PSEUDO-HORIZONTALLY HOMOTHETIC MAPS ON SEMI-RIEMANNIAN MANIFOLDS

ON SEMI-RIEMANNIAN MANIFOLDS

NECULAE DINUT¸ ˘A

We extend the notions of pseudo-harmonic morphism and of pseudo-horizontally homothetic map to the case of semi-Riemannian manifolds, and present some properties of these maps.

AMS 2000 Subject Classification: 53C43, 58E20.

Key words: harmonic map, pseudo-harmonic morphism, pseudo-horizontally con- formal map, pseudo-horizontally homothetic map, semi-Riemannian manifold.

1. INTRODUCTION

The study of harmonic morphisms developed in the last thirty years.

In 1997, Loubeau [6] and Chen [4] introduced and studied independently the notion of pseudo-harmonic morphism defined on a Riemannian manifold with values in a K¨ahler manifold. In 1999-2000, Aprodu, Aprodu and Brˆınz˘anescu [1] introduced and studied the notion of pseudo-horizontally homothetic map which, if it is also a harmonic map, is an important particular case of a pseudo- harmonic morphism with interesting geometric properties.

In this paper we extend the notion of pseudo-horizontally homothetic map to the case of a semi-Riemannian manifold, and study its properties. In Section 2 we present the definition of pseudo-horizontally homothetic maps and give the local description of these maps, when the domain is a semi-Riemannian manifold. In Section 3 we study properties of pseudo-horizontally homothetic submersions from a semi-Riemannian manifold to a K¨ahler manifold.

2. PSEUDO-HORIZONTALLY HOMOTHETIC MAPS

Let (M^m, g_M) be a semi-Riemannian manifold, (N²ⁿ, J, g_N) a K¨ahler manifold and ϕ: (M^m, gM)→ (N²ⁿ, J, gN) a smooth map with the property that for any x∈M, the subspace ker(dϕ_x)⊂T_xM is non-degenerate.

The smooth mapϕ: (M^m, gM)→(N²ⁿ, J, gN) is calledpseudo-horizon- tally(weakly)conformal atx∈M if and only if dϕx◦d^?_ϕ

x commutes withJ_ϕ(x).

MATH. REPORTS10(60),1 (2008), 37–42

The mapϕis called pseudo-horizontally (weakly) conformal, PHWC for short, (cf. [6]) if it is pseudo-horizontally (weakly) conformal at every point of M.

The local description of the PHWC condition is given by the result below (see [6], [3]).

Lemma2.1. Let (xi)i=1,...,m be real local coordinates onM,(ZA)A=1,...,n

complex local coordinates on N and ϕ^A=Z_A◦ϕ for A= 1, . . . , n. Then the PHWC condition for ϕ is equivalent to

m

X

i=1

g^ij_M∂ϕ_A

∂xi

·∂ϕ_B

∂xj

= 0 for all A, B= 1, . . . , n.

The smooth map ϕ : (M^m, g_M) → (N²ⁿ, J, g_N) is said to be pseudo- harmonic morphism (PHM for short) if and only if it pulls back local holo- morphic functions onN to local harmonic maps fromM toC.

In analogy with the Riemannian case (see [6], [3]) we have the following result.

Theorem 2.2. The map ϕis a pseudo-harmonic morphism if and only if ϕ is a harmonic map and a pseudo-horizontally(weakly) conformal map.

Now, we shall introduce a special class of pseudo-horizontally (weakly) conformal maps, calledpseudo-horizontally (weakly) homothetic maps. Let us notice that if X is a (local) section of the pull back bundle ϕ⁻¹T N, then dϕ^?(X) is a (local) horizontal vector field onM. Similarly to [1] we have

Definition2.3. A mapϕ: (M^m, gM)→(N²ⁿ, J, gN) from a semi-Rieman- nian manifold to a K¨ahler manifold, which is PHWC at a pointx∈M, is called pseudo-horizontally homothetic at x if and only if

(2.1) d_ϕx((∇^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 of ϕ(x). If the map ϕ is PHCW, then ϕis called pseudo-horizontally homothetic on M if and only if

(2.2) dϕ(∇^M_Xdϕ^?(J Y)) =Jdϕ(∇^M_Xdϕ^?(Y))

for any horizontal vector field X on M and any vector fieldY an N.

