Directory UMM :Journals:Journal_of_mathematics:BJGA:

maps into GL(2r)/Sp(2r)

Lars Sch¨afer

Abstract. In this work we use symplectic (para-)tt∗_{-bundles to obtain}

a geometric construction of (para-)pluriharmonic maps into the pseudo-Riemannian symmetric space GL(2r,R_)/Sp(R2r_{). We prove, that these}

(para-)pluriharmonic maps are exactly the admissible (para-)pluriharmonic maps. Moreover, we construct symplectic (para-)tt∗_{-bundles from}

(para-)harmonic bundles and analyse the related (para-)pluriharmonic maps.

M.S.C. 2000: 53C43,58E20.

Key words: symplectic (para-)tt∗_{-geometry, (para-)pluriharmonic maps,}

(para-)harmonic bundles, pseudo-Riemannian symmetric spaces

1

Introduction

The first motivation of this work is the study of metric (para-)tt∗_{-bundles (E, D, S, g)}

over a (para-)complex manifold (M, Jǫ_{) and their relation to admissible} (para-)pluri-harmonic maps fromM into the space of (pseudo-)metrics. Roughly speaking there exists a correspondence between these objects. For metrictt∗_{-bundles (with positive}

definite metric) on the tangent bundle of a complex manifold this result was shown by Dubrovin [8]. In [17, 19] we generalised it to the case of metrictt∗_{-bundles on abstract}

vector bundles with metrics of arbitrary signature and to para-complex geometry. So-lutions of (metric) (para-)tt∗_{-bundles are for example given by special (para-)complex}

and special (para-)K¨ahler manifolds (cf. [3, 19]) and by (para-)harmonic bundles [18, 22]. The related (para-)pluriharmonic maps are described in the given references. The analysis [20, 21] of tt∗_{-bundles (E = T M, D, S) on the tangent bundle of an}

almost (para-)complex manifold (M, Jǫ_{) shows that there exists a second interesting} class of (para-)tt∗_{-bundles (E=T M, D, S, ω), carrying symplectic formsωinstead of}

metricsg.These will be called symplectic (para-)tt∗_{-bundles. Examples are given by}

Levi-Civita flat nearly (para-)K¨ahler manifolds (Here non-integrable (para-)complex structures appear.) and by (para-)harmonic bundles which are discussed later in this work. A constructive classification of Levi-Civita flat nearly (para-)K¨ahler manifolds

∗

Balkan Journal of Geometry and Its Applications, Vol.13, No.2, 2008, pp. 86-101. c

is subject of [4, 5].

In the context of the above mentioned correspondence it arises the question if one can use these techniques to construct (para-)pluriharmonic maps out of symplectic (pa-ra-)tt∗_{-bundles and if one can characterise the obtained (para-)pluriharmonic maps.}

In this paper we answer positively to this question: We associate an admissible (cf. definition 4) (para-)pluriharmonic map fromM intoGL(2r,R_)/Sp(R2r_{) to a}

symplec-tic (para-)tt∗_{-bundle and show that an admissible (para-)pluriharmonic map induces}

a symplectic (para-)tt∗_{-bundle onE=M×R}2r_{.This is the analogue of the} correspon-dence discussed in the first paragraph. In other words we characterise in a geometric fashion the class of admissible (para-)pluriharmonic maps intoGL(2r,R_)/Sp(R2r_).In

the sequel we construct symplectic (para-)tt∗_{-bundles from (para-)harmonic bundles}

and analyse the relation between the (para-)pluriharmonic maps which are obtained from these symplectic (para-)tt∗_{-bundles and the (para-)pluriharmonic maps which}

were found in [18, 22]. We restrict to simply connected manifoldsM, since the case of general fundamental group can be obtained like in [17, 19]. In the general case all (para-)pluriharmonic maps have to be replaced by twisted (para-)pluriharmonic maps.

2

Para-complex differential geometry

We shortly recall some notions and facts of para-complex differential geometry. For a more complete source we refer to [2].

In para-complex geometry one replaces the complex structureJ withJ2_{=−1(on a}

finite dimensional vector spaceV) by thepara-complex structureτ ∈End(V) satisfying

τ2_{=1and one requires that the±1-eigenspaces have the same dimension. Analmost} para-complex structureon a smooth manifoldM is an endomorphism-fieldτ,which is a point-wise para-complex structure. If the eigen-distributions T±_{M are integrable} τ is called para-complex structure on M andM is called a para-complex manifold. As in the complex case, there exists a tensor, also called Nijenhuis tensor, which is the obstruction to the integrability of the para-complex structure.

