Harmonic Subtangent Structures

The concept of harmonic subtangent structures on almost subtangent metric manifolds is introduced and a Bochner-type formula is proved for this case. Conditions for a subtangent harmonic structure to be preserved by harmonic maps are also given. 1. 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 K？hler manifolds [4], Ianus et al. the quaternionic K？hler 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 the Lie algebra of vector fields on , and by the -module of tensor fields of -type on . An element of is usually called vector 1-form or affinor. Recall the concept of almost tangent geometry. Definition 1 (see [10]). is called almost tangent structure on if it has a constant rank and The pair is called almost tangent manifold. The name is motivated by the fact that (1) implies the nilpotence exactly as the natural tangent structure of tangent bundles. Denoting it results in . 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 defines a foliation;(ii)there exist local coordinates on such that ; that is, We call canonical coordinates and the change of canonical coordinates 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 . The natural almost tangent structure of is an example of tangent structure having exactly the expression (3) if are the