The local description of the pseudo-horizontally homothetic condition at a given point u0 ∈ M for semi-Riemannian manifolds is similar to that for Riemannian manifolds (see [1]). We namely, have

Lemma 2.4. Suppose that the linear tangent map of ϕ is surjective and let u₀ ∈ M be a given point. Let (u_i)_i=1,...,m be local normal coordinates at u₀ such that the vectors _∂u^∂

2n+1(u₀), . . . ,_∂u^∂

m(u₀) are tangent to the fibre

through u0, (ZA)A=1,...,n local normal complex coordinates at ϕ(u0)∈N, and ϕ^A=Z_A◦ϕ. Then ϕ is pseudo-horizontally homothetic atu₀ if and only if (2.3)

m

X

i=1

εi

∂ϕ^A

∂ui

(u0)·∂ϕ^B

∂ui

(u0) = 0 and

(2.4)

m

X

j=1

ε_j ∂²ϕ^A

∂uj∂uk

(u₀)·∂ϕ^B

∂uj

(u₀) = 0 for all A, B= 1, . . . , n, k= 1, . . . ,2n.

Proof. LetZ_A=x_A+iy_A,ϕ^A₁ =x_A◦ϕ,ϕ^A₂ =y_A◦ϕ. The first relation is exactly the PHWC condition at u₀ given by Lemma 2.1.

Remark that dϕ

∂

∂x_A

=X

i

∂ϕ^A₁

∂u_i · ∂

∂u_i and dϕ ∂

∂y_A

=X

i

∂ϕ^A₂

∂u_i · ∂

∂u_i.

Then at u0 we have dϕ

∇^M∂

∂uk

dϕ^? ∂

∂xA

= dϕ ∇^M∂

∂uk

X

i

∂ϕ^A₁

∂ui

· ∂

∂ui

!!

=

=X

i

ε_idϕ

∂²ϕ^A₁

∂ui∂uk

· ∂

∂ui

=X

i,B

ε_i ∂²ϕ^A₁

∂ui∂uk

∂ϕ^B₁

∂ui

· ∂

∂xB

+∂ϕ^B₂

∂ui

· ∂

∂yB

, where εi =±1.

Similarly dϕ

∇^M_∂

∂uk

dϕ^?

J ∂

∂xA

=X

i,B

ε_i ∂²ϕ^A₂

∂ui∂uk

∂ϕ^B₁

∂ui

· ∂

∂xB

+∂ϕ^B₂

∂ui

· ∂

∂yA

.

For given kand A, the condition dϕ

∇^M∂

∂uk

dϕ^?

J ∂

∂xA

=Jdϕ

∇^M∂

∂uk

dϕ^? ∂

∂xA

is equivalent to the relations X

j

εj

∂²ϕ^A₁

∂uj∂u_k ·∂ϕ^B₁

∂uj

− ∂²ϕ^A₂

∂uj∂u_k ·∂ϕ^B₂

∂uj

= 0 and

X

j

εj

∂²ϕ^A₂

∂u_j∂u_k ·∂ϕ^B₁

∂u_j − ∂²ϕ^A₁

∂u_j∂u_k ·∂ϕ^B₂

∂u_j

= 0,

for B= 1, . . . , n, which are equivalent to the relation X

j

εj

∂²ϕ^A₂

∂uj∂u_k(u0)·∂ϕ^B

∂uj

(u0) = 0

for allB = 1, . . . , n. Since the horizontal space throughu₀ is generated by the vectors _∂u^∂

k(u0) for k= 1, . . . ,2n, the conclusion follows.

3. PSEUDO-HORIZONTALLY HOMOTHETIC SUBMERSIONS

In this section we shall prove some properties of pseudo-horizontally ho- mothetic submersions from a semi-Riemannian manifold to a K¨ahler manifold.

Proposition 3.1. Let ϕ: (M^m, g_M)→(N²ⁿ, J, g_N),n≥2, m≥2n, be a horizontally conformal submersion from a semi-Riemannian manifold to a K¨ahler manifold. Thenϕis horizontally homothetic if and only if it is pseudo- horizontally homothetic.

Proof. Let us suppose that ϕ is horizontally homothetic and choose a horizontal vector fieldXonM and an arbitrary vector fieldY onN. By using the property of horizontally conformal submersions of [2], we get the relation

dϕ(∇^M_Xdϕ^?(J Y)) =∇e_X(dϕ◦dϕ^?)(J Y),

where ∇e is the connection on the pull back bundle ϕ⁻¹T N induced by the connection on N.

By using the facts that horizontal conformality implies pseudo-horizontal conformality and N is K¨ahler, we get the relation

dϕ(∇^M_Xdϕ^?(J Y)) =J∇e_X(dϕ◦dϕ^?)(Y), hence ϕis pseudo-horizontally homothetic.