The real algebra, which is generated by 1 and by thepara-complex unitewithe2_{= 1,}

is called thepara-complex numbers and denoted by C. For all z =x+ey ∈ C with

x, y∈R _{we define the para-complex conjugationas ¯·: C →C, x+ey 7→x−ey and}

therealandimaginary partsof z byR_{(z) :=x, ℑ(z) :=y.The freeC-moduleC}n _{is a} para-complex vector space where its para-complex structure is just the multiplication with e and the para-complex conjugation ofCextends to ¯·:Cn _→Cn_,v7→¯v. Note, thatzz¯=x2−y2.Therefore the algebraCis sometimes called thehypercomplex

numbers. The circle S1 _{= {z = x+iy ∈} C_|x2_+y2 _{= 1} is replaced by the four}

hyperbolas{z=x+ey∈C|x2_−y2_{=±1}.We define ˜S}1 _{to be the hyperbola given}

by the one parameter group{z(θ) = cosh(θ) +esinh(θ)|θ∈R_}.

A para-complex vector space (V, τ) endowed with a pseudo-Euclidean metric g is called para-hermitian vector space, if g is τ-anti-invariant, i.e. τ∗_{g = −g. The} para-unitary group ofV is defined as the group of automorphisms

Uπ(V) := Aut(V, τ, g) :={L∈GL(V)|[L, τ] = 0 andL∗g=g}

and its Lie-algebra is denoted by uπ(V). For Cn _{= R}n _⊕eRn _{the standard}

g= diag(1,−1) (cf. Example 7 of [2]). The corresponding para-unitary group is given

There exist two bi-gradings on the exterior algebra: The one is induced by the splitting in T±_{M and denoted by Λ}k_T∗_{M =} L

k=p+q

Λp+,q−_T∗_{M and induces an obvious}

bi-grading on exterior forms with values in a vector bundle E. The second is induced by the decomposition of thepara-complexifiedtangent bundleT MC _{=T M⊗}

RCinto

the subbundles T1,0

p M and Tp0,1M which are defined as the ±e-eigenbundles of the para-complex linear extension ofτ.This induces a bi-grading on theC-valued exterior forms noted Λk_T∗_MC₌ L

k=p+q

Λp,q_T∗_{M and finally on theC-valued differential forms}

on M Ωk

Λp,q_T∗_{M are para-complex vector bundles in the following sense: A para-complex} vector bundleof rankr over a para-complex manifold (M, τ) is a smooth real vector bundle π : E → M of rank 2r endowed with a fiber-wise para-complex structure

τE _{∈Γ(End (E)). We denote it by (E, τ}E_{). In the following text we always identify} the fibers of a para-complex vector bundleE of rankr with the free C-module Cr_. One has a notion of para-holomorphic vector bundles, too. These were extensively studied in a common work with M.-A. Lawn-Paillusseau [14].

Let us transfer some notions of hermitian linear algebra (cf. [22]) : Apara-hermitian sesquilinear scalar productis a non-degenerate sesquilinear formh:Cr_×Cr_→C,i.e. it satisfies (i)his non-degenerate: Givenw∈Cr _{such that for allv∈C}r_{h(v, w) = 0,}

GL(r, C).We remark, that there isnonotion of para-hermitian signature, since from

h(v, v) =−1 for an elementv∈Cr_{we obtainh(ev, ev) = 1.}

Proposition 1. Given an element C of End(Cr_{) then it holds (Cz, w)Cr =} (z, Ch_{w)Cr, ∀z, w∈C}r_{.The set herm(C}r_{)is a real vector space. There is a bijective}

correspondence between Herm(Cr_{)and para-hermitian sesquilinear scalar products h}

onCr _{given byH 7→h(·,·) := (H·,·)Cr.}

Apara-hermitian metrichon a para-complex vector-bundleE over a para-complex manifold (M, τ) is a smooth fiber-wise para-hermitian sesquilinear scalar product.

i.e. ˆi :=e, forǫ= 1, and ˆi =i, forǫ=−1. Further we introduce C_{ǫ with} C_{1 = C}

andC₋₁₌C_andS1

ǫ withS11= ˜S1 andS1−1=S1.In the rest of this work we extend

our language by the followingǫ-notation: If a word has a prefixǫwithǫ∈ {±1},i.e. is of the formǫX, this expression is replaced by

ǫX :=

(

X, forǫ=−1,

para-X, forǫ= 1.

