A GEOMETRIC CHARACTERIZATION OF FINSLER MANIFOLDS OF CONSTANT CURVATURE K = 1

Vol. 23, No. 6 (2000) 399–407 S0161171200002170

© Hindawi Publishing Corp.

A GEOMETRIC CHARACTERIZATION OF FINSLER MANIFOLDS OF CONSTANT CURVATURE K = 1

A. BEJANCU and H. R. FARRAN (Received 7 December 1998)

Abstract.We prove that a Finsler manifoldF^mis of constant curvatureK=1 if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundle IMofF^m.

Keywords and phrases. Finsler manifold of constant curvature, Killing vector field, indi- catrix bundle, horizontal Liouville vector field.

2000 Mathematics Subject Classification. Primary 53B40; Secondary 53B15.

1. Introduction. The geometry of Finsler manifolds of constant curvature is one of the fundamental subjects in Finsler geometry. Akbar-Zadeh [1] proved that, under some conditions on the growth of the Cartan tensor, a Finsler manifold of constant curvature K is locally Minkowskian if K =0 and Riemannian if K= −1. Recently, Bryant [5] has constructed interesting Finsler metrics of positive constant curvature on the sphere S². Shen [9] has also investigated the geometric structure of Finsler manifolds of positive constant curvature via the RiemannianY-metrics. Some special Finsler metrics of constant curvature have been intensively studied by Matsumoto [7, 8], Shibata-Kitayama [10], and Wei [11].

The purpose of the present paper is to obtain a geometric characterization of Finsler manifolds of positive constant curvature. More precisely, we prove thatF^m=(M,F) is a Finsler manifold of constant curvature K=1 if and only if the unit horizontal Liouville vector fieldξ=(yⁱ/F)δ/δxⁱis a Killing vector field on the indicatrix bundle IMofF^m. To achieve this result, we consider the Sasaki-Finsler metricGonT Mand prove that the linear connection of the Cartan connection onF^mis just the projection of the Levi-Civita connection∇with respect toG on the vertical vector bundle (see Theorem 2.1). This enables us to express the local coefficients of∇in terms of the local coefficients of the Cartan connection ofF^m(see Theorem 2.2). Finally, a necessary and sufficient condition forξto be a Killing vector field onIMleads to the proof of the main result stated in Theorem 3.3.

2. The Levi-Civita connection with respect to a Sasaki-Finsler metric. In the present section, we show that the linear connection of the Cartan connection is the projection of the Levi-Civita connection with respect to the Sasaki-Finsler metric on the vertical vector bundle. Then we express the local coefficients of the Levi-Civita connection in terms of the local coefficients of the Cartan connection.

Throughout the paper we use the Einstein convention, that is, repeated indices with one upper index and one lower index denotes summation over their range. Also, for any smooth manifoldN, we denote byᏲ(N)the algebra of smooth functions onNand byΓ(E)theᏲ(N)-module of smooth sections of a vector bundleE overN. For some Finsler tensor fields we put the indexoto denote the contraction by the supporting elementyⁱ, as for example,Tio=Tijy^j.

LetF^m=(M,F)be a Finsler manifold, whereM is a realm-dimensional smooth manifold andFis the fundamental function ofF^m(see Antonelli-Ingarden-Matsumoto [2, page 36]). ConsiderT M^◦=T M\{0}and denote byV T M^◦the vertical vector bundle overT M^◦, that is,V T M^◦=kerπ∗, whereπ∗is the tangent mapping of the canonical projectionπ:T M^◦ →M. We may think of the Finsler metricg=(gij(x,y)), where we set

gij(x,y)=1 2

∂²F²

∂yⁱ∂y^j (2.1)

as a Riemannian metric on V T M^◦. The canonical nonlinear connection HT M^◦ = (N_i^j(x,y))ofF^mis given by

N_i^j=∂G^j

∂yⁱ, (2.2a)

G^j=1 4g^jh

∂²F²

∂y^h∂x^ky^k−∂F²

∂x^h

. (2.2b)

