QUADRATIC PSEUDO-HARMONIC MORPHISMS

QUADRATIC PSEUDO-HARMONIC MORPHISMS

NECULAE DINUT¸ ˘A

We give a description of quadratic pseudo-harmonic morphisms from the Eu- clidean spacesR^mto complex spacesCⁿ.

AMS 2000 Subject Classification: 53C43, 58E20.

Key words: harmonic map, harmonic morphism, pseudo-harmonic morphism, pseudo-horizontally (weakly) conformal map, pseudo-horizontally ho- mothetic map, quadratic pseudo-harmonic morphism.

1. INTRODUCTION

Polynomial mappings play a central role in the study of harmonic mor- phisms. For example, at a critical point x₀ of finite order of a horizontally conformal map ϕ:M^m →Nⁿ between arbitrary Riemannian manifolds, the symbolδ_xo(ϕ) :R^m→Rⁿis a horizontally conformal map between Euclidean spaces defined by homogeneous polynomials (see [4]). Quadratic harmonic morphisms were classified by Ou and Wood (see [4]).

In this paper we give a description of quadratic pseudo-harmonic mor- phisms from the Euclidean spaces Rⁿ to complex spaces Cⁿ. In Section 2 we present definitions and properties of harmonic morphisms and pseudo- harmonic morphisms. In Section 3 we study quadratic pseudo-harmonic mor- phisms and PHH maps. In Section 4 we give a complete description of these morphisms in a particular case.

2. HARMONIC MORPHISMS

AND PSEUDO-HARMONIC MORPHISMS

Let ϕ : (M^m, g_M) → (Nⁿ, g_N) be a smooth map between Riemannian manifolds and∇^M, ∇^N their Levi-Civita connections. The connection on N induces a connection ˜∇on the pull back bundle ϕ⁻¹T N.

A map ϕ is called harmonic (cf. [7], [16], [5], [4], [3], [10]) if and only if, for any compact set Ω ⊂ M, its restriction to Ω is a critical point of the

REV. ROUMAINE MATH. PURES APPL.,52(2007),3, 329–339

2-energy functional

E₂(ϕ,Ω) = 1 2

Ω|dϕ|².

A standard fact states that this property can be expressed by the van- ishing of thetension field ofϕdefined by

T(ϕ) = m k=1

1 g_M(e_k, e_k)

∇˜ekdϕ(e_k)−dϕ

∇^M_eke_k ,

where (e_k)_k=1,...,m is an orthogonal frame field defined locally on M. In local coordinates (x_k)_k=1,...,m and (y_γ)_γ=1,...,n on M and N, respectively, denoting ϕ^α =y_α◦ϕ, the harmonicity of the mapϕ amounts to the equations

(1)

m k,l=1

g_M^kl

∂²ϕ^γ

∂x_k∂x_l −^m

t=1

MΓ^t_kl∂ϕ^γ

∂x_t + n α,β=1

NΓ^γ_αβ∂ϕ^α

∂x_k ·∂ϕ^β

∂x_l

= 0 for any γ= 1, . . . , n.

An important class of harmonic maps is the class of harmonic morphisms.

A map ϕ is called a harmonic morphism if and only if it pulls back local harmonic functions onN to local harmonic functions on M. Let us mention (cf. [8], [11]) that ifm < n, then the only harmonic morphisms fromM to N are the constant maps, thus the significant cases only occur when m ≥ n.

Fuglede [8] and Ishihara [11] (see, also [2], [4], [9]) proved that harmonic morphisms are precisely the harmonic mapsϕwhich arehorizontally (weakly) conformali.e.

(1)dϕ_x = 0 for any critical pointx ofϕ, and

(2) there exists a non-vanishing smooth functionλdefined on the set of smooth points ofϕsuch that at any such pointx we have

λ²(x)g_M(X, Y) =g_N(dϕ(X), dϕ(Y))

for any horizontal vector fields X and Y on M. The function λ is called the dilation function ofϕ.

Ifϕhas no critical points, then we shall give up using the word “weakly”.

In local coordinates, horizontal weakly conformality can be expressed as (2)

m k,l=1

g_M^kl∂ϕ^α

∂x_k ·∂ϕ^β

∂x_l =λ²g_N^αβ for allα, β = 1, . . . , n.

A particular class of horizontally weakly conformal maps is that of hor- izontally homothetic maps, which are given by the extra-condition X(λ) = 0 for any horizontal vector fieldX on M.

One can generalize the notion of harmonic morphism as follows (cf.