Theǫunitary group and its Lie-algebra are

Uǫ_{(p, q) :=}

where, for p+q = r, Hermp,q(Cr_{) are the hermitian matrices of hermitian} signa-ture (p, q) and hermp,q(Cr_{) are the hermitian matrices with respect to the} stan-dard hermitian product of hermitian signature (p, q) onCr_{.The standardǫhermitian} sesquilinear scalar product is (z, w)Cr

ǫ :=z·w¯ =

For the convenience of the reader we recall the definition of anǫtt∗_{-bundle given in}

[3, 17, 19] and the notion of a symplecticǫtt∗_{-bundle [20, 21]:}

Definition 1. Anǫtt∗_{-bundle(E, D, S) over anǫcomplex manifold (M, J}ǫ_{) is a real} vector bundle E → M endowed with a connection D and a section S ∈ Γ(T∗_{M ⊗}

EndE) satisfying theǫtt∗_-equation

(3.1) Rθ_{= 0 for all θ∈R,}

whereRθ _{is the curvature tensor of the connectionD}θ _{defined by}

Remark 1.

1) It is obvious that every ǫtt∗_{-bundle (E, D, S) induces a family of ǫtt}∗_-bundles

(E, D, Sθ_{),forθ∈R, with}

(3.4) Sθ:=Dθ−D= cosǫ(θ)S+ sinǫ(θ)SJǫ.

The same remark applies to symplecticǫtt∗_-bundles.

2) Notice that a symplectic ǫtt∗_{-bundle (E, D, S, ω) of rank 2r carries a D-parallel}

volume given byω∧. . .∧ω

| {z }

r times

.

The next proposition gives explicit equations forD andS, such that (E, D, S) is anǫtt∗_-bundle.

Proposition 2. (cf. [17, 19]) Let E be a real vector bundle over an ǫcomplex manifold(M, Jǫ_{)endowed with a connectionD and a sectionS ∈Γ(T}∗_M⊗EndE). Then(E, D, S)is anǫtt∗_{-bundle if and only ifDandSsatisfy the following equations:}

(3.5) RD_{+S∧S= 0, S∧S is of type (1,1), d}D_{S= 0 and d}D_S Jǫ= 0.

4

Pluriharmonic maps into

GL

(2

r,

R

₎

_/Sp

₍

R

2r

₎

In this section we present the notion ofǫpluriharmonic maps and some properties of

ǫpluriharmonic maps into the target spaceS=S(2r) :=GL(2r,R_)/Sp(R2r_).

The following notion was introduced in [1] for holomorphic and in [14] for para-holomorphic vector bundles.

Definition 2. Let (M, Jǫ_{) be an ǫcomplex manifold. A connection D on T M is} calledadaptedif it satisfies

(4.1) DJǫ_YX =JǫD_YX

for all vector fields which satisfyL_XJǫ_{= 0 (i.e. for whichX+ǫˆiJ}ǫ_{X isǫholomorphic).}

Definition 3. Let (M, Jǫ_{) be anǫcomplex manifold and (N, h) a pseudo-Riemannian} manifold with Levi-Civita connection∇h_{,Dan adapted connection onM and∇ the} connection onT∗_M⊗f∗_{T N which is induced byDand∇}h_{and considerα=∇df∈} Γ(T∗_{M ⊗T}∗_M⊗f∗_{T N).Thenf isǫpluriharmonic if and only ifαis of type (1,1),}

i.e.

α(X, Y)−ǫα(Jǫ_{X, J}ǫ_{Y) = 0}

for allX, Y ∈T M.

Remark 2.

1. Note, that an equivalent definition ofǫpluriharmonicity is to say, thatf isǫ pluri-harmonic if and only iff restricted to everyǫcomplex curve is harmonic. For a short discussion the reader is referred to [3, 17, 19].

2. One knows, that every ǫcomplex manifold (M, Jǫ_{) can be endowed with a} torsion-freeǫcomplex connectionD(cf. [12] in the complex and [19] Theorem 1 for the para-complex case), i.e.D is torsion-free and satisfiesDJǫ_{= 0.Such a} connection is adapted. In the rest of the paper, we assume, that the connection

The harmonic analogue of the following proposition is well-known.

Proposition 3. Let(M, Jǫ_{)be anǫcomplex manifold,X, Y be pseudo-Riemannian}

manifolds andΨ :X →Y a totally geodesic immersion. Then a map f :M →X is ǫpluriharmonic if and only ifΨ◦f :M →Y isǫpluriharmonic.

Theǫpluriharmonic maps obtained by our construction are exactly the admissible

ǫpluriharmonic maps in the sense of the following general definition:

Definition 4. Let (M, Jǫ_{) be anǫcomplex manifold and G/K be a locally symmetric} space with associated Cartan-decompositiong=k⊕p.A map f : (M, Jǫ_{)→G/K is} calledadmissibleif theǫcomplex linear extension of its differentialdf mapsT1,0_{M to}

an Abelian subspace ofpC_ǫ.

