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.
Inspired by the paper of Jianming , 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 , Bejan and Benyounes the almost para-Hermitian manifolds , Sahin the locally conformal Kähler manifolds , Ianus et al. the quaternionic Kähler manifolds , Jaiswal the Sasakian manifolds , Fetcu the complex Sasakian manifolds , Li the Finsler manifolds , and so forth. Fotiadis studied the noncompact case, describing the problem of finding a harmonic map between noncompact manifolds .
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 ). 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  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  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 coordinates on and are the coordinates in the fibers of . A class of examples is obtained by duality : if is an (integrable) endomorphism with , then its dual , given by for , is (integrable) endomorphism with .
If the condition in the Definition 1 is weakened, requiring that only squares to , we call almost subtangent structure. In this case, .
2. Harmonic Subtangent Structures
Let be an almost subtangent metric manifold of dimension , that is, a -dimensional smooth manifold endowed with an almost subtangent structure which is compatible with a pseudo-Riemannian metric (i.e., , for any , and let be the Levi-Civita connection associated with . Consider the exterior differential and codifferential operators defined for any tangent bundle-valued -form by for an orthonormal frame field and the Hodge-Laplace operator on :
Definition 2. An almost subtangent structure is called harmonic if .
If is compact, from the definition it follows that is harmonic if and only if and which is equivalent to , for any , and , 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).
Proof. Let , . Then As implies , we get which shows the integrability of .
Remark 4. As expected, the harmonicity of an almost subtangent structure is not always preserved under conformal transformations. Indeed, let be a harmonic subtangent structure (with respect to ) and for a smooth positive function on the -dimensional manifold , let . Then the Levi-Civita connection associated with is , for any , . The necessary and sufficient condition for to be harmonic (with respect to ) is
so the first relation is equivalent to
Taking an orthonormal frame field on with , , and computing the second relation is equivalent to
In conclusion, is also harmonic with respect to if and only if
Now we want to see how a Bochner-type formula can be written on an almost subtangent metric manifold.
We know that for any tangent bundle-valued differential form, , the following Weitzenböck formula holds : where and , , for an orthonormal frame field and , , , the Riemann curvature tensor field. We will also use the notations and , , , , . Now, on the almost subtangent metric manifold , taking equal to , for any vector field , we have
We can state the following theorem.
Theorem 5. Let be an almost subtangent metric manifold and assume that is harmonic subtangent structure. Then a Bochner-type formula reduces to for an orthonormal frame field on with , .
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 (i.e., those with ) on the spheres, Rosendo and Gadea  proved that the only spheres that admit such structures are and . Moreover, they proved that the only spheres that admit almost tangent structures (of different orders) are (of order ), (of order ), and (of order or ). For these cases, let and . Computing and taking into account that , for any , from Corollary 6, we get
3. Harmonic Maps and Harmonic Subtangent Structures
Let and be two almost subtangent metric manifolds - and -dimensional, respectively. Denote by and , respectively, the Levi-Civita connections associated with and , respectively.
Consider a smooth map and let be the tension field of , where is an orthonormal frame field on .
Proposition 8. Let be a smooth map between almost subtangent metric manifolds such that . Then for an orthonormal frame field on the -dimensional manifold .
Proof. Express and replace it in the left side of the relation.
Proposition 9. Let be a smooth map between almost subtangent metric manifolds such that . If for any , , then
Proof. For any , , and for ,
Definition 10. A smooth map is said to be harmonic if its tension field vanishes.
Proposition 11. Let be a smooth map between almost subtangent metric manifolds such that . If is harmonic map, then
for an orthonormal frame field on the -dimensional manifold .
Moreover, if for any , , then for an orthonormal frame field on the -dimensional manifold .
Corollary 12. Let be a smooth map between almost subtangent metric manifolds such that and is harmonic subtangent structure. (1)If for any , , then Moreover, if is surjective submersion, then is harmonic subtangent structure, too.(2)If is harmonic map, then for an orthonormal frame field on the -dimensional manifold .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author thanks the referees for the valuable suggestions they made in order to improve the paper. She also acknowledges the support by the Research Grant PN-II-ID-PCE-2011-3-0921.
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