Then on any coordinate neighborhoodᐁ⊂T M^◦the vector fields δ

δxⁱ = ∂

∂xⁱ−N_i^j ∂

∂y^j, i∈ {1,...,m}, (2.3) form a basis forΓ(HT M_|^◦_ᐁ). By straightforward calculations using (2.3) we obtain the following Lie brackets:

δ δxⁱ, ∂

∂y^j

=G_{i j}^k ∂

∂y^k (2.4)

δ δxⁱ, δ

δx^j

=R^kij ∂

∂y^k, (2.5)

where we set

G_{i j}^k =∂N_i^k

∂y^j (2.6a)

R^kij=δN_i^k δx^j −δN_j^k

δxⁱ. (2.6b)

OnT M^◦we consider the almost product structureQlocally given by

Q ∂

∂yⁱ

= δ

δxⁱ and Q δ

δxⁱ

= ∂

∂yⁱ. (2.7)

Then by means of the pair(g,Q), we define a Riemannian metricGonT M^◦by (cf.

Bejancu [4, page 42])

G(X,Y )=g(vX,vY )+g(QhX,QhY ) ∀X,Y∈Γ(T T M^◦), (2.8) where v and h denote the projection morphisms of T T M^◦ on VT M^◦ and HT M^◦, respectively. Clearly, we have

G δ

δxⁱ, δ δx^j

=G ∂

∂yⁱ, ∂

∂y^j

=gij, G δ

δxⁱ, ∂

∂y^j

=0, (2.9) that is,HT M^◦andV T M^◦are complementary orthogonal vector subbundles ofT T M^◦ with respect to G. As the Riemannian metricG is of Sasaki type and was obtained from a Finsler metric, we call it theSasaki-Finsler metriconT M^◦.

The Levi-Civita connection∇onT M^◦with respect toGis given by the well-known formula

2G(∇XY ,Z)=X G(Y ,Z)

+Y G(X,Z)

−Z G(X,Y ) +G [X,Y ],Z

+G [Z,X],Y

−G [Y ,Z],X

, (2.10)

for anyX,Y ,Z∈Γ(T T M^◦).

On the other hand, the Cartan connection ofF^mis the pairFC=(HT M^◦,∇^◦), where HT M^◦is the canonical nonlinear connection given by (2.2) and∇^◦is a linear connec- tion onVT M^◦whose local coefficientsCik

jandFik

jare given by

∇^◦_∂/∂yj ∂

∂yⁱ=Cik j ∂

∂y^k, (2.11a)

Cik j=1

2g^kh∂ghi

∂y^j (2.11b)

and

∇^◦_δ/δxj ∂

∂yⁱ=Fik j ∂

∂y^k, (2.12a)

F_{i j}^k =1 2g^kh

δghi

δx^j +δghj

δxⁱ −δgij

δx^h

, (2.12b)

respectively. Theh- andv-covariant derivatives of a Finsler tensor field T=(T_i···^j···) are denoted byT_i···|k^j··· andT_i···k^j··· , respectively.

In order to get an interrelation between the Levi-Civita connection∇and the linear connection∇^◦of the Cartan connection we setGj=gjhG^h, and by direct calculations using (2.1) and (2.2b), we deduce that

∂

∂y^k ∂Gi

∂y^j−∂Gj

∂yⁱ

=∂gik

∂x^j −∂gjk

∂xⁱ. (2.13)

Now, we state the following result.

Theorem2.1. The linear connection∇^◦of the Cartan connectionFCis the projec- tion of the Levi-Civita connection∇onV T M^◦, i.e., we have

∇^◦_XY=v∇XY , (2.14)

for anyX∈Γ(T T M^◦)andY ∈Γ(VT M^◦).