Loubeau [13] and Chen [6]). A smooth mapϕ: (M^m, g_M)→(N²ⁿ, J, g_N) from

a Riemannian manifold to a K¨ahler manifold is said to be apseudo-harmonic morphism (shortening PHM) if and only if it pulls back local holomorphic functions onN to local harmonic maps fromM to C.

For any x ∈ M let us denote by dϕ_x : T_ϕ(x)N → T_xM the adjoint map of a tangent linear map dϕ_x : T_xM → T_ϕ(x)N. We say that ϕ is pseudo-horizontally(weakly) conformalatx∈M if and only ifdϕ_x◦dϕ_x com- mutes withJ_ϕ(x). It is calledpseudo-horizontally (weakly) conformal(PHWC) (cf. [13]) if it is pseudo-horizontally (weakly) conformal at every point ofM. One can prove (cf [6], [13]) that pseudo-harmonic morphisms are precisely the harmonic maps which are pseudo-horizontally (weakly) conformal.

In local coordinates, i.e., (x_k)_k=1,...,m real local coordinates on M and (z_A)_A=1,...,n complex local coordinates on N, denoting ϕ^A = z_A◦ϕ for any A= 1, . . . , n, the PHWC condition forϕis equivalent to

(3)

m k,l=1

g^kl_M∂ϕ^A

∂x_k ·∂ϕ^B

∂x_l = 0 for allA, B = 1, . . . , n.

The notion of pseudo-horizontally homothetic map was introduced by Aprodu, Aprodu and Brˆınz˘anescu [1].

A mapϕ: (M^m, g_M)→(N²ⁿ, J, g_N) which is PHWC at a point x∈M is calledpseudo-horizontally homothetic at x if and only if

dϕ_x

∇^M_v dϕ(J Y)

x

=J_ϕ(x)dϕ_x

∇^M_v dϕ(Y)

x

for any horizontal tangent vector v ∈ T_xM and any vector field Y locally defined on a neighborhood ofϕ(x).

A mapϕthat is PHWC, is calledpseudo-horizontally homothetic (PHH) if and only if it is pseudo-horizontally homothetic at any point of M, i.e., if and only if

(4) dϕ

∇^M_Xdϕ(J Y)

=J dϕ

∇^M_Xdϕ(Y)

for any horizontal vector fieldX on M and any vector fieldY on N.

If condition (4) is satisfied for every vector fieldsX on M and Y on N, we shall say that the mapϕis strongly pseudo-horizontally homothetic.

The local description of the pseudo-horizontally homothetic condition at a given pointu₀ ∈M can be expressed by using local normal coordinates on M andN, respectively, as follows (see [1], Lemma 3.2).

Lemma 2.1. With the notation above, suppose ϕ is submersive. Let u₀ ∈ M be a given point, (u_k)_k=1,...,m local normal coordinates at u₀ such that the vectors _∂u^∂

2n+1(u₀), . . . ,_∂u^∂_m(u₀) are tangent to the fibre through u₀, (z_A)_A=1,...,n local normal complex coordinates at ϕ(u₀)∈N, and ϕ^A=z_A◦ϕ.

Thenϕ is pseudo-horizontally homothetic atu₀ if and only if m

k=1

∂ϕ^A

∂u_k(u₀)·∂ϕ^B

∂u_k(u₀) = 0

and m

j=1

∂²ϕ^A

∂u_j∂u_k(u₀)·∂ϕ^B

∂u_j (u₀) = 0 for allA, B= 1, . . . , n, k= 1, . . . ,2n.

From the proof of Lemma 2.1 (cf. [1]) one can see that ϕ is strongly pseudo-horizontally homothetic atu₀ if and only if

(5)

m j=1

∂²ϕ^A

∂u_j∂u_k(u₀)·∂ϕ^B

∂u_j (u₀) = 0 for allA, B = 1, . . . , n,k= 1, . . . ,2n.

3. QUADRATIC PSEUDO-HARMONIC MORPHISMS Polynomial mappings play a central role in the study of harmonic mor- phisms. For example, at a critical point x₀ of finite order of a horizontally conformal map ϕ : M^m → Nⁿ between arbitrary Riemannian manifolds, σ_x0(ϕ) :R^m →Rⁿ is a horizontally conformal map between Euclidean spaces defined by homogeneous polynomials (see [4],§5.7). Quadratic harmonic mor- phisms were classified be Ou and Wood (see [14], [15]; for a general presenta- tion see Chapter 5 of [4]).

