Almost complex structures that are harmonic maps
aa r X i v : . [ m a t h . DG ] N ov ALMOST COMPLEX STRUCTURES THAT ARE HARMONICMAPS
JOHANN DAVIDOV, ABSAR UL HAQ, OLEG MUSHKAROV
Abstract.
We find geometric conditions on a four-dimensional almost Her-mitian manifold under which the almost complex structure is a harmonic mapor a minimal isometric imbedding of the manifold into its twistor space.2010
Mathematics Subject Classification . Primary 53C43, Secondary 58E20,53C28
Key words: almost complex structures, twistor spaces, harmonic maps Introduction
Recall that an almost complex structure on a Riemannian manifold (
M, g ) iscalled almost Hermitian if it is g -orthogonal. If a Riemannian manifold admits analmost Hermitian structure J , it has many such structures. One way to see this isto consider the twistor bundle π : Z → M whose fibre at a point p ∈ M consists ofall g -orthogonal complex structures I p : T p M → T p M ( I p = − Id ) on the tangentspace of M at p yielding the same orientation as J p . The fibre is the compactHermitian symmetric space SO (2 n ) /U ( n ) and its standard metric − T race I ◦ I is K¨ahler-Einstein. The twistor space admits a natural Riemannian metric h suchthat the projection map π : ( Z , h ) → ( M, g ) is a Riemannian submersion withtotally geodesic fibres. Consider J as a section of the bundle π : Z → M and takea section V with compact support K of the bundle J ∗ V → M , the pull-back under J of the vertical bundle V → Z . There exists ε > I ofthe compact set J ( K ), the exponential map exp I is a diffeomorphism of the ε -ballin T I Z . The function || V || h is bounded on M , so there exists a number ε ′ > || tV ( p ) || h < ε for every p ∈ M and t ∈ ( − ε ′ , ε ′ ). Set J t ( p ) = exp J ( p ) [ tV ( p )]for p ∈ M and t ∈ ( − ε ′ , ε ′ ). Then J t is a section of Z , i.e. an almost Hermitianstructure on ( M, g ) (such that J t = J on M \ K ).Thus it is natural to seek for ”reasonable” criteria that distinguish some of thealmost Hermitian structures on a given Riemannian manifold (cf., for example,[6, 23, 24, 4]). Motivated by the harmonic map theory, C. Wood [23, 24] hassuggested to consider as ”optimal” those almost Hermitian structures J : ( M, g ) → ( Z , h ) that are critical points of the energy functional under variations throughsections of Z , i.e. that are harmonic sections of the twistor bundle. In general,these critical points are not harmonic maps, but, by analogy, they are referred toas ”harmonic almost complex structures” in [23, 24]. It is more appropriate in the The first and the third named authors are partially supported by the National Science Fund,Ministry of Education and Science of Bulgaria under contract DFNI-I 02/14. context of this article to call such structures ”harmonic sections”, a term used alsoin [23].The almost Hermitian structures that are critical points of the energy functionalunder variations through all maps M → Z are genuine harmonic maps and thepurpose of this paper is to find geometric conditions on a four-dimensional almostHermitian manifold ( M, g, J ) under which the almost complex structure J is aharmonic map of ( M, g ) into ( Z , h ). We also find conditions for minimality of thesubmanifold J ( M ) of the twistor space. As is well-known, in dimension four, thereare three basic classes in the Gray-Hervella classification [14] of almost Hermitianstructures - Hermitian, almost K¨ahler (symplectic) and K¨ahler structures. If ( g, J )is K¨ahler, the map J : ( M, g ) → ( Z , h ) is a totally geodesic isometric imbedding.In the case of a Hermitian structure, we express the conditions for harmonicityand minimality of J in terms of the Lee form, the Ricci and star-Ricci tensors of( M, g, J ), while for an almost K¨ahler structure the conditions are in terms of theRicci, star-Ricci and Nijenhuis tensors. Several examples illustrating these resultsare discussed in the last section of the paper, among them a Hermitian structurethat is a harmonic section of the twistor bundle and a minimal isometric imbeddingin it but not a harmonic map.
Acknowledgment . The authors would like to thank the referee for his/her re-marks. 2.
Preliminaries
Let (
M, g ) be an oriented Riemannian manifold of dimension four. The metric g induces a metric on the bundle of two-vectors π : Λ T M → M by the formula g ( v ∧ v , v ∧ v ) = 12 det [ g ( v i , v j )] . The Levi-Civita connection of (
M, g ) determines a connection on the bundle Λ T M ,both denoted by ∇ , and the corresponding curvatures are related by R ( X ∧ Y )( Z ∧ T ) = R ( X, Y ) Z ∧ T + Z ∧ R ( X, Y ) T for X, Y, Z, T ∈ T M . The curvature operator R is the self-adjoint endomorphismof Λ T M defined by g ( R ( X ∧ Y ) , Z ∧ T ) = g ( R ( X, Y ) Z, T )Let us note that we adopt the following definition for the curvature tensor R : R ( X, Y ) = ∇ [ X,Y ] − [ ∇ X , ∇ Y ].The Hodge star operator defines an endomorphism ∗ of Λ T M with ∗ = Id .Hence we have the orthogonal decompositionΛ T M = Λ − T M ⊕ Λ T M where Λ ± T M are the subbundles of Λ T M corresponding to the ( ± ∗ .Let ( E , E , E , E ) be a local oriented orthonormal frame of T M . Set s = E ∧ E + E ∧ E , s = E ∧ E + E ∧ E , s = E ∧ E + E ∧ E . (1)Then ( s , s , s ) is a local orthonormal frame of Λ T M defining an orientation onΛ T M , which does not depend on the choice of the frame ( E , E , E , E ). LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 3