Proof. First, we put v∇_∂/∂y^j ∂

∂yⁱ=Aik j ∂

∂y^k and v∇_δ/δx^j ∂

∂yⁱ=Bik j ∂

∂y^k. (2.15) Then in (2.10) we replace(X,Y ,Z)in turn by(∂/∂y^j,∂/∂yⁱ,∂/∂y^k)and(δ/δx^j,∂/∂yⁱ,

∂/∂y^k)and by using (2.1), (2.4), (2.9), and (2.11b), we obtain

A_{i j}^k =C_{i j}^k (2.16)

and

B_{i j}^k =1

2g^khδghi

δx^j +gthGjt

i−gtiGjt h

. (2.17)

Furthermore, by using (2.2a), (2.3) and (2.13), we derive gthGjt

i−gtiGjt

h=gth ∂²G^t

∂yⁱ∂y^j−gti ∂²G^t

∂y^h∂y^j

= ∂

∂y^j ∂Gh

∂yⁱ−∂Gi

∂y^h

−N_i^t∂gth

∂y^j +N_h^t∂gti

∂y^j

= ∂ghj

∂xⁱ −N_i^t∂ghj

∂y^t

− ∂gij

∂x^h−N_h^t∂gij

∂y^t

=δghj

δxⁱ −δgij

δx^h.

(2.18)

Finally, by using (2.18) in (2.17) and taking into account (2.12b) we deduce that Bik

j=Fik

j, which together with (2.16) proves the assertion of the theorem.

Next, in order to get the local coefficients of∇, we consider the local frame field {δ/δxⁱ,∂/∂yⁱ}onT M^◦and set

∇_δ/δx^j δ

δxⁱ=X_{i j}^k ∂

∂y^k+Y_{i j}^k δ

δx^k, (2.19)

∇_∂/∂yj ∂

∂yⁱ=Z_{i j}^k ∂

∂y^k+U_{i j}^k δ

δx^k, (2.20)

∇_δ/δx^j ∂

∂yⁱ=V_{i j}^k ∂

∂y^k+W_{i j}^k δ

δx^k. (2.21)

Taking into account that∇is torsion free and using (2.4) and (2.21), we infer that

∇_∂/∂yi δ δx^j =Vik

j ∂

∂y^k+Wik j δ

δx^k−Gik j ∂

∂y^k. (2.22)

Now, we replace(X,Y ,Z)from (2.10) in turn by(δ/δx^j,δ/δxⁱ,∂/∂y^h)and(δ/δx^j, δ/δxⁱ,δ/δx^h)and using (2.4), (2.5), (2.9), and (2.19), we obtain

X_{i j}^k = −C_{i j}^k −1

2R_{i j}^k Y_{i j}^k =F_{i j}^k. (2.23) Similarly, we replace (X,Y ,Z) from (2.10) in turn by (∂/∂y^j,∂/∂yⁱ,∂/∂y^h) and (∂/∂y^j,∂/∂yⁱ,δ/δx^h)and deduce that

Z_{i j}^k =C_{i j}^k 2ghkU_{i j}^h = −δgij

δx^k+ghjG_{i k}^h +gihG_{j k}^h . (2.24) AsG_{i j}^k given by (2.6a) are the local coefficients of the Berwald connection, we obtain

2ghkU_{i j}^h = −gij;k, (2.25)

wheregij;kis theh-covariant derivative ofgijwith respect to the Berwald connection.

Next, by equation (18.24) in Matsumoto [6], we have

gij;k= −2Cijk|o (2.26) and hence

U_{i j}^k =2C_{i j|o}^k . (2.27)

Finally, replace(X,Y ,Z)from (2.10) in turn by(δ/δx^j,∂/∂yⁱ,∂/∂y^h)and(δ/δx^j,

∂/∂yⁱ,δ/δx^h)and using (2.4), (2.5), (2.9), and Theorem 2.1, we derive that

V_{i j}^k =F_{i j}^k, W_{i j}^k =C_{i j}^k +1

2Rihjg^hk, (2.28)

where we setRihj=gitR^thj. Thus (2.19), (2.20), (2.21), (2.22), and the above calcula- tions enable us to state the following theorem.