Let ω0 be the standard symplectic form of R2r_{, i.e. ω0 =}Pr

i=1ei∧ei+r where (ei)2r

i=1 is the dual of the standard basis ofR2r. Then we define

(4.2) Sym(ω0) :={A∈GL(2r,R_{)|ω0(A·,·) =ω0(·, A·)}.}

The adjoint ofg∈GL(2r,R_{) with respect toω0will be denoted byg}†_{.HenceSym(ω0)}

are the elementsA∈GL(2r,R_{) which satisfyA}† _=A.

Every element A ∈ Sym(ω0) defines a symplectic form ωA on R2r by ωA(·,·) =

ω0(A·,·).To this interpretation corresponds an action

GL(2r,R_{)×Sym(ω0)→Sym(ω0), (g, A)7→(g}−1₎†_Ag−1_.

This action is used to identify S(2r) and Sym(ω0) by a map Ψ in the following proposition.

Proposition 4. Let Ψbe the canonical map Ψ : S(2r) ˜→Sym(ω0)⊂GL(2r,R₎ whereGL(2r,R_{) carries the pseudo-Riemannian metric induced by the Ad-invariant} trace-form. Then Ψ is a totally geodesic immersion and a map f from an ǫcomplex manifold (M, Jǫ_{) to S(2r) is ǫpluriharmonic, if and only if the map Ψ◦f : M →}

Sym(ω0)⊂GL(2r,R_{)isǫpluriharmonic.}

Proof. The proof is done by relating the map Ψ to the well-known Cartan-immersion. Additional information can be found in [10, 7, 9, 12].

1. First we study the identificationS(2r) ˜→Sym(ω0). GL(2r,R_{) operates onSym(ω0) via}

GL(2r,R_{)×Sym(ω0)→Sym(ω0), (g, B)7→g·B:= (g}−1₎†_Bg−1_.

The stabiliser of the 12r is Sp(R2r) and the action is seen to be transitive by choosing a symplectic basis. Using the orbit-stabiliser theorem we get by identifying orbits and rest-classes a diffeomorphism

Ψ :S(2r) ˜→Sym(ω0), g Sp(R2r_)7→g·12_{r= (g}−1₎†12_rg−1_{= (g}−1₎†_g−1_.

2. We recall some results about symmetric spaces (see: [7, 13]). Let G be a Lie-group and σ : G→G a group-homomorphism withσ2 _=Id

the subgroupK=Gσ _{={g∈G|σ(g) =g}.The Lie-algebragofGdecomposes} in g = h⊕p with dσIdG(h) = h, dσIdG(p) = −p. One has the following

information: The mapφ:G/K→Gwithφ : [gK]7→gσ(g−1_{) defines a totally}

geodesic immersion called the Cartan-immersion.

We want to utilise this in the case G =GL(2r,R_{) and K = Sp(}R2r_{). In this} spirit we define σ : GL(2r,R_{) → GL(2r,}R_{), g 7→ (g}−1₎†_{. The map σ is}

obviously a homomorphism and an involution withGL(2r,R₎σ_=Sp(R2r_).

By a direct calculation one getsdσIdG=−h

† _{and hence}

h={h∈gl_2r(R_)|h† _=−h}=sp(R2r_{), p={h∈gl}

2r(R)|h†=h}=: sym(ω0).

Thus we end up with φ : S(2r) → GL(2r,R_{), g 7→ gσ(g}−1_{) = gg}† _{= Ψ◦}

Λ(g).Here Λ is the map induced by Λ : G→G, h 7→(h−1₎†_{. This map is an}

isometry of the invariant metric. Hence Ψ is a totally geodesic immersion. Using proposition 3 the proof is finished.

Remark 3. Above we have identifiedS(2r) withSym(ω0) via Ψ.

Let us choose o = eSp(R2r_{) as base point and suppose that Ψ is chosen to map}

oto 12r. By construction Ψ is GL(2r,R_{)-equivariant. We identify the tangent-space} TωSym(ω0) atω∈Sym(ω0) with the (ambient) vector space ofω0-symmetric matrices ingl_2r(R₎

(4.3) TωSym(ω0) = sym(ω0).

For ˜ω∈S(2r) such that Ψ(˜ω) =ω, the tangent spaceT˜ωS(2r) is canonically identified with the vector space ofω-symmetric matrices:

(4.4) T˜ωS(2r) = sym(ω) :={A∈gl2r(R)|A†ω=ωA}.

Note that sym(12r) = sym(ω0).

