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×R2r.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-moduleCn 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 ˜S1 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 = Rn ⊕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 ΛkT∗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 ΛkT∗MC= L
k=p+q
Λp,qT∗M and finally on theC-valued differential forms
on M Ωk
Λp,qT∗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, τE). 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∈Crh(v, w) = 0,
GL(r, C).We remark, that there isnonotion of para-hermitian signature, since from
h(v, v) =−1 for an elementv∈Crwe obtainh(ev, ev) = 1.
Proposition 1. Given an element C of End(Cr) then it holds (Cz, w)Cr = (z, Chw)Cr, ∀z, w∈Cr.The set herm(Cr)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 Cǫ with C1 = C
andC−1=CandS1
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, Jǫ) 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 connectionDθ 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), dDS= 0 and dDS 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) DJǫYX =JǫDYX
for all vector fields which satisfyLXJǫ= 0 (i.e. for whichX+ǫˆiJǫX 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∇hand 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(JǫX, JǫY) = 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 mapsT1,0M to
an Abelian subspace ofpCǫ.
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·12r= (g−1)†12rg−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∈gl2r(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 ingl2r(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
−1X∈ω−1sym(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−1dfwith 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−1df : T M →gl2r(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 ǫY(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,Cǫ)/Uǫ(p, q),
where the target-space is carrying the (pseudo-)metric induced by the trace-form on GL(r,Cǫ), isǫpluriharmonic if and only if
ψ= Ψǫ◦φ : M →GL(r,Cǫ)/Uǫ(p, q) ˜→Hermǫ
p,q(Crǫ)⊂GL(r,Cǫ)
isǫpluriharmonic.
To be complete we mention the related symmetric decomposition:
h={A∈glr(Cǫ)|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−1df 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
dDS(X, Y) = DX(S
Y)−DY(SX)−S[X,Y]= 0,
(5.1)
dDSJǫ(X, Y) = DX(SJǫY)−DY(SJǫX)−SJǫ[X,Y] = 0.
(5.2)
The equation (5.2) implies
0 =dDS
Jǫ(JǫX, Y) = DJǫX(SJǫY)− ǫDY(SX)
| {z }
(5.1)
= ǫ(DX(SY)−S[X,Y])
−SJǫ[JǫX,Y]
= DJǫX(SJǫY)−ǫDX(SY) +ǫS[X,Y]−SJǫ[JǫX,Y].
In localǫholomorphic coordinate fieldsX, Y onM we get in the frames
JǫX(Ss
JǫY)−ǫX(SYs) + [SsX, SYs]−ǫ[SJsǫX, SJsǫY] = 0.
NowA=−2ǫSsgives 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, sinceT1,0Σ 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−1dfwith 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] = [SJǫX, SJǫY]
is equivalent to the vanishing of the curvature ofA=−2ǫSinterpreted as a connection onE (see proposition 6).
This shows the vanishing of the tensordDS
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, Jǫ).
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−1df =−2ǫSs.From 0 =Dθs=Ds+ǫSs we obtain that the connectionDin the frame sis just∂−ǫSs=∂+A
2.
2. We have to find aDθ-flat frames. It is Dθ=∂−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)/Uǫ(p, q).As motivation for consideringǫharmonic bundles we prove :
Proposition 10. The canonical inclusion
i:GL(r,Cǫ)/Uǫ(p, q)֒→GL(2r,R)/Sp(R2r)
is totally geodesic. Let further(M, Jǫ)be anǫcomplex manifold, then a mapα:M →
GL(r,Cǫ)/Uǫ(p, q)isǫpluriharmonic if and only ifi◦α:M →GL(2r,R)/Sp(R2r)is
ǫpluriharmonic.
Proof. Looking at the inclusion of the symmetric decompositions
glr(Cǫ) = hermp,q(Crǫ)⊕uǫ(p, q)⊂gl2rR= sym(ω0)⊕o(k, l),
we see, that hermǫp,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,Cǫ)/Uǫ(p, q) ֒→i GL(2r,R)/Sp(R2r) 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)/Uǫ(p, q) withr=p+qtoǫharmonic bundles over (M, Jǫ). 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(Λ1,0T∗M⊗E) and ¯C: Γ(E)→Γ(Λ0,1T∗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 ×Cr 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 MCǫ
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 + ¯CZ¯ for X=Z+ ¯Z∈T M andZ ∈T1,0M is a symplecticǫtt∗-bundle.
Proof. Forλ= cosǫ(α) + ˆisinǫ(α)∈S1ǫ we computeD(λ):
D(Xλ) = DX+λCZ+ ¯λC¯Z¯=DX+ cosǫ(α)(CZ+ ¯CZ¯) + sinǫ(α)(ˆiCZ−ˆiC¯Z¯)
= DX+ cosǫ(α)SX+ sinǫ(α)(CJǫZ+ ¯CJǫZ¯)
= DX+ cosǫ(α)SX+ sinǫ(α)SJǫX=DXα.
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∈T1,0M one finds
h(SX·,·) =h(CZ+ ¯CZ¯·,·) =h(·, CZ+ ¯CZ¯·) =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)∈
S1ǫ 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 ˆi coincides with the automorphism jǫ =
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 ∈ Symk,l(R2r),
which commute withjǫ, i.e. [H, jǫ] = 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 hermǫp,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 = ¯Ct = 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) = (Iǫjǫι(H)z, w)
This map is easily seen to have maximal rank and to be equivariant with respect to the followingGL(r,Cǫ)-action on Hermǫ
p,q(Crǫ):
GL(r,Cǫ)×Hermǫp,q(Crǫ)→Hermp,qǫ (Crǫ), (g, H)7→(g−1)hHg−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,Cǫ)/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