For every a ∈ Λ T M , define a skew-symmetric endomorphism K a of T π ( a ) M by g ( K a X, Y ) = 2 g ( a, X ∧ Y ) , X, Y ∈ T π ( a ) M. (2)Note that, denoting by G the standard metric − T race P Q on the space of skew-symmetric endomorphisms, we have G ( K a , K b ) = 2 g ( a, b ) for a, b ∈ Λ T M . If σ ∈ Λ T M is a unit vector, then K σ is a complex structure on the vector space T π ( σ ) M compatible with the metric and the orientation of M . Conversely, the 2-vector σ dual to one half of the fundamental 2-form of such a complex structure isa unit vector in Λ T M . Thus the unit sphere subbunlde Z of Λ T M parametrizesthe complex structures on the tangent spaces of M compatible with its metric andorientation. This subbundle is called the twistor space of M .The Levi-Civita connection ∇ of M preserves the bundles Λ ± T M , so it induces ametric connection on each of them denoted again by ∇ . The horizontal distributionof Λ T M with respect to ∇ is tangent to the twistor space Z . Thus we have thedecomposition T Z = H ⊕ V of the tangent bundle of Z into horizontal and verticalcomponents. The vertical space V τ = { V ∈ T τ Z : π ∗ V = 0 } at a point τ ∈ Z is the tangent space to the fibre of Z through τ . Considering T τ Z as a subspaceof T τ (Λ T M ) (as we shall always do), V τ is the orthogonal complement of τ inΛ T π ( τ ) M . The map V ∋ V τ → K V gives an identification of the vertical spacewith the space of skew-symmetric endomorphisms of T π ( τ ) M that anti-commutewith K τ . Let s be a local section of Z such that s ( p ) = τ where p = π ( τ ).Considering s as a section of Λ T M , we have ∇ X s ∈ V τ for every X ∈ T p M since s has a constant length. Moreover, X hτ = s ∗ X − ∇ X s is the horizontal lift of X at τ . Denote by × the usual vector cross product on the oriented 3-dimensional vectorspace Λ T p M , p ∈ M , endowed with the metric g . Then it is easy to check that g ( R ( a ) b, c ) = g ( R ( b × c ) , a ) (3)for a ∈ Λ T p M , b, c ∈ Λ T p M . It is also easy to show that for every a, b ∈ Λ T p MK a ◦ K b = − g ( a, b ) Id + K a × b . (4)For every t >
0, define a Riemannian metric h t by h t ( X hσ + V, Y hσ + W ) = g ( X, Y ) + tg ( V, W )for σ ∈ Z + , X, Y ∈ T π ( σ ) M , V, W ∈ V σ .The twistor space Z admits two natural almost complex structures that arecompatible with the metrics h t . One of them has been introduced by Atiyah,Hitchin and Singer who have proved that it is integrable if and only if the basemanifold is anti-self-dual [2]. The other one, introduced by Eells and Salamon,although never integrable, plays an important role in harmonic map theory [11].The action of SO (4) on Λ R preserves the decomposition Λ R = Λ R ⊕ Λ − R . Thus, considering S as the unit sphere in Λ R , we have an action ofthe group SO (4) on S . Then, if SO ( M ) denotes the principal bundle of theoriented orthonormal frames on M , the twistor space Z is the associated bundle SO ( M ) × SO (4) S . It follows from the Vilms theorem (see, for example, [3, Theorem9.59]) that the projection map π : ( Z , h t ) → ( M, g ) is a Riemannian submersionwith totally geodesic fibres (this can also be proved by a direct computation).
JOHANN DAVIDOV, ABSAR UL HAQ, OLEG MUSHKAROV
Let (
N, x , ..., x ) be a local coordinate system of M and let ( E , ..., E ) be anoriented orthonormal frame of T M on N . If ( s , s , s ) is the local frame of Λ T M defined by (1), then e x a = x a ◦ π , y j ( τ ) = g ( τ, ( s j ◦ π )( τ )), 1 ≤ a ≤
4, 1 ≤ j ≤ T M on π − ( N ).The horizontal lift X h on π − ( N ) of a vector field X = X a =1 X a ∂∂x a is given by X h = X a =1 ( X a ◦ π ) ∂∂ e x a − X j,k =1 y j ( g ( ∇ X s j , s k ) ◦ π ) ∂∂y k . (5)Hence [ X h , Y h ] = [ X, Y ] h + X j,k =1 y j ( g ( R ( X ∧ Y ) s j , s k ) ◦ π ) ∂∂y k (6)for every vector fields X and Y on N . Let τ ∈ Z . Using the standard identification T τ (Λ T π ( τ ) M ) ∼ = Λ T π ( τ ) M , we obtain from (6) the well-known formula[ X h , Y h ] τ = [ X, Y ] hτ + R p ( X ∧ Y ) τ, p = π ( τ ) . (7)Denote by D the Levi-Civita connection of ( Z , h t ). Then, using the Koszulformula for the Levi-Civita connection, identity (7), and the fact that the fibers of Z are totally geodesic submanifolds, it is easy to show the following. Lemma 1. ([8]) If X, Y are vector fields on M and V is a vertical vector field on Z , then ( D X h Y h ) τ = ( ∇ X Y ) hτ + 12 R p ( X ∧ Y ) τ, (8)( D V X h ) τ = H ( D X h V ) τ = − t R p ( τ × V ) X ) hτ (9) where τ ∈ Z , p = π ( τ ) , and H means ”the horizontal component”. Let (
M, g, J ) be an almost Hermitian manifold of dimension four. Define asection J of Λ T M by g ( J , X ∧ Y ) = 12 g ( JX, Y ) , X, Y ∈ T M.
Note that the section 2 J is dual to the fundamental 2-form of ( M, g, J ). Consider M with the orientation induced by the almost complex structure J . Then J takesits values in the twistor space Z of the Riemannian manifold ( M, g ).Let J and J be the Atiyah-Hitchin-Singer and Eells-Salamon almost complexstructures on the twistor space Z . It is well-known (and easy to see) that the map J : ( M, J ) → ( Z , J ) is holomorphic if and only if the almost complex structure J isintegrable, while J : ( M, J ) → ( Z , J ) is holomorphic if and only if J is symplectic(i.e. ( g, J ) is an almost K¨ahler structure).In this note we are going to find geometric conditions under which the map J : ( M, g ) → ( Z , h t ) that represents J is harmonic. LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 5
Let J − T Z → M be the pull-back of the bundle T Z → Z under the map J : M → Z . Then we can consider the differential J ∗ : T M → T Z as a section ofthe bundle Hom ( T M, J − T Z ) → M . Denote by D ( J ) the connection on J − T Z induced by the Levi-Civita connection D on T Z . The Levi Civita connection ∇ on T M and the connection D ( J ) on J − T Z induce a connection e ∇ on the bundle Hom ( T M, J − T Z ). The map J : ( M, g ) → ( Z , h t ) is harmonic iff T race e ∇ J ∗ = 0 . (cf., for example, [10]). Recall also that the map J : ( M, g ) → ( Z , h t ) is totallygeodesic iff e ∇ J ∗ = 0. Proposition 1.