Proposition 5. The differential of ϕ:= Ψ−1 _{atω∈Sym(ω0)is given by}

(4.5) sym(ω0)∋X 7→ −1

2ω

−1_X∈ω−1_{sym(ω0) = sym(ω).}

Using this proposition we relate now the differentials

(4.6) dfx:TxM →sym(ω0)

of a mapf :M →Sym(ω0) atx∈M and

(4.7) df˜x:TxM →sym(f(x))

of a map ˜f =ϕ◦f :M →S(2r):df˜x=dϕ dfx=−1₂f(x)−1dfx.

We interpret the one-formA=−2df˜=f−1_{dfwith values ingl}

2r(R) as connection form on the vector bundle E = M ×R2r_{. We note, that the definition of A is the}

Proposition 6. Let f : M → GL(2r,R_{) be a C}∞_{-mapping and A := f}−1_{df :} T M →gl_2r(R_{). Then the curvature of Avanishes, i.e. forX, Y ∈Γ(T M)it is}

(4.8) Y(AX)−X(AY) + [AY, AX] +A[X,Y] = 0.

In the next proposition we recall the equations forǫpluriharmonic maps from an

ǫcomplex manifold toGL(2r,R_):

Proposition 7. (cf. [17, 19]) Let (M, Jǫ_{) be an ǫcomplex manifold, f : M →}

GL(2r,R_)aC∞_{-map andA defined as in proposition 6.} Theǫpluriharmonicity of f is equivalent to the equation

(4.9) Y(AX) + 1

2[AY, AX]−ǫJ ǫ_Y(A

JǫX)−ǫ1

2[AJǫY, AJǫX] = 0,

for allX, Y ∈Γ(T M).

With a similar argument as in proposition 4 we have shown in [18, 22]:

Proposition 8. Let (M, Jǫ_{)be an ǫcomplex manifold. A map}

φ : M →GL(r,C_ǫ)/Uǫ_{(p, q),}

where the target-space is carrying the (pseudo-)metric induced by the trace-form on GL(r,C_{ǫ), isǫpluriharmonic if and only if}

ψ= Ψǫ_{◦φ : M →GL(r,Cǫ)/U}ǫ_{(p, q) ˜→Herm}ǫ

p,q(Crǫ)⊂GL(r,Cǫ)

isǫpluriharmonic.

To be complete we mention the related symmetric decomposition:

h={A∈gl_r(C_ǫ)|Ah_=−A}=uǫ_{(p, q), p={A∈gl}

r(Cǫ)|Ah=A}=: hermǫp,q(Crǫ).

5

tt

∗

-geometry and pluriharmonic maps

In this section we are going to state and prove the main results. Like in section 4 one regards the mappingA=f−1_{df as a mapA:T M →gl}

2r(R).

Theorem 1. Let(M, Jǫ_{)be a simply connectedǫcomplex manifold. Let(E, D, S, ω)}

be a symplecticǫtt∗_{-bundle where E has rank2randM dimension n.}

The matrix representationf : M →Sym(ω0) of ω in aDθ_{-flat frame of E induces}

an admissibleǫpluriharmonic map f˜: M →f Sym(ω0) ˜→S(2r), where S(2r) carries the (pseudo-Riemannian) metric induced by the trace-form on GL(2r,R_{). Let s}′ _be anotherDθ_{-flat frame. Thens}′ _{=s·U for a constant matrix and the ǫpluriharmonic} map associated tos′ _isf′_=U†_{f U.}

Proof. Thanks to remark 1.1) we can restrict to the caseθ=πforǫ=−1 andθ= 0 forǫ= 1.

Let s := (s1, . . . , s2r) be a Dθ_{-flat frame of E (i.e. Ds = −ǫSs), f the matrix}

X(f) = Xω(s, s) =ω(DXs, s) +ω(s, DXs) = −ǫ[ω(SXs, s) +ω(s, SXs)]

= −2ǫω(SXs, s) =−2ǫf·SXs.

It followsAX =−2ǫSsX.We now prove theǫpluriharmonicity using

dD_{S(X, Y) = DX(S}

Y)−DY(SX)−S[X,Y]= 0,

(5.1)

dDSJǫ(X, Y) = DX(S_Jǫ_Y)−D_Y(S_Jǫ_X)−S_Jǫ_[X,Y] = 0.

(5.2)

The equation (5.2) implies

0 =dD_S

Jǫ(JǫX, Y) = D_JǫX(S_Jǫ_Y)− ǫD_Y(SX)

| {z }

(5.1)

= ǫ(DX(SY)−S[X,Y])

−SJǫ_[Jǫ_X,Y]

= DJǫX(S_Jǫ_Y)−ǫDX(S_Y) +ǫS[_X,Y]−S_Jǫ_[Jǫ_X,Y].

