PSEUDO-RIEMANNIAN MANIFOLDS

PSEUDO-RIEMANNIAN MANIFOLDS

D. A. CATALANO

Received 1 March 2006; Revised 24 July 2006; Accepted 8 August 2006

We give here a geometric proof of the existence of certain local coordinates on a pseudo- Riemannian manifold admitting a closed conformal vector field.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

A vector fieldVon a pseudo-Riemannian manifold (M,g) is called conformal if

ᏸVg=2λg (1.1)

for a scalar fieldλ, whereᏸdenotes the Lie derivative onM. It is easy to see that ifV is locally a gradient field, then (1.1) is equivalent to

∇XV=λX for every vector fieldX. (1.2) Here∇ denotes the Levi-Civita connection of g. We call vector fields satisfying (1.2) closed conformal vector fields. They appear in the work of Fialkow [3] about conformal geodesics, in the works of Yano [7–11] about concircular geometry in Riemannian man- ifolds, and in the works of Tashiro [6], Kerbrat [4], K¨uhnel and Rademacher [5], and many other authors.

IfVis lightlike on (M,g), then from (1.2), we get

Xg(V,V)=2g_∇XV,V₌2λg(X,V)=0 (1.3) for every vector fieldX. Thusλ≡0 andVis parallel. About lightlike parallel vector fields, we have the following theorem.

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 36545, Pages1–8

DOI 10.1155/IJMMS/2006/36545

Theorem 1.1 (Brinkmann [2]). If (M,g) admits a lightlike parallel vector fieldV, then there are local coordinatesu¹,u²,. . .,uⁿ(n:=dimM >2) such thatV=∂/∂u¹and

gi j

=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

0 1 0 ··· 0

1 0 0 ··· 0

0 0

... ... g_αβ 0 0

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

, (1.4)

whereα,β∈ {3,. . .,n}and∂g_αβ/∂u¹=0.

Brinkmann’s proof is purely analytical. We will give, in the next section, geometric tools which will allow us to generalize Brinkmann’s theorem.

2. Geometric constructions

Let (M,g) be a connected pseudo-Riemannian manifold of dimensionnand signature (k,n−k) with 0< k < n. Given a vector fieldWonM, we denote byWthe one-form defined byW(X)=g(W,X). ThenWis locally a gradient field if and only ifdW=0.

In the following, a vector fieldWsatisfying∇WW=0 will be called geodesic.

Lemma 2.1. IfWis a geodesic vector field, thendWis invariant under the flow ofW.

Proof. Let (∇W)(X,Y)=(∇XW)(Y)=g(∇XW,Y). Then, from the fact thatWis ge- odesic, it follows that

ᏸW∇W(X,Y)=Wg∇XW,Y−g∇[W,X]W,Y−g∇XW, [W,Y]

=gR(W,X)W,Y+g∇XW,∇YW, (2.1) whereRdenotes the Riemannian curvature tensor,

R(X,Y)Z= ∇X∇YZ− ∇Y∇XZ− ∇[X,Y]Z. (2.2) Sinceg(R(W,X)W,Y) is symmetric with respect toX,Y, from

dW(X,Y)=

∇W(X,Y)−

∇W(Y,X), (2.3)

we get (ᏸWdW)(X,Y)=(ᏸW∇W)(X,Y)−(ᏸW∇W)(Y,X)=0.

Lemma 2.2. IfWis a lightlike geodesic vector field, thendW(X,W)=0.

Proof. We have the following.

Wlightlike ⇒(∇W)(X,W)=g(∇XW,W)=0 Wgeodesic ⇒(∇W)(W,X)=g(∇WW,X)=0

⇒dW(X,W)=0.

A nontangent vector fieldW on a pseudo-Riemannian hypersurfaceMcan be ex- tended to a geodesic vector fieldW in a neighbourhood ofMin the following way. Let c(s,p) be the geodesic starting atp=c(0,p)∈Mwith ˙c(0,p)=W(p) and W(c(s,p)) :=

˙

c(s,p). Then, taking into account the fact thatWis transversal (i.e. nontangent) toM, we conclude thatW is a geodesic vector field on a neighbourhood ofMextendingW.

