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 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 on M, by Γ(TM) the Lie algebra of vector fields on M, and by Tsr(M) the C∞(M)-module of tensor fields of (r,s)-type on M. An element of T11(M) is usually called vector 1-form or affinor.
Recall the concept of almost tangent geometry.
Definition 1 (see [10]).
J∈T11(M) is called almost tangent structure on M if it has a constant rank and
(1)ImJ=kerJ.
The pair (M,J) is called almost tangent manifold.
The name is motivated by the fact that (1) implies the nilpotence J2=0 exactly as the natural tangent structure of tangent bundles. Denoting rankJ=n it results in m=2n. If in addition, we assume that J is integrable, that is,
(2)NJ(X,Y)∶=[JX,JY]-J[JX,Y]-J[X,JY]+J2[X,Y]=0,
then J is called tangent structure and (M,J) is called tangent manifold.
From [11] we deduce some aspects of tangent manifolds:
the distribution ImJ(=kerJ) defines a foliation;
there exist local coordinates (x,y)=(xi,yi)1≤i≤n on M such that J=∂/∂yi⊗dxi; that is,
(3)J(∂∂xi)=∂∂yi,J(∂∂yi)=0.
We call (x,y) canonical coordinates and the change of canonical coordinates (x,y)→(x~,y~) is given by
(4)x~i=x~i(x),y~i=∂x~i∂xaya+Bi(x).
So another description can be obtained in terms of G-structures. Namely, a tangent structure is a G-structure with [12]
(5)G={C=(AOnBA)∈GL(2n,R);hhhA∈GL(n,R),B∈gl(n,R)(AOnBA)}
and G is the invariance group of matrix J=(OnOnInOn); that is, C∈G if and only if C·J=J·C.
The natural almost tangent structure J of M=TN is an example of tangent structure having exactly the expression (3) if (xi) are the coordinates on N and (yi) are the coordinates in the fibers of TN→N. A class of examples is obtained by duality [12]: if J is an (integrable) endomorphism with J2=0, then its dual J*:Γ(T*M)→Γ(T*M), given by J*α:=α∘J for α∈Γ(T*M), is (integrable) endomorphism with (J*)2=0.
If the condition in the Definition 1 is weakened, requiring that only J squares to 0, we call J almost subtangent structure. In this case, ImJ⊂kerJ.
2. Harmonic Subtangent Structures
Let (M,g,J) be an almost subtangent metric manifold of dimension 2n, that is, a 2n-dimensional smooth manifold M endowed with an almost subtangent structure J which is compatible with a pseudo-Riemannian metric g (i.e., g(JX,Y)=g(X,JY), for any X, Y∈Γ(TM)) and let ∇ be the Levi-Civita connection associated with g. Consider the exterior differential and codifferential operators defined for any tangent bundle-valued p-form T∈Γ(ΛpT*M⊗TM) by
(6)dT(X0,…,Xp)∶=∑i=0p(-1)i(∇XiT)(X0,…,Xi^,…,Xp),δT(X1,…,Xp-1)∶=-∑i=12n(∇EiT)(Ei,X1,…,Xp-1),
for {Ei}1≤i≤2n an orthonormal frame field and the Hodge-Laplace operator on Γ(ΛpT*M⊗TM):
(7)Δ:=d∘δ+δ∘d.
Jianming studied in [1] some properties of harmonic complex structures and we discussed in [13] the paracosymplectic case.
Definition 2.
An almost subtangent structure J is called harmonic if ΔJ=0.
If M is compact, from the definition it follows that J is harmonic if and only if dJ=0 and δJ=0 which is equivalent to (∇XJ)Y=(∇YJ)X, for any X, Y∈Γ(TM) and trace(∇J)=0, ∇ being the Levi-Civita connection associated with the pseudo-Riemannian structure g.
Proposition 3.
On a compact almost subtangent manifold, any harmonic almost subtangent structure J is integrable (i.e., it is a subtangent structure).
Proof.
Let X, Y∈Γ(TM). Then
(8)(dJ)(X,Y)∶=(∇J)(X,Y)-(∇J)(Y,X)=[X,JY]+∇JYX-[Y,JX]-∇JXY-J[X,Y].
As ΔJ=0 implies dJ=0, we get
(9)0=(dJ)(JX,Y)+(dJ)(X,JY)=[JX,JY]+J2[X,Y]-J[JX,Y]-J[X,JY],
which shows the integrability of J.
Remark 4.