In localǫholomorphic coordinate fieldsX, Y onM we get in the frames

Jǫ_X(Ss

Jǫ_Y)−ǫX(S_Ys) + [Ss_X, S_Ys]−ǫ[S_Jsǫ_X, S_Jsǫ_Y] = 0.

NowA=−2ǫSs_{gives equation (4.9) and proves the ǫpluriharmonicity off.}

UsingAX =−2ǫSXs =−2df˜(X), we find the property of the differential, asS∧S is of type (1,1) by theǫtt∗_{-equations, see proposition 2. The last statement is obvious.}

Theorem 2. Let (M, Jǫ_{) be a simply connected ǫcomplex manifold and put}

E = M ×R2r_{. Then an ǫpluriharmonic map f}˜ _{: M → S(2r) gives rise to an}

ǫpluriharmonic mapf :M →f˜ S(2r) ˜→Sym(ω0)֒→i GL(2r,R_).

If the mapf˜is admissible, then the mapf induces a symplecticǫtt∗_{-bundle(E, D=} ∂−ǫS, S=ǫdf , ω˜ =ω0(f·,·))on M where∂ is the canonical flat connection on E.

Remark 4. We observe, that forǫRiemannian surfacesM = Σ everyǫpluriharmonic map is admissible, sinceT1,0_{Σ is one-dimensional.}

Proof.

Let ˜f :M →S(2r) be anǫpluriharmonic map. Then due to proposition 4 we know, thatf :M→Sym˜ (ω0)֒→i GL(2r,R_{) isǫpluriharmonic.}

Since E = M ×R2r_{, we want to regard sections of E as 2r-tuples of C}∞_(M,R

)-functions.

As in section 4 we consider the one-formA=−2df˜=f−1_{dfwith values ingl}

2r(R) as a connection onE.The curvature of this connection vanishes (proposition 6). First, the constraints onω are fulfilled:

Proof. This is a direct computation withX∈Γ(T M) andv, w∈Γ(E): To finish the proof, we have to check theǫtt∗_{-equations. The secondǫtt}∗_-equation

(5.3) −ǫ[SX, SY] = [SJǫ_X, S_Jǫ_Y]

is equivalent to the vanishing of the curvature ofA=−2ǫSinterpreted as a connection onE (see proposition 6).

This shows the vanishing of the tensordD_S

Jǫ.It remains to show the curvature

equa-tion forD.We observe, thatD+ǫS =∂−ǫS+ǫS =∂and that the connection∂is flat,

In the situation of theorem 2 the two constructions are inverse.

Proposition 9.

1. Given a symplectic ǫtt∗_{-bundle (E, D, S, ω) on an ǫcomplex manifold (M, J}ǫ_).

Let f˜be the associated admissibleǫpluriharmonic map constructed to aDθ_-flat

framesin theorem 1. Then the symplecticǫtt∗_-bundle(M×Rr_{,S,˜ ω˜)associated}

2. Given an admissibleǫpluriharmonic mapf˜: (M, Jǫ_{)→S(2r),then one obtains}

via theorem 2 a symplecticǫtt∗_-bundle(M×R2r_{, D, S, ω). The ǫpluriharmonic}

map associated to this symplectic ǫtt∗_{-bundle is conjugated to the map f}˜_{by a} constant matrix.

Proof. Using again remark 1.1) we can set θ =π in the complex andθ = 0 in the para-complex case.

1. The mapf is obviouslyω in the frame s and in the computations of theorem 1 one getsA=−2df˜=f−1_{df =−2ǫS}s_{.From 0 =D}θ_{s=Ds+ǫSs we obtain} that the connectionDin the frame sis just∂−ǫSs_=∂+A

2.

2. We have to find aDθ_{-flat frames. It is D}θ_{=∂−ǫS+ǫS =∂. Hence we can} take s as the standard-basis of R2r _{and we get f. Every other basis gives a} conjugated result.

6

Harmonic bundle solutions

In this section we use the notationHǫ_{(p, q) :=GL(r,Cǫ)/U}ǫ_{(p, q).As motivation for} consideringǫharmonic bundles we prove :

Proposition 10. The canonical inclusion

i:GL(r,C_ǫ)/Uǫ_{(p, q)֒→GL(2r,}R_)/Sp(R2r₎

is totally geodesic. Let further(M, Jǫ_{)be anǫcomplex manifold, then a mapα:M →}

GL(r,C_ǫ)/Uǫ_{(p, q)isǫpluriharmonic if and only ifi◦α:M →GL(2r,R)/Sp(R}2r_)is

ǫpluriharmonic.