Conversely, suppose thatϕis pseudo-horizontally homothetic and choose a vector field X on N. By using again [2] we have

dϕ(∇^M_dϕ?(X)dϕ^?(J X)) =J∇e_dϕ?(X)(dϕ◦dϕ^?)(X)−

−2λdϕ^?(X)(λ)J X+λg_N(X, X)Jdϕ(gradλ),

whereλis the function from the definition of horizontal conformality (see, for example, [5]). Then,

dϕ^?(J X)(λ)X= dϕ^?(X)(λ)J X −g_N(X, X)Jdϕ(gradλ) which gives

dϕ(gradλ) = 1

g_N(X, X)(dϕ^?(X)(λ)X+ dϕ^?(J X)(λ)J X).

SinceNis K¨ahler andn≥2, we can find another vector fieldY onN such that X, J X,Y, andJ Y are linearly independent. By using the last formula for X and Y, we get dϕ(grad (λ)) = 0, so ϕis horizontally homothetic.

Theorem 3.2. Let ϕ : (M^m, gM) → (N²ⁿ, J, gN), n ≥ 2, m ≥ 2n, be a pseudo-horizontally homothetic submersion. Then ϕ is a harmonic map if and only if it is has minimal fibres.

Proof. We shall use a convenient Gramm-Schmidt method: start with a non-vanishing sectione1inϕ⁻¹T N, then dϕ^?(e1) and dϕ^?(J e1) are orthogonal;

at each step kpicke_k orthogonal on (dϕ◦dϕ^?)(ei) and on (dϕ◦dϕ^?)(J ei) for any i≤k−1. In this way, {dϕ^?(e₁), . . . ,dϕ^?(e_n),dϕ^?(J e₁), . . . ,dϕ^?(J e_n)} is an orthogonal system on ϕ⁻¹T N. DenoteE_k= dϕ^?(e_k) andE_k⁰ = dϕ^?(J e_k).

Now, take an orthonormal basis{u₁, . . . , u_s}for the vertical distribution, where s=m−2n≥0.

By computation, we get

∇e_E⁰

idϕ(E_i⁰) =J∇_E⁰

idϕ(Ei) =J(∇e_Eidϕ(E_i⁰) + dϕ[E_i⁰, Ei]) =

=−∇e_Eidϕ(Ei) +Jdϕ[E_i⁰, Ei], which implies

1

gM(Ei, Ei) ·∇e_Eidϕ(Ei) + 1

gM(E_i⁰, E_i⁰) ·∇e_E⁰

idϕ(E_i⁰) =

= 1

g_M(Ei, Ei)J·dϕ[E_i⁰, E_i⁰], where g_M(Ei, Ei) =±1.

Analogously, we have dϕ(∇^M_E0

iE_i⁰) =−dϕ(∇^M_E

iE_i) +Jdϕ[E_i⁰, E_i], which implies

1

gM(Ei, Ei) ·Jdϕ[E_i⁰, Ei] = 1

gM(Ei, Ei)·dϕ(∇^M_E

iEi) + 1

gM(E_i⁰, E_i⁰)dϕ(∇^M_E0 iE_i⁰).

The formula of the tension torsion yields then τ(ϕ) =−dϕ

X

j

∇^M_u

ju_j

,

Thus,τ(ϕ) = 0 if and only ifP

j

∇^M_u

juj is vertical, i.e., the fibres ofϕare minimal (see [3]).

REFERENCES

[1] M.A. Aprodu, M. Aprodu and V. Brˆınz˘anescu, A class of harmonic submersions and minimal submanifolds.Int. J. Math.11(2000), 1177–1191.

[2] P. Baird and S. Gudmundsson,p-harmonic maps and minimal submanifolds.Math. Ann.

294(1992), 611–624.

[3] P. Baird and J.C. Wood,Harmonic Morphisms between Riemannian Manifolds.Claren- don Press, Oxford, 2003.

[4] J. Chen, Structures of certain harmonic maps into K¨ahler manifolds. Int. J. Math. 8 (1997), 573–581.

[5] N. Dinut¸˘a,Quadratic pseudo-harmonic morphisms. Rev. Roumaine Math. Pures Appl.

52(2007),3, 329–339.

[6] E. Loubeau,Pseudo-harmonic morphisms.Int. J. Math.7(1997), 943–957.

Received 23 October 2007 University of Pite¸sti

Faculty of Educational Sciences Doaga Street

110440 Pite¸sti, Romania rxn dnt@yahoo.com