].
Let  be a smooth, -dimensional real manifold for which we denote by  ∞ () the real algebra of smooth real functions on , by Γ() the Lie algebra of vector fields on , and by    () the  ∞ ()-module of tensor fields of (, )type on .An element of  1 1 () is usually called vector 1form or affinor.
Recall the concept of almost tangent geometry.
The name is motivated by the fact that (1) implies the nilpotence  2 = 0 exactly as the natural tangent structure of tangent bundles.Denoting rank  =  it results in  = 2.If in addition, we assume that  is integrable, that is, then  is called tangent structure and (, ) is called tangent manifold.
From [11] we deduce some aspects of tangent manifolds: (i) the distribution Im (= ker ) defines a foliation; (ii) there exist local coordinates (, ) = (  ,   ) 1≤≤ on  such that  = /  ⊗   ; that is, We call (, ) canonical coordinates and the change of canonical coordinates (, ) → ( x, ỹ) is given by So another description can be obtained in terms of structures.Namely, a tangent structure is a -structure with [12] and  is the invariance group of matrix  = (         ); that is,  ∈  if and only if  ⋅  =  ⋅ .
If the condition in the Definition 1 is weakened, requiring that only  squares to 0, we call  almost subtangent structure.In this case, Im  ⊂ ker .
If  is compact, from the definition it follows that  is harmonic if and only if  = 0 and  = 0 which is equivalent to (∇  ) = (∇  ), for any ,  ∈ Γ() and trace(∇) = 0, ∇ being the Levi-Civita connection associated with the pseudo-Riemannian structure .Proposition 3. On a compact almost subtangent manifold, any harmonic almost subtangent structure  is integrable (i.e., it is a subtangent structure).
Taking {  } 1≤≤2 an orthonormal frame field on (, ) with ∇     = 0, 1 ≤ ,  ≤ 2, and computing the second relation is equivalent to In conclusion,  is also harmonic with respect to Δ if and only if trace Now we want to see how a Bochner-type formula can be written on an almost subtangent metric manifold.
Proof .A similar computation like in [1] leads us to (       ,   ) . (20) Therefore, as  is harmonic if Δ = 0, from (17), we obtain Notice that if  is only almost subtangent structure, from the proof of the theorem, we deduce that If  is compact, integrating this relation with respect to the canonical measure, we obtain the following characterization of a harmonic almost subtangent structure.Corollary 6.Let (, , ) be a compact almost subtangent metric manifold.Then the almost subtangent structure  is harmonic if and only if Example 7. Concerning the existence of almost tangent structures of order 2 (i.e., those  with  2 = 0) on the spheres, Rosendo and Gadea [15] proved that the only spheres that admit such structures are  3 and  15 .Moreover, they proved that the only spheres that admit almost tangent structures (of different orders) are  3 (of order 2),

Journal of Mathematics
∇ and ∇, respectively, the Levi-Civita connections associated with  and g, respectively.