On Generalized Douglas-Weyl Spaces

On Generalized Douglas-Weyl Spaces

A. Tayebi, H. Sadeghi and E. Peyghan November 29, 2011

Abstract

In this paper, we show that the class of R-quadratic Finsler spaces is a proper subset of the class of generalized Douglas-Weyl spaces. Then we prove that all generalized Douglas-Weyl spaces with vanishing Landsberg curvature have vanishing the non-Riemannian quantityH, generalizing result previously only known in the case of R-quadratic metric. Also, this yields an extension of well-known Numata’s Theorem.

Keywords: Landsberg metric, R-quadratic metric,H-curvature.¹

1 Introduction

In Finsler geometry, there are several well-known projective invariants of Finsler metrics namely, Douglas curvature, Weyl curvature, and another projective in- variant which is due to Akbar-Zadeh [2][14][15]. Douglas curvature is a non- Riemannian projective invariant constructed from the Berwald curvature [8].

The notion of Douglas curvature was proposed by B´acs´o and Matsumoto as a generalization of Berwald curvature [4]. The Douglas curvature vanishes for Riemannian spaces, therefore it is plays a role only outside the Riemannian world [9]. Finsler metrics withDⁱ_jkl= 0 are calledDouglas metrics and Finsler metrics with W_kⁱ = 0 are called Weyl metrics. There is another projective in- variant in Finsler geometry, namely Dⁱ_jkl|my^m =T_jklyⁱ that is hold for some tensorTjkl, whereDⁱ_jkl|mdenotes the horizontal covariant derivatives ofDⁱ_jkl with respect to the Berwald connection of F. This equation is equivalent to that for any linearly parallel vector fields u = u(t), v = v(t) and w = w(t) along a geodesic c(t), there is a function T = T(t) such that _dt^d

Dc˙(u, v, w)

=Tc.˙ The geometric meaning of this identity is that the rate of change of the Douglas curvature along a geodesic is tangent to the geodesic [12].

For a manifold M, letGDW(M) denote the class of all Finsler metrics satis- fying in above relation for some tensorT_jkl (T_jkl not fixed). In [6], B´acs´o-Papp show that GDW(M) is closed under projective changes.

A natural question is: how large isGDW(M) and what kind of interesting metrics does it have? It is obvious that all Douglas metrics belong to this class.

On the other hand, all Weyl metrics (metrics of scalar flag curvature) also belong to this class. The later is really a surprising result, due to Sakaguchi [17].

12000 Mathematics subject Classification: 53C60, 53C25.

In this paper, we show that the class of generalized Douglas-Weyl metrics contains the class of R-quadratic metrics as a special case.

Theorem 1.1. Every R-quadratic Finsler metric is a generalized Douglas-Weyl metric.

A Finsler metric is said to be R-quadratic if its Riemann curvature Ry is quadratic iny∈TxM [5]. The notion of R-quadratic metric was introduced by Shen [18]. There are many non-Riemann R-quadratic Finsler metrics. For ex- ample, all Berwald metrics are R-quadratic. Some non-Berwaldian R-quadratic Finsler metrics have been constructed in [10]. Thus R-quadratic Finsler metrics form a rich class of Finsler spaces.

In [2], Akbar-Zadeh considered a non-Riemannian quantityHwhich is ob- tained from the mean Berwald curvature by the covariant horizontal differen- tiation along geodesics. In the class of Weyl metrics, vanishing this quantity reasults that the Finsler metric is of constant flag curvature and this fact clarifies its geometric meaning [2][13].

Recently Li-Shen prove that every R-quadratic Randers metric has constant non-Riemannian invariantS-curvature, hence it has vanishing non-Riemannian invariantH[10]. Then Mo extend their result and show that every R-quadratic Finsler metric has vanishing H-curvature [11]. In this paper, we get an exten- sion of these results and prove that every generalized Douglas-Weyl space with vanishing Landsberg curvature satisfiesH= 0.

Theorem 1.2. Let (M, F) be a generalized Douglas-Weyl space. Suppose that F is a Landsberg metric. Then H= 0.