Proof. Looking at the inclusion of the symmetric decompositions

gl_r(C_{ǫ) = hermp,q(}Cr_ǫ)⊕uǫ(p, q)⊂gl_2rR_{= sym(ω0)⊕}o(k, l),

we see, that hermǫ_p,q(Cr

ǫ)⊂sym(ω0) is a Lie-triple system, i.e.

[hermǫp,q(Crǫ),[hermǫp,q(Crǫ),hermǫp,q(Crǫ)]]⊂hermǫp,q(Crǫ)

and that therefore the inclusionGL(r,C_ǫ)/Uǫ_{(p, q) ֒→}i _{GL(2r,R)/Sp(R}2r_{) is totally} geodesic. The second statement follows from proposition 3.

In [18, 22] we related ǫpluriharmonic maps from an ǫcomplex manifold (M, Jǫ₎ intoHǫ_{(p, q) =GL(r,Cǫ)/U}ǫ_{(p, q) withr=p+qtoǫharmonic bundles over (M, J}ǫ_). First we recall the definition of anǫharmonic bundle:

Definition 5. Anǫharmonic bundle (E →M, D, C,C, h¯ ) consists of the following data:

An ǫcomplex vector bundle E over an ǫcomplex manifold (M, Jǫ_{), an ǫhermitian} metric h, a metric connection D with respect to h and two C∞_{-linear maps C :}

Γ(E)→Γ(Λ1,0_T∗_{M⊗E) and ¯C: Γ(E)→Γ(Λ}0,1_T∗_{M⊗E),such that the connection}

D(λ)=D+λC+ ¯λC¯

Remark 5. In the complex case with positive definite metric h this definition is equivalent to the definition of a harmonic bundle in Simpson [23]. Equivalent struc-tures in the complex case with metrics of arbitrary signature have been also regarded in Hertling’s paper [11].

The relation ofǫharmonic bundles toǫpluriharmonic maps is stated in the following theorem.

Theorem 3. (cf. [18, 22])

(i) Let (E → M, D, C,C, h¯ ) be an ǫharmonic bundle over the simply connected ǫcomplex manifold (M, Jǫ_{). Then the representation of h in a D}(λ)_{-flat frame} defines anǫpluriharmonic mapφh:M →Hermp,qǫ (Crǫ). The mapφh induces an

admissible ǫpluriharmonic mapφ˜h= Ψǫ◦φh:M →Hǫ(p, q)(cf. proposition 8

forΨǫ_{) .}

(ii) Let (M, Jǫ_{) be a simply connected ǫcomplex manifold andE =M ×C}r ǫ. Given

an admissible ǫpluriharmonic mapφ˜h :M →Hǫ(p, q), then (E, D=∂−ǫC−

ǫC, C¯ =ǫ(dφ˜h)1,0_{, h= (φ}

h·,·)Cr

ǫ) defines an ǫharmonic bundle, where ∂ is the

ǫcomplex linear extension onT MC_ǫ

of the flat connection onE=M×Cr ǫ.

In the complex case of signature(r,0)and(0, r)every pluriharmonic mapφ˜h is

admissible.

The last theorem and proposition 10 yieldǫpluriharmonic maps toGL(2r,R_)/Sp(R2r_).

We are going to identify the related symplecticǫtt∗_{-bundles. Therefore we construct}

symplecticǫtt∗_{-bundles fromǫharmonic bundles, via the next proposition.}

Proposition 11. Let (E → M, D, C,C, h¯ ) be an ǫharmonic bundle over the ǫcomplex manifold (M, Jǫ_{), then (E, D, S,Ω = Imh) with S}

X := CZ + ¯C_Z¯ for X=Z+ ¯Z∈T M andZ ∈T1,0_{M is a symplecticǫtt}∗_-bundle.

Proof. Forλ= cosǫ(α) + ˆisinǫ(α)∈S1_{ǫ we computeD}(λ)_:

D(_Xλ) = DX+λCZ+ ¯λC¯_Z¯=DX+ cosǫ(α)(CZ+ ¯C_Z¯) + sinǫ(α)(ˆiCZ−ˆiC¯_Z¯)

= DX+ cosǫ(α)SX+ sinǫ(α)(CJǫ_Z+ ¯C_Jǫ_Z¯)

= DX+ cosǫ(α)SX+ sinǫ(α)SJǫ_X=D_Xα.

Hence we see

(6.1) Dα=D(λ)

andDα _{is flat if and only ifD}(λ) _{is flat.}

Further we claim, thatS is Ω-symmetric. With X=Z+ ¯Z forZ∈T1,0_{M one finds}

h(SX·,·) =h(CZ+ ¯C_Z¯·,·) =h(·, CZ+ ¯C_Z¯·) =h(·, SX·).

This yields the symmetry ofS with respect to Ω = Imh.