Now, we shall study quadratic pseudo-harmonic morphisms. By a qua- dratic map we mean one which is defined by homogeneous polynomials of degree 2. A quadratic mapϕ:R^m→Cⁿ can always be written in the form (6) ϕ(x) = (x^tA₁x, . . . , x^tA_nx),

wherexdenotes a column vector inR^m,x^tits transpose, and theA_k ∈ M_m(C) are complex symmetric m×m matrices, which are called thecomponent ma- trices of ϕ.

OnR^m we shall take the standard Euclidean metric and on Cⁿwe shall take the standard Hermitian metric. We shall also consider the inner product onCⁿ given by

(7) z, w=

n k=1

z_k·w_k,

wherez= (z₁, . . . , z_n) andw= (w₁, . . . , w_n) are elements ofCⁿ.

We have the following first result.

Proposition 3.1 (Criterion for a quadratic PHWC map). A quadratic map ϕ:R^m →Cⁿ given by (6) is a PHWC map if and only if its component matrices satisfy the conditions

(8) A_k·A_l+A_l·A_k= 0

for allk, l= 1, . . . , n.

Proof. We can writeϕ(x) = (ϕ¹(x), . . . , ϕⁿ(x)). By using conditions (3) we get thatϕis a PHWC map if and only if

(9) gradϕ^k,gradϕ^l= 0, k, l= 1, . . . , n at all points x∈R^m. But

gradϕ^k= 2·A_k·x and equations (9) are equivalent to

(10) x^t·A_k·A_l·x= 0

for all points x∈ R^m and all k, l = 1, . . . , n (since the matrices A_k are sym- metric). Taking the transpose of (10) we get

(x^t·A_k·A_l·x)^t = 0 and sox^t·A_l·A_k·x= 0.

It follows that equations (9) are equivalent to the equations (11) x^t·(A_k·A_l+A_l·A_k)·x= 0 for allx∈R^m and allk, l = 1, . . . , n.

Let us denoteM_kl =A_k·A_l+A_l·A_k. SinceM_kl^t =M_kl, the matrixM_klis symmetric. LetM_kl=U_kl+ iV_kl where U_kl, V_kl ∈ M_m(R) are real symmetric matrices. Then equations (11) are equivalent to

(12) x^t·U_kl·x= 0 and x^t·V_kl·x= 0.

SinceU_kl, V_kl are real symmetric matrices we get the equivalent equations

(13) U_kl= 0 and V_kl = 0,

which are equivalent toM_kl= 0 for allk, l= 1, . . . , n, i.e., conditions (8) from the statement.

Theorem3.1 (Automatic harmonicity). Letϕ:R^m→Cⁿbe a quadratic PHWC map. The ϕis harmonic, and so is a pseudo-harmonic morphism.

Proof. Write ϕ(x) = (ϕ¹(x), . . . , ϕⁿ(x)). It suffices to prove that every component ϕ^k : R^m → C is a harmonic function. For l = k, by (8) we get

A²_k= 0 for all k= 1, . . . , n, and so the matrixA_k is nilpotent. It follows that the trace ofA_k is zero. But

TrA_k =a^(k)₁₁ +· · ·+a^(k)_mm = ∂²ϕ^k

∂x²₁ +· · ·+∂²ϕ^k

∂x²_m = 0 and, by (1), ϕ^k is a harmonic function for all k= 1, . . . , n.

Corollary3.1. A quadratic mapϕ:R^m→Cⁿgiven by(6)is a pseudo- harmonic morphism if and only if its component matrices satisfy the condi- tions(8).

Proposition 3.2. Let ϕ:R^m →Cⁿ be a quadratic map given by(6). If we writeA_k =B_k+ iC_k, where B_k, C_k are real symmetric matrices, then ϕ is a pseudo-harmonic morphism if and only if for all k, l= 1, . . . , n we have (14) B_k·B_l+B_l·B_k=C_k·C_l+C_l·C_k

and

(15) (B_k·C_l+B_l·C_k) + (C_k·B_l+C_l·B_k) = 0.

Proof. Direct computation in (8).

Remark. If ϕ^k : R^m → C is a component of a quadratic PHWC map ϕ : R^m → Cⁿ with ϕ(x) = (x^tA₁x, . . . , x^tA_nx) and A_k = B_k+iC_k, with B_k, C_k real symmetric matrices, then by (14) and (15) we get the equations B_k²=C_k² and B_k·C_k+C_k·B_k= 0.