According to Theorem 1.2, every Landsberg metricF of scalar flag curvature K satisfies H = 0 and then F is of constant flag curvature. By Akbar-Zadeh Theorem, the Cartan tensor ofF satisfies ¨A_ijk+KA_ijk = 0 [1]. SinceF is a Landsberg metric, then KA_ijk= 0. If we suppose that F is of non-zero scalar flag curvature, then F is Riemannian. Therefore, we get the following.

Corollary 1.1. Every Landsberg metric of non-zero scalar flag curvature is Riemannian.

The Corollary 1.1, was proved by Numata [16]. Then Theorem 1.2 can be regarded as a generalization of the Numata Theorem.

The converse of Theorem 1.2 is not true. For example, consider following Finsler metric on the unit ball Bⁿ⊂Rⁿ,

F(y) :=

p|y|²−(|x|²|y|²−< x, y >²)

1− |x|² +< x, y >

1− |x|², y∈T_xBⁿ=Rⁿ where|.|and<, >denote the Euclidean norm and inner product inRⁿ, respec- tively. F is called theFunk metric which is a Randers metric onBⁿ [19]. Funk metric is a generalized Douglas-Weyl metric satisfiesH= 0 whileL6= 0.

There are many connections in Finsler geometry [20][21]. In this paper, we set the Berwald connection on Finsler manifolds. The h- and v- covariant derivatives of a Finsler tensor field are denoted by “|” and “, ” respectively.

2 Preliminaries

Let M be a n-dimensional C^∞ manifold. Denote by TxM the tangent space at x ∈ M, by T M = ∪_x∈MTxM the tangent bundle of M and by T M0 = T M \ {0} the slit tangent bundle of M. A Finsler metric onM is a function F :T M →[0,∞) which has the following properties: (i)F isC^∞ onT M0; (ii) F is positively 1-homogeneous on the fibers of tangent bundleT M, and (iii) for eachy∈TxM, the following quadratic form gy onTxM is positive definite,

gy(u, v) := 1 2

∂²

∂s∂t

F²(y+su+tv)

|s,t=0, u, v∈TxM.

Let x ∈ M and F_x := F|_TxM. To measure the non-Euclidean feature ofF_x, defineC_y:T_xM⊗T_xM ⊗T_xM →Rby

C_y(u, v, w) := 1 2

d

dt[g_y+tw(u, v)]|_t=0, u, v, w∈T_xM.

The familyC:={Cy}_{y∈T M0} is called the Cartan torsion. It is well known that C=0 if and only ifF is Riemannian.

Given a Finsler manifold (M, F), then a global vector fieldGis induced by F onT M0, which in a standard coordinate (xⁱ, yⁱ) forT M0 is given by

G=yⁱ ∂

∂xⁱ −2Gⁱ(x, y) ∂

∂yⁱ.

where Gⁱ(x, y) := ¹₄g^il(x, y){[F²]_xky^ly^k −[F]²_xl}. G is called the associated spray to (M, F). The projection of an integral curve ofGis called a geodesic in M. In local coordinates, a curvec(t) is a geodesic if and only if its coordinates (cⁱ(t)) satisfy ¨cⁱ+ 2Gⁱ( ˙c) = 0.

Fory ∈TxM0, define By : TxM ⊗TxM ⊗TxM →TxM andEy : TxM ⊗ T_xM →Rby

B_y(u, v, w) :=Bⁱ_jkl(y)u^jv^kw^l ∂

∂xⁱ|_x, E_y(u, v) :=E_ij(y)uⁱv^j,

where Bⁱ_jkl(y) := _∂yj^∂_∂y^3G_kⁱ_∂yl(y),Eij(y) := ¹₂B^m_ijm(y), u=u^{i ∂}_∂xi|x,v =v^{i ∂}_∂xi|x

andw=w^{i ∂}_∂xi|x. BandEare called the Berwald curvature and mean Berwald curvature respectively. A Finsler metric is called a Berwald metric and weakly Berwald metric ifB= 0 andE= 0, respectively [19][22].