Moreover, ifWis lightlike, then so isW. Denoting withW,W^⊥the tangent and nor- mal component ofW, for vector fields X,Y onMtangent toM, we have the following lemma.

Lemma 2.3. dW(X,Y)=d(W)(X,Y).

Proof. The statement follows fromg(∇XW^⊥,Y)−g(∇YW^⊥,X)= −g(W^⊥, [X,Y])=0.

The following remark will be used in the proof of the next proposition.

Remark 2.4. LetVbe a vector field and letϕbe a function onM. At a pointp0∈M, the gradient of the solutions ofV f=ϕspan an affine hyperplaneHofTp0M. Letv:=V(p0), thenH= {x∈T_p0M|g(x,v)=ϕ(p0)}and

(a) ifϕ(p0)=0, thenHcontains lightlike, spacelike, and timelike vectors,

(b) ifϕ(p0)=0, thenHcontains only lightlike vectors and the zero vector if and only ifn=2 andvis lightlike.

Proposition 2.5. IfV is a closed conformal vector field on (M,g), then in a neighbour- hood of a pointp0whereV(p0)=0, there is a lightlike geodesic gradient fieldWsuch that g(V,W)=1.

Proof. We divide the proof into two cases.

Case 1. n >2 orn=2 andV(p0) is nonlightlike.

Letube a solution ofV u=0 withg(p0)(∇u,∇u)=0 (here∇udenotes the gradient ofu). According toRemark 2.4(b), such a solution exists. Letᐁbe an open neighbour- hood ofp0on whichg(∇u,∇u)=0, and letMbe the pseudo-Riemannian hypersurface u⁻¹(u(p0))∩ᐁ. Then∇uis a normal vector field onMand, fromV u=0, we have that V :=V|Mis a tangent vector field onM. Let f:M→Rbe a solution ofVf=1 such thatg(p0)(∇f,∇f) andg(p0)(∇u,∇u) have opposite sign (seeRemark 2.4(a)). Without loss of generality, we assume thatg(∇f,∇f)=0 onM. Setting W:= ∇f+h∇u, where h²:= −g(∇f,∇f)/g(∇u,∇u)>0, we get

g(W, W) =g(∇f,∇f) +h²g(∇u,∇u)=0, g(V,W) =Vf=1. (2.4) Let nowWbe the geodesic vector field extending W in a neighbourhood ofM. Then Wis lightlike. FromWg(V,W)=g(∇WV,W) +g(V,∇WW)=0 andg(V,W) =1, we conclude thatg(V,W)=1. It remains to show thatWis locally a gradient.

For vector fieldsX,YonM(not necessarily tangent toM), we can write

X=X+αW, Y=Y+βW, (2.5)

whereαandβare certain functions onMandX,Yare tangent toM. UsingLemma 2.2, we get

0=dW(X,W)=dWX+αW,W=dWX,W. (2.6) In the same way, we getdW(W,Y)=0, and thereforedW(X,Y)=dW(X,Y).

NowLemma 2.3andW= ∇f imply thatdW(X,Y)=0 onM. Using Lemma 2.1, we conclude thatdW=0.

Case 2. n=2 andV(p0) is lightlike.

According toRemark 2.4(b), we cannot proceed as inCase 1since the gradient at p0

of a solution ofV u=0 is a lightlike vector. Remarking that along an integral curveα ofV through p0 V is lightlike, we setM:=Imα. Let nowW be a lightlike vector field alongαsuch thatV andW are linearly independent. Then, sinceg is nondegenerate, g(V,V)g(W, W) −g(V,W) ²= −g(V,W) ²=0. Therefore we can assume thatg(V,W) = 1. SinceWis not tangent toα, we can extend it to a geodesic vector fieldWon a neigh- bourhoodᐁof p0. ThenWg(W,W)=0 which, together withW lightlike, impliesW lightlike, andWg(V,W)=g(∇WV,W)=0 which, together withg(V,W) =1, implies g(V,W)=1. Since every vector field onᐁcan be written as a linear combination ofV andW, we haveg(∇XW,Y)−g(∇YW,X)=0 for every vector fieldX,Y onᐁif and only ifg(∇VW,W)−g(∇WW,V)=0.