By Proposition 5.5.1 in [4], ϕ^k : R^m → C is a horizontally (weakly) conformal map for all k = 1, . . . , n, and so, by Theorem 5.2.3 in [4], ϕ^k is harmonic. This gives another proof of Theorem 3.1 above.

Remark. One can try to classify the quadratic pseudo-harmonic mor- phisms from the Euclidean spaces R^m to complex spaces Cⁿ by using condi- tions (14) and (15), which only contain real symmetric matrices, in a similar way as for quadratic harmonic morphism in [4]. But the computation is too complicate and we shall use conditions (8).

Remark. Let nowϕ:R^m →Cⁿbe a quadratic strongly pseudo-horizontally homothetic morphism. Since gradϕ^k= 2·A_k·x, hence

∂

∂x_j(gradϕ^k) = 2A_k,j

when A_k,j is the j-column of the matrix A_k. By computation, from (5) in Lemma 2.1, we obtain the conditions

(16) A_k·A_l= 0, k, l= 1, . . . , n for a quadratic strongly PHH morphism.

Definition 3.1. An n-tuple (P₁, . . . , P_n) of symmetric endomorphisms P_k:C^m →C^m ofC^m is called a complex Clifford system onC^m, if

(17) P_k◦P_l+P_l◦P_k= 0, k, l= 1, . . . , n.

We have

Proposition3.3. There is a one-to-one correspondence between the set of quadratic pseudo-harmonic morphismsϕ:R^m→Cⁿ and the set of complex Clifford system (P₁, . . . , P_n) on C^m.

Proof. In the canonical basis on C^m, the endomorphismsP₁, . . . , P_n are represented by symmetric matrices A₁, . . . , A_n. From (17) we get conditions (8), thus we have a quadratic pseudo-harmonic morphismϕ:R^m →Cⁿ.

Conversely, any matrixA_kdefines a symmetric endomorphismP_k:C^m→ C^m and from conditions (8) we get (17), hence a complex Clifford system.

Denote V =Cⁿ and consider a quadratic formq :V → Con V (associ- ated with aC-bilinear form not with a Hermitian form). The Clifford algebra of the pair (V, q), denoted by Cl (V, q), is generated by the elements ofV and the unit element 1∈C=⊗⁰V modulo the relations

(18) v·v=−q(v)·1, ∀v∈V.

There exists a natural isomorphism ofC-vector spaces

(19) ΛV →˜ Cl (V, q)

where ΛV is the exterior Grassmann algebra of V. This isomorphism of C- vector spaces is an isomorphism of algebras if and only if the quadratic form q= 0 (see [12]). A representation of the Grassmann algebraG_n= ΛV on the vector spaceC^m is a homomorphism ofC-algebrasρ:G_n→End (C^m).

We have the following interpretation of a quadratic pseudo-harmonic morphism.

Theorem 3.2. Every quadratic pseudo-harmonic morphism ϕ : R^m → Cⁿ defines a representation ρ_ϕ of the Grassmann algebra G_n= ΛV.

Proof. Letϕ:R^m→Cⁿ be given by

ϕ(x) = (x^tA₁x, . . . , x^tA_nx), x∈R^m.

By Proposition 3.3 any matrix A_k, k = 1, . . . , n defines a symmetric endomorphismP_k:C^m →C^m, hence a complex Clifford system (P₁, . . . , P_n).

If{e₁, . . . , e_n}is the canonical basis ofC^m, the Grassmann algebraG_n = ΛV, V = C^m, is generated as C-algebra by e₁, . . . , e_n and 1 ∈ C with the relations

e_k∧e_l+e_l∧e_k= 0, k, l= 1, . . . , n.

Define the representationρ_ϕ on the generators by

ρ_ϕ(e_k) =P_k, k= 1, . . . , n, ρ_ϕ(1) =I_m. Since (P₁, . . . , P_n) is a complex Clifford system, we have

P_k◦P_l+P_l◦P_k= 0, k, l= 1, . . . , n, hence

ρ_ϕ(e_k∧e_l+e_l∧e_k) =P_k◦P_l+P_l◦P_k= 0 and the applicationρ_ϕ is well-defined.

Remark. In the real case, i.e., for quadratic harmonic morphisms, there is a complete classification (see [14], [15]) because we know all the representa- tions of a Clifford algebra. In our (complex) case, i.e., for quadratic pseudo- harmonic morphisms, the description of all representations of a Grassmann algebra is lacking (see [17], for the case G₂). Thus the converse of Theo- rem 3.2 is a difficult problem that is equivalent to the problem of canonical form ofnmatrices A_k,k= 1, . . . , n, satisfying conditions (8).