DefineB˜y:TxM⊗TxM⊗TxM →TxM andHy:TxM ⊗TxM →Rby B˜y(u, v, w) := ˜Bⁱ_jkl(y)u^jv^kw^l ∂

∂xⁱ|x, Hy(u, v) :=Hij(y)uⁱv^j,

where ˜Bⁱ_jkl :=Bⁱ_jkl|sy^s and Hij :=E_ij|sy^s. Then B˜y and Hy are defined as the covariant derivative ofBandEalong geodesics, respectively [13].

DefineDy :TxM⊗TxM⊗TxM →TxMbyDy(u, v, w) :=Dⁱ_jkl(y)uⁱv^jw^{k ∂}_∂xi|x

where

Dⁱ_jkl:=Bⁱ_jkl− 2

n+ 1{Ejkδⁱ_l+Ejlδ_kⁱ +Eklδⁱ_j+Ejk,lyⁱ}.

We callD:={D_y}_{y∈T M0} the Douglas curvature. A Finsler metric withD= 0 is called a Douglas metric. The notion of Douglas metrics was proposed by B´acs´o-Matsumoto as a generalization of Berwald metrics [4].

DefineLy:TxM⊗TxM⊗TxM →RandL˜y:TxM⊗TxM⊗TxM →Rby L_y(u, v, w) :=L_ijk(y)uⁱv^jw^k andL˜_y(u, v, w) := ˜L_ijk(y)uⁱv^jw^k where

Lijk:=C_ijk|sy^s and ˜Lijk :=L_ijk|sy^s.

The family L :={Ly}y∈T M₀ is called the Landsberg curvature. F is called a Landsberg metric ifL= 0 [23].

Theorem 2.1. ([3]) For a Douglas metricF on a manifoldM, ifL= 0, then B= 0.

By using the notion of Landsberg curvature, we defineΣy :TxM⊗TxM⊗ T_xM ⊗T_xM →RbyΣ_y(u, v, w, z) := Σ_ijkl(y)uⁱv^jw^kz^l where

Σ_ijkl := 2(L_ijk|l−L_ijl|k).

A Finsler metric is said to be stretch metric ifΣ= 0. In [7], L. Berwald showed that stretch curvature vanishes if and only if the length of a vector remains unchanged under the parallel displacement along an infinitesimal parallelogram.

3 Proof of Theorem 1.1

The notion of Riemann curvature for Riemann metrics can be extended to Finsler metrics. For a vector y ∈TxM0, the Riemann curvature Ry :TxM → TxM is defined byRy(u) :=Rⁱ_k(y)u^{k ∂}_∂xi, where

Rⁱ_k(y) = 2∂Gⁱ

∂x^k − ∂²Gⁱ

∂x^j∂y^ky^j+ 2G^j ∂²Gⁱ

∂y^j∂y^k −∂Gⁱ

∂y^j

∂G^j

∂y^k.

The family R := {Ry}y∈T M0 is called the Riemann curvature [19]. A Finsler metricF is said to be R-quadratic ifR_y is quadratic in y∈T_xM at each point x∈M. Let

Rⁱ_jkl(x, y) :=1 3

∂

∂y^j ∂Rⁱ_k

∂y^l −∂Rⁱ_l

∂y^k ,

whereRⁱ_jklis the Riemann curvature of Berwald connection. We have Rⁱ_k =Rⁱ_jkl(x, y)y^jy^l.

ThenRⁱ_k is quadratic iny∈TxM if and only if Rⁱ_jkl are functions of position alone.

In this section, we prove that every R-quadratic Finsler metric is a general- ized Douglas-Weyl metric. To prove this, we need the following.

Lemma 3.1.