For every
X, Y ∈ T p M , p ∈ M , e ∇ J ∗ ( X, Y ) = 12 [ ∇ XY J − g ( ∇ XY J , J ) J ( p ) + ∇ Y X J − g ( ∇ Y X J , J ) J ( p ) − t ( R p ( J ( p ) × ∇ X J ) Y ) hσ − t ( R p ( J ( p ) × ∇ Y J ) X ) hσ ] where ∇ XY J = ∇ X ∇ Y J − ∇ ∇ X Y J is the second covariant derivative of the section J of Λ T M . Proof . Extend X and Y to vector fields in a neighbourhood of the point p . Takean oriented orthonormal frame E , ..., E near p such that E = JE , E = JE ,so J = s . Define coordinates (˜ x a , y j ) as above by means of this frame and acoordinate system of M at p . Set V = (1 − y ) − / ( y ∂∂y − y ∂∂y ) ,V = (1 − y ) − / ( − y y ∂∂y + (1 − y ) ∂∂y − y y ∂∂y ) . Then V , V is a g -orthonormal frame of vertical vector fields in a neighbourhood ofthe point σ = J ( p ) such that V ◦ J = s , V ◦ J = s . Note also that [ V , V ] σ = 0.This and the Koszul formula imply ( D V k V l ) σ = 0 since D V k V l are vertical vectorfields, k, l = 1 ,
2. Thus D W V l = 0, l = 1 ,
2, for every vertical vector W at σ .Considering J as a section of Λ T M , we have J ∗ ◦ Y = Y h ◦ J + ∇ Y J = Y h ◦ J + X k =1 g ( ∇ Y J , s k )( V k ◦ J ) , hence D ( J ) X ( J ∗ ◦ Y ) = ( D J ∗ X Y h ) ◦ J + P k =1 g ( ∇ Y J , s k )( D J ∗ X V k ) ◦ J + P k =1 [ g ( ∇ X ∇ Y J , s k ) + g ( ∇ Y J , ∇ X s k )]( V k ◦ J )This, in view of Lemma 1, implies D ( J ) X p ( J ∗ ◦ Y ) = ( ∇ X Y ) hσ + 12 R ( X, Y ) σ − t R p ( J ( p ) × ∇ X J ) Y ) hσ − t R p ( J ( p ) × ∇ Y J ) X ) hσ + P k =1 g ( ∇ X ∇ Y J , s k ) s k ( p ) + P k =1 [ g ( ∇ Y J , s k )[ X h , V k ] σ + g ( ∇ Y J , ∇ X s k ) s k ( p )] . An easy computation using (5) gives[ X h , V ] σ = g ( ∇ X s , s ) s ( p ) , [ X h , V ] σ = g ( ∇ X s , s ) s ( p ) . JOHANN DAVIDOV, ABSAR UL HAQ, OLEG MUSHKAROV
These identities imply X k =1 [ g ( ∇ Y J , s k )[ X h , V k ] σ + g ( ∇ Y J , ∇ X s k ) s k ( p )] = 0 . Since g ( ∇ X J , J ) = 0 for every X ∈ T p M , we have g ( ∇ Y ∇ X J , J ) = − g ( ∇ X J , ∇ Y J ) = g ( ∇ X ∇ Y J , J )Hence X k =1 g ( ∇ X ∇ Y J , s k ) s k ( p ) = ∇ X ∇ Y J − g ( ∇ X ∇ Y J + ∇ Y ∇ X J , J ) J ( p ) . It follows that e ∇ J ∗ ( X, Y ) = D ( J ) X p ( J ∗ ◦ Y ) − ( ∇ X Y ) hσ − ∇ ∇ X Y J = 12 [ ∇ X ∇ Y J − ∇ ∇ X Y J − g ( ∇ X ∇ Y J , J ) J ( p )+ ∇ Y ∇ X J − ∇ ∇ Y X J − g ( ∇ Y ∇ X J , J ) J ( p ) − t ( R p ( J ( p ) × ∇ X J ) Y ) hσ − t ( R p ( J ( p ) × ∇ Y J ) X ) hσ ] . Corollary 1. If ( M, g, J ) is a K¨ahler surface, the map J : ( M, g ) → ( Z , h t ) is atotally geodesic isometric imbedding. Remark 1 . By a result of C. Wood [23, 24], J is a harmonic section of the twistorspace ( Z , h t ) → ( M, g ) if and only if [ J, ∇ ∗ ∇ J ] = 0 where ∇ ∗ ∇ is the roughLaplacian. Taking into account that ∇ ∗ ∇ J = − T race ∇ J , one can see that thelatter condition is equivalent to g ( T race ∇ J , X ∧ Y − JX ∧ JY ) = 0 , X, Y ∈ T M, which is equivalent to V T race ∇ J = 0 . Thus, by Proposition 1, J is a harmonicsection if and only if V T race e ∇ J ∗ = 0 . Harmonicity of J Let Ω(
X, Y ) = g ( JX, Y ) be the fundamental 2-form of the almost Hermitianmanifold (
M, g, J ). Denote by N the Nijenhuis tensor of JN ( Y, Z ) = − [ Y, Z ] + [
JY, JZ ] − J [ Y, JZ ] − J [ JY, Z ] . It is well-known (and easy to check) that2 g (( ∇ X J )( Y ) , Z ) = d Ω( X, Y, Z ) − d Ω( X, JY, JZ ) + g ( N ( Y, Z ) , JX ) , (10)for all X, Y, Z ∈ T M . LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 7
The case of integrable J . Suppose that the almost complex structure J is integrable. This is equivalent to ( ∇ X J )( Y ) = ( ∇ JX J )( JY ), X, Y ∈ T M [13,Corollary 4.2]. Let B be the vector field on M dual to the Lee form θ = − δ Ω ◦ J with respect to the metric g . Then (10) and the identity d Ω = θ ∧ Ω imply thefollowing well-known formula2( ∇ X J )( Y ) = g ( JX, Y ) B − g ( B, Y ) JX + g ( X, Y ) JB − g ( JB, Y ) X. (11)We have g ( ∇ X J , Y ∧ Z ) = 12 g (( ∇ X J )( Y ) , Z )and it follows that ∇ X J = 12 ( JX ∧ B + X ∧ JB ) . (12)The latter identity implies ∇ XY J = 12 [( ∇ X J )( Y ) ∧ B + Y ∧ ( ∇ X J )( B ) + JY ∧ ∇ X B + Y ∧ J ∇ X B ] . (13)Now recall that the ∗ -Ricci tensor ρ ∗ of the almost Hermitian manifold ( M, g, J )is defined by ρ ∗ ( X, Y ) = trace { Z → R ( JZ, X ) JY } . Note that ρ ∗ ( JX, JY ) = ρ ∗ ( Y, X ) , (14)in particular ρ ∗ ( X, JX ) = 0.Denote by ρ the Ricci tensor of the Riemannian manifold ( M, g ). Theorem 1.