As expected, the harmonicity of an almost subtangent structure is not always preserved under conformal transformations. Indeed, let J be a harmonic subtangent structure (with respect to Δ) and for a smooth positive function f on the 2n-dimensional manifold M, let g~=f2g. Then the Levi-Civita connection associated with g~ is ∇~XY=∇XY+(1/f)X(f)·Y+(1/f)Y(f)·X-(1/f)g(X,Y)gradg(f), for any X, Y∈Γ(TM). The necessary and sufficient condition for J to be harmonic (with respect to Δ~) is
(10)(∇~XJ)Y=(∇~YJ)X,foranyX,Y∈Γ(TM),trace~(∇~J)=0,
but
(11)(∇~XJ)Y=(∇XJ)Y+1f[(gradg(f))(JY)(f)·X-Y(f)·JXddddddddddddd-g(X,JY)gradg(f)ddddddddddddd+g(X,Y)J(gradg(f))],
so the first relation is equivalent to
(12)X(f)·JY-(JX)(f)·Y=Y(f)·JX-(JY)(f)·X,foranyX,Y∈Γ(TM).
Taking {Ei}1≤i≤2n an orthonormal frame field on (M,g) with ∇EiEj=0, 1≤i,j≤2n, and computing
(13)trace~(∇~J)=trace(∇J)+1f∑i=12n[gradg(f)(JEi)(f)·Ei-Ei(f)·JEiggggggggggggggggggggggggggg+J(gradg(f))ggggggggggggggggggggggggggg-g(Ei,JEi)gradg(f)],
the second relation is equivalent to
(14)∑i=12n[Ei(f)·JEi-(JEi)(f)·Ei]=∑i=12n[J(gradg(f))-g(Ei,JEi)gradg(f)].
In conclusion, J is also harmonic with respect to Δ~ if and only if
(15)df⊗J-J⊗df=(df∘J)⊗I-I⊗(df∘J)(16)trace[df⊗J-(df∘J)⊗I]=2nfJ(gradg(f))-trace[g∘(I×J)]·gradg(f).
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, T∈Γ(Λ1T*M⊗TM), the following Weitzenböck formula holds [14]:
(17)ΔT=-∇2T-S,
where ∇2T:=∑i=12n∇Ei∇EiT-∇∇EiEiT and S(X):=∑i=12n(REiXT)Ei, X∈Γ(TM), for {Ei}1≤i≤2n an orthonormal frame field and RXY:=∇X∇Y-∇Y∇X-∇[X,Y], X, Y∈Γ(TM), the Riemann curvature tensor field. We will also use the notations RXYZ=:RXYZ and RXYZW=:g(RXYZ,W), X, Y, Z, W∈Γ(TM). Now, on the almost subtangent metric manifold (M,g,J), taking T equal to J, for any vector field X, we have
(18)S(X)∶=∑i=12n(REiXJ)Ei=∑i=12nREiXJEi-∑i=12nJ(REiXEi)=∑i=12n[REiXJEi-J(REiXEi)].
We can state the following theorem.
Theorem 5.
Let (M,g,J) be an almost subtangent metric manifold and assume that J is harmonic subtangent structure. Then a Bochner-type formula reduces to
(19)|∇J|2=∑1≤i,j≤2nREiEjJEiJEj,
for {Ei}1≤i≤2n an orthonormal frame field on (M,g) with ∇EiEj=0, 1≤i,j≤2n.
Proof .
A similar computation like in [1] leads us to
(20)〈∇2J,J〉=∑i=12n〈∇Ei∇EiJ,J〉=-|∇J|2,〈S,J〉=∑j=12n〈SEj,JEj〉=∑1≤i,j≤2ng(REiEjJEi,JEj).
Therefore, as J is harmonic if ΔJ=0, from (17), we obtain
(21)0=〈ΔJ,J〉=-〈∇2J,J〉-〈S,J〉=|∇J|2-∑1≤i,j≤2ng(REiEjJEi,JEj)=|∇J|2-∑1≤i,j≤2nREiEjJEiJEj.
Notice that if J is only almost subtangent structure, from the proof of the theorem, we deduce that
(22)〈ΔJ,J〉=|∇J|2-∑1≤i,j≤2nREiEjJEiJEj.
If M 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 (M,g,J) be a compact almost subtangent metric manifold. Then the almost subtangent structure J is harmonic if and only if
(23)∫M(|∇J|2-∑1≤i,j≤2nREiEjJEiJEj)dμg=0.
Example 7.