2ˆiXΩ(e, f) = X.(h(e, f)−h(f, e)) = (Z+ ¯Z).(h(e, f)−h(f, e))

From theorem 1 one obtains the next corollary.

Corollary 1. Let (E → M, D, C,C, h¯ ) be an ǫharmonic bundle over the simply connectedǫcomplex manifold(M, Jǫ_{),then the representation ofΩ =Imhin aD}(λ) -flat frame defines anǫpluriharmonic mapΦΩ:M →GL(R2r)/Sp(R2r).

Proof. This follows from the identity (6.1), i.e.D(_Xλ)=Dα

Xforλ= cosǫ(α)+ˆisinǫ(α)∈

S1_{ǫ and from proposition 11 and theorem 1.}

Our aim is to understand the relations between theǫpluriharmonic maps in theo-rem 3 and corollary 1. Therefore we need to have a closer look at the maph7→Imh.

First, we identify Cr

ǫ with Rr⊕ˆiRr = R2r. In this model the multiplication with ˆ_{i coincides with the automorphism j}ǫ ₌

µ

ǫ) decomposes in its real-partAand its imaginary part

B,i.e.C=A+ˆi BwithA, B∈End(Rr_{). In the above modelCis given by the matrix} conjugatedCh _{ofC correspond to}

ι( ¯C) = complex case) coincide with the subset of symmetric matrices H ∈ Sym_k,l(R2r_),

which commute withjǫ_{, i.e. [H, j}ǫ_{] = 0,where the pair (k, l) is}

(k, l) =

(

(2p,2q) forǫ=−1,

Likewise, TIk,lHerm

ǫ

p,q(Crǫ) consists of symmetric matrices h ∈ symk,l(R2r), which commute withjǫ_{, i.e. theǫhermitian matrices in sym}

k,l(R2r) which we have denoted by hermǫ_p,q(Cr

ǫ).

An ǫhermitian sesquilinear scalar product h (of signature (p, q) for ǫ = −1) cor-responds to an ǫhermitian matrix H ∈ Hermǫ

p,q(Crǫ) (of hermitian signature (p, q) for ǫ = −1) defined by h(·,·) = (H·,·)Cr

ǫ. The condition C

h _{= ¯C}t _{= C, i.e. C is}

ǫhermitian, means in our model, thatChas the form

ι(C) =

Using this information we find the explicit representation of the map which cor-responds to taking the imaginary part Imh of h. This is the map ℑ satisfying Imh= (ℑ(H)·,·)R2r, where (·,·)_R2r is the Euclidean standard scalar product onR2r.

standard scalar product onR2r_.

This yields Imh(z, w) = (Iǫ_jǫ_{ι(H)z, w)}

This map is easily seen to have maximal rank and to be equivariant with respect to the followingGL(r,C_{ǫ)-action on Herm}ǫ

p,q(Crǫ):

GL(r,C_ǫ)×Hermǫ_p,q(Cr_{ǫ)→Hermp,q}ǫ ₍Cr_{ǫ), (g, H)7→(g}−1₎h_Hg−1

and theGL(2r,R_{)-action on Sym(ω0) which was considered in section 4. Summarising}

we have the commutative diagram in which all maps apartℑof the square were shown to be totally geodesic:

Proposition 12. A map h: M → Hermǫ

p,q(Crǫ) is ǫpluriharmonic, if and only if Ω =Imh:M →Sym(ω0)isǫpluriharmonic.

Proof. As discussed above, the mapℑis a totally geodesic immersion and therefore we are in the situation of proposition 3.

From this proposition it follows:

Proposition 13. Let (E → M, D, C,C, h¯ ) be an ǫharmonic bundle over the ǫcomplex manifold(M, Jǫ_{),(E, D, S,Ω =Imh)the symplecticǫtt}∗_{-bundle constructed} in proposition 11 andΦ˜Ω:M →GL(R2r)/Sp(R2r)the ǫpluriharmonic map given in corollary 1. ThenΦ˜Ω= [i]◦Φh˜ and these ǫpluriharmonic maps are admissible.

Proof. This follows using the definition of (E, D, S,Ω) (cf. proposition 11) from corol-lary 1 and proposition 12. For the second part one observes, that the differential of [i] is a homomorphism of Lie-algebras.

This describes the ǫpluriharmonic maps coming from symplecticǫtt∗_-bundles

in-duced byǫharmonic bundles. Conversely, this gives an Ansatz to constructǫharmonic bundles fromǫpluriharmonic maps to GL(r,C_ǫ)/Uǫ_{(p, q).For metricǫtt}∗_{-bundles we}

have gone this way in [18, 22] to obtain theorem 3.

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Author’s address:

Lars Sch¨afer