Suppose that the almost complex structure J is integrable. Thenthe map J : ( M, g ) → ( Z , h t ) is harmonic if and only if dθ is a (1 , -form and ρ ( X, B ) = ρ ∗ ( X, B ) for every X ∈ T M . Proof . According to Proposition 1 the map J is harmonic if and only if V T race e ∇ J ∗ = V T race ∇ J = 0and π ∗ H T race e ∇ J ∗ = T race { T M ∋ X → R ( J × ∇ X J ) X } = 0 . It follows from identity (13) that for every
X, Y ∈ T M g ( T race ∇ J , X ∧ Y ) = − g ( ∇ JX B, Y ) + g ( ∇ JY B, X ) − g ( ∇ X B, JY ) + g ( ∇ Y B, JX ) + || B || g ( X, JY )= − dθ ( JX, Y ) − dθ ( X, JY ) + || B || g ( X, JY ) . (15)Take an orthonormal frame E , E = JE , E , E = JE so that J = E ∧ JE + E ∧ JE . Then (15) implies4 g ( T race ∇ J , J ) g ( J , X ∧ Y ) = −|| B || g ( JX, Y ) . Therefore4 g ( V T race ∇ J , X ∧ Y ) = 4[ g ( T race ∇ J , X ∧ Y ) − g ( T race ∇ J , J ) g ( J , X ∧ Y )]= − dθ ( JX, Y ) − dθ ( X, JY ) . (16) JOHANN DAVIDOV, ABSAR UL HAQ, OLEG MUSHKAROV
Thus V T race ∇ J = 0 if and only if dθ ( JX, Y )+ dθ ( X, JY ) = 0, which is equivalentto dθ being of type (1 , J × ∇ X J = g ( ∇ X J , s ) s − g ( ∇ X J , s ) s = g ( ∇ JX J , s ) s + g ( ∇ JX J , s ) s = ∇ JX J = −
12 ( X ∧ B − JX ∧ JB ) . It follows that for every X ∈ T M g ( π ∗ H T race e ∇ J ∗ , X ) = 2 X i =1 g ( R ( J × ∇ E i J ) E i , X ) = − ρ ( X, B ) + ρ ∗ ( X, B )(17)This proves the theorem.Remark 1 and formula (16) in the proof of Theorem 1 give the following.
Corollary 2.
The map J : ( M, g ) → ( Z , h t ) representing an integrable almostHermitian structure J on ( M, g ) is a harmonic section if and only if the -form dθ is of type (1 , . Remark 2 . Note that the 2-form dθ of a Hermitian surface ( M, g, J ) is of type(1 ,
1) if and only if the ⋆ -Ricci tensor ρ ∗ is symmetric.Indeed, let s = T race ρ and s ∗ = T race ρ ∗ be the scalar and ∗ -scalar curvatures.Set L ( X, Y ) = ( ∇ X θ )( Y ) + 12 θ ( X ) θ ( Y ) , X, Y ∈ T M.
Formula (3.4) in [22] implies the following identity ([19]) ρ ( X, Y ) − ρ ∗ ( X, Y ) = 12 [ L ( JX, JY ) − L ( X, Y )] + s − s ∗ g ( X, Y ) . It follows that, when ρ is J -invariant, ρ ∗ is symmetric if and only if L ( JX, JY ) − L ( JY, JX ) = L ( X, Y ) − L ( Y, X ). This identity is equivalent to dθ ( JX, JY ) = dθ ( X, Y ), which means that dθ is of type (1 , The case of symplectic J . Recall that an almost Hermitian manifold iscalled almost K¨ahler (or symplectic) if its fundamental 2-form is closed.Denote by Λ T M the subbundle of Λ T M orthogonal to J (thus Λ T p M = V J ( p ) ). Under this notation we have the following. Theorem 2.
Let ( M, g, J ) be an almost K¨ahler -manifold. Then the map J :( M, g ) → ( Z , h t ) is harmonic if and only if the ∗ -Ricci tensor ρ ∗ is symmetric and T race { Λ T M ∋ τ → R ( τ )( N ( τ )) } = 0 . Proof . The 2-form Ω is harmonic since d Ω = 0 and ∗ Ω = Ω, so by theWeitzenb¨ock formula(
T race ∇ Ω)(
X, Y ) =
T race { Z → ( R ( Z, Y )Ω)(
Z, X ) − ( R ( Z, X )Ω)(
Z, Y ) } ,X, Y ∈ T M (see, for example, [10]). We have( R ( Z, Y )Ω)(
Z, X ) = − Ω( R ( Z, Y ) Z, X ) − Ω( Z, R ( Z, Y ) X )= g ( R ( Z, Y ) Z, JX ) + g ( R ( Z, Y ) X, JZ ) . Hence (
T race ∇ Ω)(
X, Y ) = ρ ( Y, JX ) − ρ ( X, JY ) + 2 ρ ∗ ( X, JY ) . LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 9
Thus2 g ( T race ∇ J , X ∧ Y ) = ( T race ∇ Ω)(
X, Y ) = ρ ( Y, JX ) − ρ ( X, JY ) + 2 ρ ∗ ( X, JY ) . (18)Let E , ..., E be an orthonormal basis of a tangent space T p M such that E = JE , E = JE . Define s , s , s by (1). Then J ( p ) = s and V J ( p ) = span { s , s } . ByProposition 1 and identity (18) g ( T race e ∇ J ∗ , s ) = g ( T race ∇ J , s ) = ρ ∗ ( E , E ) − ρ ∗ ( E , E ) ,g ( T race e ∇ J ∗ , s ) = g ( T race ∇ J , s ) = − ρ ∗ ( E , E ) − ρ ∗ ( E , E ) . (19)It follows, in view of (14), that V T race e ∇ J ∗ = 0 if and only if ρ ∗ ( E i , E j ) = ρ ∗ ( E j , E i ), i, j = 1 , ..., π ∗ H T race e ∇ J ∗ = T race { T M ∋ X → R ( J × ∇ X J ) X } wefirst note that, by (10), g (( ∇ X J )( Y ) , Z ) = 12 g ( N ( Y, Z ) , JX ) . Then g ( ∇ X J , Y ∧ Z ) = 12 g (( ∇ X J )( Y ) , Z ) = 14 g ( N ( Y, Z ) , JX )The Nijenhuis tensor N ( Y, Z ) is skew-symmetric, so it induces a linear map Λ T M → T M which we denote again by N . It follows that, for every a ∈ Λ T M and X ∈ T π ( a ) M , g ( ∇ X J , a ) = 14 g ( N ( a ) , JX ) . (20)Take an orthonormal basis E , ..., E of T p M , p ∈ M , with E = JE , E = JE .Define s , s , s by (1). Then J = s and J × ∇ X J = g ( ∇ X J , s ) s − g ( ∇ X J , s ) s = −
14 [ g ( JN ( s ) , X ) s − g ( JN ( s ) , X ) s ]= 14 [ g ( N ( s ) , X ) s + g ( N ( s ) , X ) s ] = −∇ JX J , in view of (20) and the identities N ( JX, Y ) = N ( X, JY ) = − JN ( X, Y ) , X, Y ∈ T p M. (21)Therefore g ( T race { T M ∋ X → R ( J × ∇ X J ) X } , Y ) = − P i =1 g ( R ( ∇ JE i J ) E i , Y )= P i =1 [ g ( ∇ JE i J , s ) g ( R ( s ) Y, E i ) + g ( ∇ JE i J , s ) g ( R ( s ) Y, E i )]= g ( ∇ JR ( s ) Y J , s ) + g ( ∇ JR ( s ) Y J , s )= −
14 [ g ( N ( s ) , R ( s ) Y ) + g ( N ( s ) , R ( s ) Y )]= 14 [ g ( R ( s )( N ( s )) , Y ) + g ( R ( s )( N ( s )) , Y )] . Thus4 g ( T race { T M ∋ X → R ( J × ∇ X J ) X } , Y ) = g ( T race { Λ T M ∋ τ → R ( τ )( N ( τ )) } , Y ) . (22)This proves the theorem. Minimality of J The map J : M → Z is an imbedding and, in this section, we discuss the problemwhen J ( M ) is a minimal submanifold of ( Z , h t ).Let D ′ be the Levi-Civita connection of the metric on J ( M ) induced by themetric h t on Z . Let Π be the second fundamental form of the submanifold J ( M ).Then, as is well-known (and easy to see), for every vector fields X and Y on M e ∇ J ∗ ( X, Y ) = D ′ J ∗ X ( J ∗ ◦ Y ◦ J − ) + Π( J ∗ X, J ∗ Y ) − J ∗ ( ∇ X Y ) . Thus Π( J ∗ X, J ∗ Y ) is the normal component of e ∇ J ∗ ( X, Y ), in particular J ( M ) is aminimal submanifold if and only if the normal component of T race e ∇ J ∗ vanishes.4.1. The case of integrable J .Theorem 3. Suppose that the almost complex structure