Concerning the existence of almost tangent structures of order 2 (i.e., those J with J2=0) on the spheres, Rosendo and Gadea [15] proved that the only spheres that admit such structures are S3 and S15. Moreover, they proved that the only spheres that admit almost tangent structures (of different orders) are S3 (of order 2), S7 (of order 6), and S15 (of order 2 or 4). For these cases, let JEi=∑1≤k≤2nJikEk and traceJ=∑1≤k≤2nJkk. Computing REiEjJEiJEj=JiiJjj-JijJji and taking into account that ∑1≤k≤2nJikJkj=0, for any i,j∈{1,…,2n}, from Corollary 6, we get
(24)∫S3|∇J|2dμg(traceJ)2=4π3,∫S15|∇J|2dμg(traceJ)2=2·(2π)715!!.
3. Harmonic Maps and Harmonic Subtangent Structures
Let (M,g,J) and (M~,g~,J~) be two almost subtangent metric manifolds 2n- and 2n~-dimensional, respectively. Denote by ∇ and ∇~, respectively, the Levi-Civita connections associated with g and g~, respectively.
Consider Φ:(M,g,J)→(M~,g~,J~) a smooth map and let
(25)τ(Φ)∶=∑i=12n[∇~Φ*EiΦ*Ei-Φ*(∇EiEi)]
be the tension field of Φ, where {Ei}1≤i≤2n is an orthonormal frame field on (M,g).
Proposition 8.
Let Φ:(M,g,J)→(M~,g~,J~) be a smooth map between almost subtangent metric manifolds such that Φ*∘J=J~∘Φ*. Then
(26)J~(τ(Φ))-Φ*(
trace
(∇J))+
trace
((∇~J~)∘(Φ*×Φ*))=∑i=12n[∇~Φ*EiΦ*(JEi)-Φ*(∇EiJEi)],
for {Ei}1≤i≤2n an orthonormal frame field on the 2n-dimensional manifold (M,g).
Proof.
Express trace(∇J)=∑i=12n(∇J)(Ei,Ei)=∑i=12n[∇EiJEi-J(∇EiEi)] and replace it in the left side of the relation.
Proposition 9.
Let Φ:(M,g,J)→(M~,g~,J~) be a smooth map between almost subtangent metric manifolds such that Φ*∘J=J~∘Φ*. If for any X∈Γ(TM), Φ*∘∇XJ=(∇~Φ*XJ~)∘Φ*, then
(27)
trace
((∇~J~)∘(Φ*×Φ*))=Φ*(
trace
(∇J)).
Proof.
For any X, Y∈Γ(TM), ∇~Φ*XJ~(Φ*Y)-Φ*(∇XJY)=J~(∇~Φ*XΦ*Y)-Φ*(J(∇XY)) and for X=Y=Ei,
(28)J~(τ(Φ))-Φ*(trace(∇J))+trace((∇~J~)∘(Φ*×Φ*))=J~(∑i=12n[∇~Φ*EiΦ*Ei-Φ*(∇EiEi)])=J~(τ(Φ)).
Definition 10.
A smooth map Φ:(M,g,J)→(M~,g~,J~) is said to be harmonic if its tension field τ(Φ) vanishes.
Proposition 11.
Let Φ:(M,g,J)→(M~,g~,J~) be a smooth map between almost subtangent metric manifolds such that Φ*∘J=J~∘Φ*. If Φ is harmonic map, then
(29)Φ*(
trace
(∇J))=
trace
((∇~J~)∘(Φ*×Φ*))-∑i=12n[∇~Φ*EiΦ*(JEi)-Φ*(∇EiJEi)],
for {Ei}1≤i≤2n an orthonormal frame field on the 2n-dimensional manifold (M,g).
Moreover, if for any X∈Γ(TM), Φ*∘∇XJ=(∇~Φ*XJ~)∘Φ*, then
(30)∑i=12n[∇~Φ*EiΦ*(JEi)-Φ*(∇EiJEi)]=0,
for {Ei}1≤i≤2n an orthonormal frame field on the 2n-dimensional manifold (M,g).
Corollary 12.
Let Φ:(M,g,J)→(M~,g~,J~) be a smooth map between almost subtangent metric manifolds such that Φ*∘J=J~∘Φ* and J is harmonic subtangent structure.
If for any X∈Γ(TM), Φ*∘∇XJ=(∇~Φ*XJ~)∘Φ*, then
(31)
trace
((∇~J~)∘(Φ*×Φ*))=0.
Moreover, if Φ is surjective submersion, then J~ is harmonic subtangent structure, too.
If Φ is harmonic map, then
(32)
trace
((∇~J~)∘(Φ*×Φ*))=∑i=12n[∇~Φ*EiΦ*(JEi)-Φ*(∇EiJEi)],
for {Ei}1≤i≤2n an orthonormal frame field on the 2n-dimensional manifold (M,g).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
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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