Theorem2.2. The Levi-Civita connection∇ on T M^◦ with respect to the Sasaki- Finsler metric G is locally expressed in terms of the local coefficients of the Cartan connection ofF^mas follows:

∇_δ/δx^j δ δxⁱ= −

C_{i j}^k +1

2R^kij

∂

∂y^k+F_{i j}^k δ

δx^k, (2.29)

∇_∂/∂yj ∂

∂yⁱ=C_{i j}^k ∂

∂y^k+2C_{i j|o}^k δ

δx^k, (2.30)

∇_δ/δx^j ∂

∂yⁱ=F_{i j}^k ∂

∂y^k+

C_{i j}^k +1 2Rihjg^hk

δ δx^k

= ∇_∂/∂yi δ

δx^j+G_{i j}^k ∂

∂y^k.

(2.31)

3. The main result. It is well known that, on the tangent bundleT M, there exists a globally defined vector fieldL=yⁱ(∂/∂yⁱ)called theLiouville vector field. By means of the almost product structureQ, we obtain another vector fieldQL=yⁱ(δ/δxⁱ) which we call thehorizontal Liouville vector fieldofF^m. Clearly,ξ=-ⁱ(δ/δxⁱ), where -ⁱ=yⁱ/F is a unit vector field with respect toG. To state the next theorem, we recall that the angular metric ofF^mhas the local components

hij=gij−-i-j; -i=gij-^j= ∂F

∂yⁱ. (3.1)

Also, we recall that the Lie derivative ofG with respect toξ is given by (cf. Yano- Kon [12, page 41])

LξG

(X,Y )=G(∇Xξ,Y )+G(∇Yξ,X) ∀X,Y∈Γ(T T M^◦). (3.2) Now we prove the following theorem.

Theorem3.1. The Lie derivative ofGwith respect toξsatisfies the equations LξG vX,vY

= LξG

(hX,hY )=0, (3.3) LξG hX,vY

=1

F(hij−Rioj)XⁱY^j (3.4) for anyX,Y∈Γ(T T M^◦), wherehX=Xⁱ(δ/δxⁱ)andvY=Yⁱ(∂/∂yⁱ).

Proof. First, by using (2.31), (2.9) and taking into account thatN_i^k=y^jFik j, we obtain

G

∇_∂/∂y^jξ, ∂

∂yⁱ

=-^k

F_{j k}^h −G_{j k}^h ghi

=1 F



N_j^h−y^k∂N_j^h

∂y^k



ghi

=0,

(3.5)

sinceN_j^hare positively homogeneous of degree 1 with respect to(y^k). Next, by using (2.29) and (2.9), we deduce that

G

∇_δ/δx^jξ, δ δxⁱ

=gki-^k|j=0, (3.6)

since-^k|j=0. Taking into account (3.2), we see that (3.5) and (3.6) yield (3.3). Finally, substitutingX andY from (3.2) byδ/δx^j and∂/∂yⁱ, respectively, and using (2.29), (2.31), and (2.9), we infer that

LξG δ δxⁱ, ∂

∂y^j

=-ij−1

FRjoi=1

F(hij−Rioj), (3.7) since by equations (17.30) and (17.21) in Matsumoto [6] we have-ij=(1/F)hij and Rioj=Rjoi. As (3.7) implies (3.4), the proof is complete.

Next, for a fixed pointx∈Mwe consider the indicatrixIxatx, which is a hyper- surface of the fibreT M_x^◦ given by the equationF(x,y)=1. Then denote byIMthe hypersurface ofT M^◦consisting of indicatrices at all points ofMand call it theindica- trix bundleoverF^m. It is easy to show thatQξ=-ⁱ(∂/∂yⁱ)is the unit normal vector field with respect to the Sasaki-Finsler metric. Indeed, if the local equations ofIMin T M^◦are

xⁱ=xⁱ(u^α), yⁱ=yⁱ(u^α), α∈ {1,...,2m−1}, (3.8) then, we have

∂F

∂xⁱ

∂xⁱ

∂u^α+ ∂F

∂yⁱ

∂yⁱ

∂u^α =0. (3.9)