J is integrable. Then themap J : M → ( Z , h t ) is a minimal isometric imbedding if and only if dθ is a (1 , form and ρ ( X, B ) = ρ ∗ ( X, B ) for every X ⊥ { B, JB } . Proof . Let p ∈ M and suppose that B p = 0. Then ∇ J | p = 0 by (12). Hence J ∗ ( X ) = X h J ( p ) for every X ∈ T p M . Thus the tangent space of J ( M ) at the point J ( p ) is the horizontal space H J ( p ) , while the normal space is the vertical space V J ( p ) . Let E , E = JE , E , E = JE be an orthonormal basis of T p M anddefine s , s , s by formula (1). Then J ( p ) = s and V J ( p ) = span { s , s } Hencethe normal component at J ( p ) of T race e ∇ J ∗ vanishes if and only if g ( V T race e ∇ J ∗ , s ) = g ( V T race e ∇ J ∗ , s ) = 0 . Applying (16), we see that this is equivalent to dθ ( E , E ) = − dθ ( E , E ) , dθ ( E , E ) = dθ ( E , E ) . The latter identities are equivalent to ( dθ ) p being of type (1 , B p = 0. Then we can find an orthonormal basis of T p M of theform E, JE, || B p || − B p , || B p || − JB p . It follows from (12) that E h J ( p ) − t || B p || ∇ E J , ( JE ) h J ( p ) − t || B p || ∇ JE J is a h t -orthogonal basis of the normal space of J ( M ) at J ( p ). Therefore, accordingto Proposition 1, the normal component of the vertical part of T race e ∇ J ∗ at s ( p )vanishes if and only if g ( V T race ∇ J , ∇ E J ) = g ( V T race ∇ J , ∇ JE J ) = 0 . It follows from (12) and (16) that the latter identities hold if and only if dθ ( X, B ) = dθ ( JX, JB ) for every X ⊥ { B p , JB p } , which is equivalent to dθ being of type (1 , T race e ∇ J ∗ vanishes if and only if − ρ ( E, B ) + ρ ∗ ( E, B ) = − ρ ( JE, B ) + ρ ∗ ( JE, JB ) = 0 , or, equivalently, ρ ( X, B ) = ρ ∗ ( X, B ) for every X ∈ T p M , X ⊥ { B p , JB p } . Corollary 3. If J is integrable and the Ricci tensor is J -invariant then, the map J : M → ( Z , h t ) is a minimal isometric imbedding. LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 11
Proof . According to [19, Lemma 1], the Ricci tensor ρ is J -invariant if and onlyif ρ − ρ ∗ = s − s ∗ g. Thus ρ ∗ is symmetric if ρ is J -invariant, hence dθ is of type (1 ,
1) by Remark 2.Moreover, clearly ρ ( X, B ) = ρ ∗ ( X, B ) for X ⊥ B . Thus the result follows fromTheorem 3.This proof and Corollary 2 give the following Corollary 4. If J is integrable and the Ricci tensor is J -invariant, then J :( M, g ) → ( Z , h t ) is a harmonic section. Remark 3 . By [1, Theorem 2], every compact Hermitian surface with J -invariantRicci tensor is locally conformally K¨ahler, dθ = 0. Moreover, if its first Bettinumber is even, it is globally conformally K¨ahler ([21]). It is still unknown whetherthere are compact complex surfaces with J -invariant Ricci tensor and odd firstBetti number.4.2. The case of symplectic J . Set N p = span { N ( X, Y ) :
X, Y ∈ T p M } , p ∈ M , so N p = N (Λ T p M ). Identity (21) implies that N (Λ − T p M ) = 0 and N ( J ) = 0. Hence N p = N (Λ T p M ) is a J -invariant subspace of T p M of dimension0 or 2. Theorem 4.
Let ( M, g, J ) be an almost K¨ahler -manifold. Then the map J : M → ( Z , h t ) is a minimal isometric imbedding if and only if the ⋆ -Ricci tensor ρ ∗ is symmetric and for every p ∈ MT race { Λ T p M ∋ τ → R p ( τ )( N ( τ )) } ∈ N p . Proof . Suppose that N p = 0 for a point p ∈ M . Then ∇ J | p = 0 by (20).Hence J ∗ ( X ) = X h J ( p ) for every X ∈ T p M . Thus the normal space of J ( M ) at thepoint J ( p ) is the vertical space V J ( p ) . Therefore the normal component at J ( p ) of T race e ∇ J ∗ vanishes if and only if g ( T race e ∇ J ∗ , s ) = g ( T race e ∇ J ∗ , s ) = 0 . where s , s are defined via (1) by means of an orthonormal basis E , ..., E of T p M such that E = JE , E = JE . According to (19) (and in view of (14)) the latteridentities are equivalent to ρ ∗ ( X, Y ) = ρ ∗ ( Y, X ) for every
X, Y ∈ T p M .Now assume that N p = 0. Then there exists τ ∈ Λ T p M , || τ || = 1, such that τ ⊥ J ( p ) and N ( τ ) = 0. Take a unit vector E ∈ T p M and set E = JE , E = K τ E , E = K J ( p ) × τ E . Then E , ..., E is an orthonormal basis of T p M such that J ( p ) = s , τ = s , J ( p ) × τ = s . By (21), N ( τ ) = N ( s ) = 2 N ( E , E ), N ( J ( p ) × τ ) = N ( s ) = 2 JN ( E , E ), thus N ( J ( p ) × τ ) = JN ( τ ). Now we set A = || N ( τ ) || − N ( τ ), A = JA , A = K τ A , A = K J ( p ) × τ A . Note that A = JA by (4). In view of (20), we have for every X ∈ T p Mg ( ∇ X J , ∇ A J )= 116 [ g ( N ( τ ) , JX ) g ( N ( τ ) , JA ) + g ( N ( J ( p ) × τ ) , JX ) g ( N ( J ( p ) × τ ) , JA )]= 116 || N ( τ ) || g ( A , X ) . and g ( ∇ X J , ∇ A J ) = 116 || N ( τ ) || g ( A , X ) . Note also that g ( ∇ A J , ∇ A J ) = 0 . Therefore ( A ) h J ( p ) − t || N ( τ ) || ∇ A J , ( A ) h J ( p ) − t || N ( τ ) || ∇ A J is a h t - orthogonal basis of the normal space of J ( M ) at J ( p ). It follows fromProposition 1 that the normal component of the vertical part of T race e ∇ J ∗ at s ( p )vanishes if and only if g ( T race ∇ J , ∇ A J ) = g ( T race ∇ J , ∇ A J ) = 0 . (23)Using (20), we see that ∇ A J = 14 || N ( τ ) || J ( p ) × τ = 14 || N ( τ ) || ( A ∧ A + A ∧ A ) , ∇ A J = − || N ( τ ) || τ = − || N ( τ ) || ( A ∧ A + A ∧ A ) . It follows from (18) that (23) is equivalent to the identities ρ ∗ ( A , A ) = ρ ∗ ( A , A ) , ρ ∗ ( A , A ) = ρ ∗ ( A , A ) . Now, taking into account (14), we see that (23) is equivalent to ρ ∗ ( N ( τ ) , X ) = ρ ∗ ( X, N ( τ )) for X ⊥ { N ( τ ) , JN ( τ ) } , i.e. ρ ∗ ( X, Y ) = ρ ∗ ( Y, X ) for X ⊥ N p , Y ∈N p . The subspace N p and N ⊥ p of T p M are two-dimensional and J -invariant, andit follows from (14) that ρ ∗ ( X, Y ) = ρ ∗ ( Y, X ) for
X, Y ∈ N p or X, Y ∈ N ⊥ p . Thusidentity (23) is equivalent to ρ ∗ being symmetric.In view of (22), the normal component of the horizontal part of T race e ∇ J ∗ vanishes if and only if T race { Λ T M ∋ τ → R ( τ )( N ( τ )) } ⊥ { A , A } . This proves the statement. 5.