4. EXAMPLES

Letϕ:R^m →Cⁿbe a quadratic pseudo-harmonic morphism ϕ(x) = (x^tA₁x, . . . , x^tA_nx), x∈R^m,

where the symmetric matricesA_k ∈ Mm(C) satisfy conditions (8).

Let us suppose that all matrices A_k, k = 1, . . . , n, have rank one (rank (A_k) = 1, k= 1, . . . , n). Then the matrixA=A_k has the form

(20) A=







α²₁ α₁α₂ . . . α₁α_m α₁α₂ α²₂ . . . α₂α_m . . . . . . . . . . . . α₁α_m α₂α_m . . . α²_m





,

whereα_t0 = 0 for at least one t₀= 1, . . . , n. We have A² =α·A,

where

α=α²₁+α²₂+· · ·+α²_m. SinceA² = 0 we get

(21) α²₁+α²₂+· · ·+α²_m = 0.

The form (20) is not unique. Indeed, if

A=







˜

α²₁ α˜₁α˜₂ . . . α˜₁α˜_m

˜

α₁α˜₂ α˜²₂ . . . α˜₂α˜_m . . . . . . . . . . . .

˜

α₁α˜_m α˜₂α˜_m . . . α˜²_m





, then we have

α²_k = ˜α²_k and α_kα_l= ˜α_kα˜_l, k, l= 1, . . . , m, hence we get

( ˜α₁, . . . ,α˜_m) = (α₁, . . . , α_m) or

( ˜α₁, . . . ,α˜_m) = (−α₁, . . . ,−α_m).

It follows that the matrices of the form (20) are in one-to-one correspondence with the points of the set

Qˆ =Q\ {0}/{±Id},

whereQ⊂C^m is the hyperquadric given by the equation (22) z₁²+z²₂+· · ·+z_m² = 0.

LetB =A_l,l=k; the matrix B has the same form

B =







β₁² β₁β₂ . . . β₁β_m β₁β₂ β₂² . . . β₂β_m . . . . . . . . . . . . β₁β_m β₂β_m . . . β_m²





. Since

A·B+B·A= 0,

if we denoteC =A·B = (c_tl),D=B·A= (d_tl), we get c_tl =α_tβ_l·

m k=1

α_kβ_k, d_tl =β_tα_l· m k=1

α_kβ_k, c_tl+d_tl= 0, hence

(23) (α_tβ_l+β_tα_l)·^m

k=1

α_kβ_k= 0, t, l= 1, . . . , n.

Since rank (A) = rank (B) = 1, there exist t₀, l₀ such that α_t0 = 0 and β_l0 = 0. Let us suppose that ^m

k=1α_kβ_k= 0. From (23) we obtain (24) α_tβ_t= 0 and α_tβ_l+β_tα_l= 0, t=l.

Ifl₀ =t₀ thenα_t0 ·β_t0 = 0, which contradicts with (24). If l₀=t₀ then from α_t0 ·β_t0 = 0 we getβ_t0 = 0 and from α_l0 ·β_l0 = 0 we get α_l0 = 0. But

α_t0β_l0 +β_t0α_l0 = 0

and we obtainα_t0β_l0 = 0, again a contradiction. It follows that (25)

m k=1

α_k·β_k= 0.

In this particular case, we obtain the complete classification of quadratic pseudo-harmonic morphisms.

Theorem 4.1. Let ϕ:R^m →Cⁿ be a quadratic pseudo-harmonic mor- phism given by

ϕ(x) = (x^tA₁x, . . . , x^tA_nx), x∈R^m.

Suppose that rank (A_k) = 1, k = 1, . . . , n. Then ϕ is completely determined by n points in the set Qˆ = Q\ {0}/{±Id}, such that the coordinates of any representative satisfy condition (25); Q ⊂ C^m is the hyperquadric given by equation (22).

Remark. From the computation above we get by (25) the conditions A_k·A_l= 0, k, l= 1, . . . , n,

hence any quadratic pseudo-harmonic morphismϕ:R^m →Cⁿwith rank (A_k) = 1,k= 1, . . . , n, is in fact a strongly PHH morphism.

Acknowledgements.We want to express our thanks to Professor Vasile Brˆınz˘anescu for his constant help and moral support.

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Received September 28, 2006 University of Pite¸sti Faculty of Educational Sciences

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