ThusWbeing lightlike and geodesic implies thatWis a gradient vector field.

It remains to show thatV is lightlike along an integral curveαthrough p0:=α(0).

This follows from (d/dt)g(V,V)=2g(∇VV,V)=2λg(V,V), since its general solution is

g(α(t))(V,V)=g(p0)(V,V)e^20tλ(u)du.

For example, letM=Rⁿ_kbe the pseudo-Euclidian space of dimensionnand signature (k,n−k) with 0< k < n, that is,x,x = −(x²₁+···+x_k²) + (x²_k+1+···+x²_n). The position vector fieldV(x)=_n

i=1xi(∂/∂xi)|xsatisfies∇XV=X, and therefore it is a closed confor- mal vector field. We will construct, following the proof ofProposition 2.5, a lightlike ge- odesic gradient fieldWwithV,W =1 in a neighbourhood of a pointx0=0 (V(x)=0 if and only ifx=0). We take for simplicityx0=(1, 0,. . ., 0), thenu(x1,. . .,x_n) :=x_n/x1is a solution ofV u=0 with∇u,∇u|x0=1. The hypersufaceM:=u⁻¹(u(x0))=u⁻¹(0) is the hyperplanex_n=0. LetV :=V|M, then f(x1,. . .,x_n−1) :=lnx1is a solution ofVf=1 with∇f,∇f|x0= −1. Defining for everyx∈Mthat

W(x) : = ∇f(x) +∇u(x)= 1 x1

− ∂

∂x1+ ∂

∂x_n

x, (2.7)

it is easy to see that

W(x) := 1 x1+xn

− ∂

∂x1

+ ∂

∂xn

x

(2.8)

is a geodesic vector field onMextendingW. Moreover Wis lightlike,V,W =1, and W= ∇ln|x1+xn|. It is clear thatW is not unique and not everywhere defined. More generally, for an arbitrary pointx0=0, we have, for instance, that

W= ∇lna,x, whereais a lightlike vector inRⁿ_kwitha,x0

=0, (2.9)

is a lightlike geodesic gradient field satisfyingV,W =1.

Finally we remark that a nontrivial conformal vector field (a vector fieldVis nontrivial if there is a pointp∈MwithV(p)=0) has isolated zeros (see [4]). This is in general not true if the conformal vector field is not closed (see, e.g., an example in [1]).

3. Local coordinates

LetV andWbe vector fields as inProposition 2.5and letE1=V−g(V,V)W,E2=W.

It is easy to see that

(i)E1,E2are linearly independent;

(ii) the distributionᏰspanned byE1,E2is integrable and the metricgis nondegen- erate onᏰ;

(iii) the distributionᏰ^⊥spanned by the vector fields orthogonal toE1,E2is integrable andgis nondegenerate onᏰ^⊥;

(iv) [E1,E2]=0.

We can now state the following theorem.

Theorem 3.1. If (M,g) admits a closed conformal vector fieldV, then in a neighbourhood of a pointp0whereV(p0)=0, there are local coordinatesu¹,u²,. . .,uⁿsuch thatV=∂/∂u¹+ a(∂/∂u²), for some functiona=a(u²), and

gi j

=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

−a 1 0 ··· 0

1 0 0 ··· 0

0 0

... ... g_αβ

0 0

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

, (3.1)

whereα,β∈ {3,. . .,n}, det(gαβ)=0, and∂gαβ/∂u¹+a(∂gαβ/∂u²)=agαβ(a:=da/du²).