As theh-covariant derivative ofF vanishes, by using (2.3), we obtain

N_i^k∂xⁱ

∂u^α+∂y^k

∂u^α

-k=0. (3.10)

The natural frame field onIMis represented by

∂

∂u^α= ∂xⁱ

∂u^α

∂

∂xⁱ+∂yⁱ

∂u^α

∂

∂yⁱ= ∂xⁱ

∂u^α δ δxⁱ+

N_i^k∂xⁱ

∂u^α+∂y^k

∂u^α ∂

∂y^k. (3.11) Then by (3.10), we deduce

G ∂

∂u^α,Qξ

=

N_i^k∂xⁱ

∂u^α+∂y^k

∂u^α

y^hghk=0. (3.12)

ThusQξ is orthogonal to any vector tangent toIM. The horizontal Liouville vector field is tangent toIMsinceG(ξ,Qξ)=0.

To state the next corollary, we recall thatξis a Killing vector field onIMwith respect toGif and only ifLξG=0 (cf. Yano-Kon [12, page 41]). Thus, by Theorem 3.1, we may state the following corollary.

Corollary3.2. The unit horizontal Liouville vector fieldξis a Killing vector field on the indicatrix bundleIMif and only if

hij(x,y)=Rioj(x,y) ∀(x,y)∈IM. (3.13) Now, we consider a Finsler vector fieldX=Xⁱ(∂/∂yⁱ)which is noncolinear to the Liouville vector fieldL. Then thecurvature (flag curvature)ofF^mfor the flag spanned by{L,X}is the function (see Equation (26.1) in Matsumoto [6] or Bao-Cheen-Shen [3])

K(x,y,X)= RiojXⁱX^j

F²hijXⁱX^j. (3.14)

In caseKis a constant we say thatF^mis a Finsler manifold of constant curvature. The above results enable us to state a geometric characterization of Finsler manifolds of constant curvature by means of the horizontal Liouville vector field, which is the main result of this paper.

Theorem3.3. The Finsler manifoldF^mis of constant curvatureK=1if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundle IM.

Proof. SupposeK=1 and from (3.14) we obtain (3.13), sinceF(x,y)=1 onIM.

Conversely, supposeξis a Killing vector field onIM. Then by using (3.13) in (3.14), we deduce thatK(x,y,X)=1 for any Finsler vectorX(x,y)and any(x,y)∈IM. Now, take a point(x,y)∈T M^◦\IM. Then there existsa∈(0,∞)\{1}such thatF(x,y)=a.

AsFis positive homogeneous of degree 1 with respect toy, we haveF(x,(1/a)y)=1.

Hence(x,(1/a)y)∈IMand by (3.13), we have hij

x,1

ay

=Rioj

x,1

ay

. (3.15)

Taking into account thathijandRiojare positively homogeneous of degrees 0 and 2, respectively, we infer that

Rioj(x,y)=F²(x,y)hij(x,y). (3.16) Thus from (3.14), we deduceK(x,y,X)=1. This completes the proof.

In the above discussions the constant curvature was taken to beK=1 for “normali- sation” purposes only. However the geometric characterization remains valid for any positive constant curvature. To be more precise, we give the following. For any real numberλ >0, we define theλ-indicatrix bundleIλMto be:

IλM=

(x,y)∈T M^◦:F(x,y)= 1

λ

. (3.17)

Then simple modifications in the calculations given earlier will show that the unit hor- izontal Liouville vector fieldξis a Killing vector field onIλMif and only ifhij(x,y)= Rioj(x,y),∀(x,y)∈IλM. So, as before, this can be used to prove the following the- orem.

Theorem3.4. The Finsler manifoldF^m is of constant positive curvatureλ if and only if the unit horizontal Liouville vector field is a Killing vector field onIλM.

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Bejancu: Department of Mathematics and Computer Science, Kuwait University, P.O.

Box5969, Safat13060, Kuwait

E-mail address:[email protected]

Farran: Department of Mathematics and Computer Science, Kuwait University, P.O. Box5969, Safat13060, Kuwait

E-mail address:[email protected]