Examples
In this section we give examples of almost Hermitian structures that determineharmonic maps into twistor spaces. We also provide an example of a Hermitianstructure that is a minimal isometric imbedding and a harmonic section of thetwistor space (in sense of C. Wood [23, 24]) but not a harmonic map.5.1.
Kodaira surfaces.
Recall that every primary Kodaira surface M can be ob-tained in the following way [17, p.787]. Let ϕ k ( z, w ) be the affine transformationsof C given by ϕ k ( z, w ) = ( z + a k , w + a k z + b k ) , where a k , b k , k = 1 , , ,
4, are complex numbers such that a = a = 0 , Im ( a a ) = mb = 0 , b = 0for some integer m >
0. They generate a group G of transformations acting freelyand properly discontinuously on C , and M is the quotient space C /G . LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 13
It is well-known that M can also be described as the quotient of C endowed witha group structure by a discrete subgroup Γ. The multiplication on C is defined by( a, b ) . ( z, w ) = ( z + a, w + az + b ) , ( a, b ) , ( z, w ) ∈ C , and Γ is the subgroup generated by ( a k , b k ), k = 1 , ..., M as the quotient of the group C by the discrete subgroupΓ. Every left-invariant object on C descends to a globally defined object on M and both of them will be denoted by the same symbol.We identify C with R by ( z = x + iy, w = u + iv ) → ( x, y, u, v ) and set A = ∂∂x − x ∂∂u + y ∂∂v , A = ∂∂y − y ∂∂u − x ∂∂v , A = ∂∂u , A = ∂∂v . These form a basis for the space of left-invariant vector fields on C . We note thattheir Lie brackets are [ A , A ] = − A , [ A i , A j ] = 0for all other i, j . It follows that the group C defined above is solvable.Denote by g the left-invariant Riemannian metric on M for which the basis A , ..., A is orthonormal.We shall show that any integrable or symplectic almost complex structures J on M compatible with the metric g and defined by a left-invariant almost complexstructure on C is a harmonic map from ( M, g ) to ( Z , h t ).Note that by [15] every complex structure on M is induced by a left-invariantcomplex structure on C . I . If J is a left-invariant almost complex structure compatible with g , we have JA i = P j =1 a ij A j where a ij are constants with a ij = − a ji . Let N be the Nijenhuistensor of J . Computing N ( A i , A j ) in terms of a ij , one can see ([18, 7]) that J isintegrable if and only if JA = ε A , JA = ε A , ε , ε = ± . Denote by θ the Lee form of the Hermitian structure ( M, g, J ) where J is definedby means of the latter identities.The non-zero covariant derivatives ∇ A i A j are ∇ A A = −∇ A A = − A , ∇ A A = ∇ A A = A , ∇ A A = ∇ A A = − A . This implies that the Lie form is θ ( X ) = − ε g ( X, A ) . Therefore B = − ε A , ∇ θ = 0 . (24)Set for short R ijk = R ( A , A j ) A k . Then the non-zero R ijk are R = − A , R = 3 A , R = A ,R = − A , R = A , R = − A . Set also ρ ij = ρ ( A i , A j ), ρ ∗ ij = ρ ∗ ( A i , A j ). Then ρ ij = 0 a nd ρ ∗ ij = 0 e xceptρ = ρ = − , ρ = 2 , ρ ∗ = ρ ∗ = − . (25) It follows form (24), (25) and Theorem 1 that the complex structure J is a harmonicmap from ( M, g ) to the twistor space ( Z , h t ).It is easy to describe explicitly the twistor space ( Z , h t ) ([7]) since Λ M admitsa global orthonormal frame defined by s = ε A ∧ A + ε A ∧ A , s = A ∧ A + ε ε A ∧ A , s = ε A ∧ A + ε A ∧ A . It induces a natural diffeomorphism F : Z ∼ = M × S , P k =1 x k s k ( p ) → ( p, x , x , x ),under which J determines the section p → ( p, , , h t we need the covariant derivatives of s , s , s with respectto the Levi-Civita connection ∇ of g . The non-zero of these are ∇ A s = − ǫ ∇ A s = − ε ε s , ε ∇ A s = −∇ A s = ε s ,ε ∇ A s = − ε ∇ A s = s . It follows that F ∗ sends the horizontal lifts A h , ..., A h at a point σ = P k =1 x k s k ( p ) ∈Z to the following vectors of T M ⊕ T S A + ε ε ( − x , , x ) , A + ε ( x , − x , , A , A + ε (0 , x , − x ) . For x = ( x , x , x ) ∈ S , set u ( x ) = ε ε ( − x , , x ) , u ( x ) = ε ( x , − x , , u ( x ) = 0 , u ( x ) = ε (0 , x , − x ) . Denote the pushforward of the metric h t under F again by h t . Then, if X, Y ∈ T p M and P, Q ∈ T x S , h t ( X + P, Y + Q ) = g ( X, Y ) + t < P − X i =1 g ( X, A i ) u i ( x ) , Q − X j =1 g ( Y, A j ) u j ( x ) > (26)where < ., . > is the standard metric of R . II . Suppose again that J is an almost complex structure on M obtained from aleft-invariant almost complex structure on C and compatible with the metric g .Set JA i = P j =1 a ij A j . Denote the fundamental 2-form of the almost Hermitianstructure ( g, J ) by Ω. The basis dual to A , ..., A is α = dx, α = dy, α = xdx + ydy + du, α = − ydx + xdy + dv. We have dα = dα = dα = 0 , dα = 2 dx ∧ dy . Hence d Ω = d X i Then E , ..., E is an orthonormal frame of T M for which JE = E and JE = E .The only non-zero Lie bracket of these fields is[ E , E ] = − ε ε cos ϕE + ε