Proof. From Frobenius theorem, we know that there are local coordinatesu¹,u²,. . .,uⁿ such that

∂

∂u^{1 =}E1, ∂

∂u^{2 =}E2, g_1α=g_2α=0, α₌3,. . .,n. (3.2)

Henceg11=g(E1,E1)=g(V,V)−2g(V,V)g(V,W)= −g(V,V),g12=g(V,W)=1,g22= g(W,W)=0 and, settingEi=∂/∂uⁱ,i=1,. . .,n, we have that

∂g_αβ

∂u¹ +a∂g_αβ

∂u^{2 =}g∇E1Eα+g(V,V)∇E2Eα,Eβ

+gE_α,∇E1E_β+g(V,V)∇E2E_β

=g∇EαE1+g(V,V)∇EαE2,Eβ +gEα,∇EβE1+g(V,V)∇EβE2

=g∇Eα

E1+g(V,V)E2

,Eβ

+gEα,∇Eβ

E1+g(V,V)E2

=g∇EαV,E_β+gE_α,∇EβV,=2λg_αβ,

(3.3)

wherea=g(V,V). FromXg(V,V)=2λg(X,V) andg(E1,V)=g(E3,V)=···=g(En,V)= 0, we conclude thata=a(u²). Furthermore

a=Wg(V,V)=2λ (3.4)

anda=0 if and only ifV is lightlike (cf. with Brinkmann’s theorem).

On the other hand, we have the following proposition.

Proposition 3.2. If on a neighbourhoodᐁof a point p0∈M, there are local coordinates as inTheorem 3.1, thenV=∂/∂u¹+a(∂/∂u²) is a closed conformal vector field onᐁ. Proof. The statement follows from

g∇EiV,E_j=g∇EiE1,E_j+aδ_2iδ_1j+ag∇EiE2,E_j

=1 2

∂g_1j

∂uⁱ +∂g_{i j}

∂u^{1 −}

∂g_1i

∂u^j +a∂g_{i j}

∂u²

+aδ_2iδ_1j

=1 2

∂gi j

∂u¹+a∂gi j

∂u²

+1

2aδ1iδ2j+δ2iδ1j ,

(3.5)

where δ is the Kronecker delta. Namely, for every pair (i,j), we get g(∇EiV,Ej)= (1/2)ag_{i j}. Moreover,Vis lightlike if and only ifa=0.

Remark 3.3. If inProposition 3.2we assume thata=0, then according to Fialkow results, see [3, formulas (12.9) and (12.10)], we must be able to prove that (ᐁ,g) is locally iso- metric to a warped product with a one-dimensional base manifold. This can be seen in

the following way: take local coordinatesu¹,. . .,uⁿinᐁsuch that

∂

∂u^{1 =} 1

|a| ∂

∂u¹+a ∂

∂u²

, ∂

∂u^{2 =}

∂

∂u¹, ∂

∂u^α=

∂

∂u^α, α=3,. . .,n. (3.6) This is reached by the coordinate transformation

u¹= |a|

a du², u²=u¹− 1

adu², u^α=u^α, α=3,. . .,n. (3.7) Then it is easy to see thata=a(u¹) and that

g_{i j}:=

g ∂

∂u¯ⁱ, ∂

∂u¯^j

=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝

±1 0 0 ··· 0

0 −a 0 ··· 0

0 0

... ... gαβ

0 0

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠

. (3.8)

Furthermore, from∂g_αβ/∂u¹+a(∂g_αβ/∂u²)=ag_αβ, we get

∂gαβ

∂u^{1 =} 1

|a| ∂gαβ

∂u¹ +a∂gαβ

∂u²

= 1 |a|

da du²gαβ=1

a da

du¹gαβ, (3.9) and thereforeg_αβ=ag_αβ, where∂g_αβ/∂u¹=0. Thus (ᐁ,g) is locally isometric to a warped product with a one-dimensional base manifold and warped factora. In these local coor- dinates, the metric of the fiber manifold is given by

⎛

⎜⎜

⎜⎜

⎝

−1 0 ··· 0 0... g_αβ 0

⎞

⎟⎟

⎟⎟

⎠ (3.10)

which means, in other words, thatu²,. . .,uⁿare Fermi coordinates on the fiber manifold.

Acknowledgment

The author wishes to thank Professor K. Voss of the Swiss Federal Institute of Technology in Zurich for helpful suggestions on the subject.

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D. A. Catalano: Departamento de Matem´atica, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal

E-mail address:[email protected]