sin ϕE ) . The non-zero covariant derivatives ∇ E i E j are ∇ E E = ∇ E E = ε ε cos ϕE , ∇ E E = ∇ E E = ε sin ϕE , ∇ E E = −∇ E E = − ε ε cos ϕE − ε sin ϕE , ∇ E E = ∇ E E = − ε ε cos ϕE , ∇ E E = ∇ E E = − ε sin ϕE . Set R ijk = R ( E i , E j ) E k . We have the following table for the non-zero componentsof the curvature tensor R : R = cos ϕE + 12 ε sin 2 ϕE , R = − cos ϕE , R = − ε sin 2 ϕE ,R = 12 ε sin 2 ϕE + sin ϕE , R = − ε sin 2 ϕE , R = − sin ϕE ,R = − E , R = 3 E ,R = cos ϕE , R = 12 ε sin 2 ϕE , R = − cos ϕE − ε sin 2 ϕE ,R = 12 ε sin 2 ϕE , R = sin ϕE , R = − ε sin 2 ϕE − sin ϕE . Define an orthonormal frame s l , l = 1 , , 3, of Λ T M by means of E , ..., E . Then s , s is a frame of Λ T M and by (21) N ( s ) = 2 N ( E , E ) = − ε ε cos ϕE + 4 ε sin ϕE ,N ( s ) = 2 N ( E , E ) = 4 ε ε cos ϕE + 4 ε sin ϕE . It follows that T race { Λ T M ∋ τ → R ( τ )( N ( τ )) } = R ( s )( N ( s )) + R ( s )( N ( s )) = 0 . Setting ρ ∗ ij = ρ ∗ ( E i , E j ), we have ρ ∗ = ρ ∗ = cos ϕ, ρ ∗ = ρ ∗ = sin ϕ, ρ ∗ = ρ ∗ = − ε sin 2 ϕ, and the other ρ ∗ ij vanish. Thus, by Theorem 2, the almost K¨ahler structure J is aharmonic map ( M, g ) → ( Z , h t ).As in the preceding case, it is easy to find an explicit description of the twistorspace Z of M and the metric h t ([7]). The frame s , s , s gives rise to an obviousdiffeomorphism F : Z ∼ = M × S under which J becomes the map p → ( p, , , s , s , s : ∇ E s = ∇ E s = ε ε cos ϕs , ∇ E s = −∇ E s = − ε sin ϕs , ∇ E s = ε ε cos ϕs , ∇ E s = − ε ε cos ϕs , ∇ E s = 0 , ∇ E s = ε sin ϕs , ∇ E s = − ε sin ϕs , ∇ E s = 0 , ∇ E s = − ε ε cos ϕs − ε sin ϕs , ∇ E s = ε sin ϕs − ε ε cos ϕs . Using this table we see that F ∗ sends the horizontal lifts E hi , i = 1 , ..., 4, to E i + u i where u ( x ) = ( x ε ε cos ϕ, x ε sin ϕ, − x ε ε cos ϕ − x ε sin ϕ ) ,u ( x ) = ( x ε ε cos ϕ, − x ε ε cos ϕ, , u ( x ) = ( x ε sin ϕ, − x ε sin ϕ, ,u ( x ) = ( − x ε sin ϕ, x ε ε cos ϕ, x ε sin ϕ − x ε ε cos ϕ ) . for x = ( x , x , x ) ∈ S . Then, if X, Y ∈ T p M and P, Q ∈ T x S , h t ( X + P, Y + Q ) = g ( X, Y ) + t < P − X i =1 g ( X, E i ) u i ( x ) , Q − X j =1 g ( Y, E j ) u j ( x ) > . (27)5.2. Four-dimensional Lie groups. We shall show that every left-invariant al-most K¨ahler structure ( g, J ) with J -invariant Ricci tensor on a 4-dimensional Liegroup M determines a harmonic map J : ( M, g ) → ( Z , h t ).These (non-integrable) structures have been determined in [12]. According tothe main result therein, for any such a structure ( g, J ), there exists an orthonormalframe of left-invariant vector fields E , ..., E such that JE = E , JE = E and [ E , E ] = 0 , [ E , E ] = sE + s t E , [ E , E ] = s − t t E − sE , [ E , E ] = − tE − sE , [ E , E ] = − sE − s − t t E , [ E , E ] = − s + t t E , where s and t = 0 are real numbers. Then we have the following table for theLevi-Civita connection ∇ E E = − sE − s − t t E , ∇ E E = − s − t t E + sE , ∇ E E = s + t t E , ∇ E E = − s − t t E + sE , ∇ E E = sE + s − t t E , ∇ E E == s + t t E , ∇ E E = sE + s − t t E , ∇ E E = s − t t E − sE , ∇ E E = s + t t E , ∇ E E = s − t t E − sE , ∇ E E = − sE − s − t t E , ∇ E E = − s + t t E , ∇ E E = ∇ E E = ∇ E E = ∇ E E = 0 . This implies the following table for the components R ijk = R ( E i , E j ) E k of thecurvature tensor; in this table λ = s + t t . R = 2 λE , R = − λE , R = 2 λE , R = − λE ,R = − λE , R = λE , R = λE , R = − λE ,R = − λE , R = − λE , R = λE , R = λE ,R = − λE , R = − λE , R = λE , R = λE ,R = λE , R = − λE , R = − λE , R = λE ,R = 2 λE , R = − λE , R = − λE , R = 4 λE . LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 17 Then the non-zero ρ ∗ ij = ρ ∗ ( E i , E j ) are ρ ∗ = ρ ∗ = 4 λ, ρ ∗ = ρ ∗ = − λ. Therefore the ∗ -Ricci tensor ρ ∗ is symmetric.Set s = E ∧ E + E ∧ E , s = E ∧ E + E ∧ E . Then N ( s ) = 2 N ( E , E ) = − sE + s − t t E ) ,N ( s ) = 2 N ( E , E ) = 8( − s − t t E + sE ) . It follows that T race { Λ T M ∋ τ → R ( τ )( N ( τ )) } = 0 . Thus, according to Theorem 4, J is a harmonic map.5.3. Inoue surfaces of type S . It has been observed in [20] that every Inouesurface M of type S admits a locally conformal K¨ahler metric g (cf. also [9]).The map J : ( M, g ) → ( Z , h t ) determined by the complex structure of M is aminimal isometric imbedding that is a harmonic section of the twistor space butnot a harmonic map. To see this let us first recall the construction of the Inouesurfaces of type S ([16]). Let A ∈ SL (3 , Z ) be a matrix with a real eigenvalue α > β and β , β = β . Choose eigenvectors ( a , a , a ) ∈ R and ( b , b , b ) ∈ C of A corresponding to α and β , respectively. Then the vectors( a , a , a ) , ( b , b , b ) , ( b , b , b ) are C -linearly independent. Denote the upper-halfplane in C by H and let Γ be the group of holomorphic automorphisms of H × C generated by g o : ( w, z ) → ( αw, βz ) , g i : ( w, z ) → ( w + a i , z + b i ) , i = 1 , , . The group Γ acts on H × C freely and properly discontinuously. Then M = ( H × C ) / Γ is a complex surface known as Inoue surface of type S . As in [20], consideron H × C the Hermitian metric g = 1 v ( du ⊗ du + dv ⊗ dv ) + v ( dx ⊗ dx + dy ⊗ dy ) , u + iv ∈ H , x + iy ∈ C . This metric is invariant under the action of the group Γ, so it descends to a Her-mitian metric on M which we denote again by g . Instead on M , we shall work withΓ-invariant objects on H × C . Let Ω be the fundamental 2-form of the Hermitianstructure ( g, J ) on H × C , J being the standard complex structure. Then d Ω = 1 v dv ∧ Ω . Hence the Lee form is θ = d ln v . In particular, dθ = 0, i.e. ( g, J ) is a locallyconformal K¨ahler structure. Set E = v ∂∂u , E = v ∂∂v , E = 1 √ v ∂∂x , E = 1 √ v ∂∂y . These are Γ-invariant vector fields constituting an orthonormal basis. Note that thevector field dual to the Lee form is B = E . The non-zero Lie brackets of E , ..., E are [ E , E ] = − E , [ E , E ] = − E , [ E , E ] = − E . Then we have the following table for the Levi-Civita connection ∇ of g : ∇ E E = E , ∇ E E = − E , ∇ E E = 12 E , ∇ E E = − E , ∇ E E = 12 E , ∇ E E = − E , and all other ∇ E i E j = 0. It follows that ρ ( E k , B ) = ρ ∗ ( E k , B ) = 0 for k = 1 , , , ρ (E , B) = 12 , ρ ∗ (E , B) = − . By Corollary 2 J : ( M, g ) → ( Z , h t ) is a harmonic section. It is also a minimalisometric imbedding by Theorem 3. But J is not a harmonic map according toTheorem 1. References [1] V. Apostolov, P. Gauduchon, The Riemannian Goldberg-Sachs theorem, Int.J.Math. (1997), 421-439.[2] M.F. Atiyah, N.J. Hitchin, I.M. Singer, Self-duality in four-dimensional Riemannian ge-ometry , Proc. R. Soc. London Ser. A, (1978), 435-461.[3] A. Besse , Einstein manifolds , Classics in Mathematics, Springer-Verlag, Berlin, 2008.[4] G. Bor, L. Hern´andez-Lamoneda, M. Salvai, Orthogonal almost-complex structures of min-imal energy , Geom. Dedicata (2007), 75-85.[5] C. Borcea, Moduli for Kodaira surfaces , Composition Math. (1984), 373-380.[6] E. Calabi and H. Gluck, What are the best almost-complex structures on the 6-sphere? ,Proc.Sym.Pure Math. (1993), part 2, 99-106.[7] J. Davidov, Twistorial construction of minimal hypersurfaces , Inter. J. Geom. Methods inModern Physics , No 6 (2014), 1459964.[8] J. Davidov, O. Mushkarov, On the Riemannian curvature of a twistor space , Acta Math.Hungarica (1991), 319-332.[9] S. Dragomir, L. Ornea, Locally conformal K¨ahler geometry , Progress in Math., v. 155,Birkh¨auser, Boston-Basel-Berlin, 1998.[10] J. Eeells,L. Lemaire, Selected topics in harmonic maps , Cbms Regional Conference Seriesin Mathematics, vol. , AMS, Providernce, Rhode Island, 1983.[11] J. Eeells, S. Salamon, Twistorial constructions of harmonic maps of surfaces into four-manifolds , Ann.Scuola Norm.Sup. Pisa, ser.IV, (1985), 589-640.[12] A. Fino, Almost K¨ahler -dimensional Lie groups with J -invariant Ricci tensor , Diff.Geom.Appl. (2005), 26-37.[13] A. Gray, Minimal varieties and almost Hermitian manifolds , Michigan Math. J. (1965),273-287.[14] A. Gray, L.M. Hervella, The sixteen classes of almost Hermitian manifolds and their linearinvariants , Ann. Mat. Pure Appl. (1980), 35-50.[15] K. Hasegawa, Complex and K¨ahler structures on compact solvmanifolds , J. SymplecticGeom. (2005), 749-767.[16] M. Inoue, On surfaces of class V II , Invent. Math. (1974), 269-310.[17] K. Kodaira, On the structure of compact complex analytic surfaces I . Amer. J. Math. (1964), 751-798.[18] O. Muˇskarov, Two remarks on Thurston’s example , In: Complex analysis and applications’85 (Varna, 1985), Publ. House Bulgar. Acad. Sci., Sofia, 1986, pp. 461-468.[19] O. Muˇskarov, On Hermitian surfaces with J -invariant Ricci tensor , J. Geom. (2001),151-156.[20] F. Tricerri, Some example of locally conformal K¨ahler manifolds , Rend. Sem. Mat. Univ.Torino (19982), 81-92.[21] I. Vaisman, On locally and globally conformal K¨ahler manifolds , Trans. Amer. Math. Soc. (1980), 533-542.[22] I. Vaisman, Some curvature properties of complex surfaces , Ann. Mat. Pura Appl. (1982), 1-18. LMOST COMPLEX STRUCTURES THAT ARE HARMONIC MAPS 19 [23] C. Wood, Instability of the nearly-K¨ahler six-sphere , J. reine angew. Math. (1993),205-212.[24] C. Wood, Harmonic almost-complex structures , Compositio Mathematica (1995), 183-212. Johann Davidov, Institute of Mathematics and Informatics, Bulgarian Academy ofSciences, Acad. G.Bonchev Str. Bl.8, 1113 Sofia, Bulgaria. E-mail address : [email protected] Absar ul Haq, Department of Mathematics, University of Management and Tech-nology Lahore, Sialkot Campus, Pakistan. E-mail address : [email protected] Oleg Mushkarov, Institute of Mathematics and Informatics, Bulgarian Academy ofSciences, Acad. G.Bonchev Str. Bl.8, 1113 Sofia, Bulgaria and South-West University”Neofit Rilski”, 2700 Blagoevgrad, Bulgaria. E-mail address ::