Rⁱ_jkl|m+Rⁱ_jlm|k+Rⁱ_jmk|l=Bⁱ_jkuR^u_lm+Bⁱ_jluR^u_km+Bⁱ_kluR^u_jm,(1)

Bⁱ_jkl|m−Bⁱ_jmk|l=Rⁱ_jml,k, (2)

Bⁱ_jkl,m=Bⁱ_jkm,l. (3)

Proof. The curvature form of Berwald connection is Ωⁱ_j=dωⁱ_j−ω^k_j∧ωⁱ_k =1

2Rⁱ_jklω^k∧ω^l−Bⁱ_jklω^k∧ω^n+l. (4) For the Berwald connection, we have the following structure equation

dgij−gjkΩ^k_i−gikΩ^k_j=−2Lijkω^k+ 2Cijkω^n+k. (5) Differentiating (5) yields the following Ricci identity

gpjΩ^p_i−gpiΩ^p_j =− 2L_ijk|lω^k∧ω^l−2Lijk,lω^k∧ω^n+l−2C_ijl|kω^k∧ω^n+l

− 2Cijl,kω^n+k∧ω^n+l−2CijpΩ^p_ly^l. (6) Differentiating of (4) yields

dΩ_i^j−ω_i^k∧Ω_k^j+ω_k^j∧Ω_i^k= 0. (7) DefineBⁱ_jkl|m andBⁱ_jkl,m by

dB_jklⁱ −B_mklⁱ ω_i^m−Bⁱ_jmlω^m_k −B_jkmⁱ ω^m_l +B_jklⁱ ω_mⁱ =B_jkl|mⁱ ω^m+B_jkl,mⁱ ω^n+m. (8) Similarly, we defineRⁱ_jkl|mandRⁱ_jkl,m by

dRⁱ_jkl−Rⁱ_mklω_i^m−Bⁱ_jmlω^m_k −Rⁱ_jkmω^m_l +Rⁱ_jklω_mⁱ =Rⁱ_jkl|mω^m+R_jkl,mⁱ ω^n+m. (9) From (6), (7), (8) and (9), one obtains the above Bianchi identity.

Proposition 3.1. Every R-quadratic Finsler metric is a generalized Douglas- Weyl metric.

Proof.

Dⁱ_jkl=Bⁱ_jkl− 2

n+ 1{Ejkδⁱ_l+Eklδⁱ_j+Eljδⁱ_k+Ejk,lyⁱ}. (10) Then

D_jkl|mⁱ y^m=Bⁱ_jkl|my^m− 2

n+ 1{Hjkδⁱ_l+Hklδⁱ_j+Hljδ_kⁱ +E_jk,l|my^myⁱ}. (11) By (2), it follows that

Bⁱ_jkl|my^m=Rⁱ_jml,ky^m, (12)

which yields

Hjk=R^p_jmp,ky^m. (13)

We obtain

hⁱ_αD_jkl|m^α y^m=hⁱ_αR^α_jml,ky^m− 2

n+ 1{R^p_jmp,khⁱ_l+R_lmp,j^p hⁱ_k+R_kmp,l^p hⁱ_j}y^m. (14) F is R-quadratic, then we have

hⁱ_αD^α_jkl|my^m= 0.

It means that F is a generalized Douglas-Weyl metric.

The following examples shows that there is a generalized Douglas-Weyl met- ric which is not R-quadratic.

Example 1. LetX = (x, y, z)∈B³(1)⊂R³andY = (u, v, w)∈TxB³(1). Put A:= (x²+y²+z²)u−2x(xu+yv+zw),

B:= 1−(x²+y²+z²)², C:=u²+v²+w². DefineF =F(x, y) by

F:=α+β =

√A²+BC

B +A

B.

The flag curvature ofF is given by K= −3u

F +x²−2y²−2z².

It means thatF is of scalar flag curvature and thenF is a generalized Douglas- Weyl metric onB³(1). It is easy to show thatF is not R-quadratic metric.

4 Proof of Theorem 1.2

In these section, we will prove a generalized version of Theorem 1.2. Indeed, we study compact generalized Douglas-Weyl spaces with vanishing stretch curva- ture and prove the following.

Theorem 4.1. Every compact generalized Douglas-Weyl space with vanishing stretch curvature satisfiesH= 0.

The most elegant importance of studying Finsler metrics, is to obtain non- Riemannian PDEs in the sence that they hold trivially for Riemannian metrics.

To prove Theorem 4.1, we find a PDE on mean Berwald curvature of generalized Douglas-Weyl metrics with vanishing stretch tensor. For this reason, we need the following:

Lemma 4.2. Let (M, F)be a generalized Douglas-Weyl space. Then B˜^l_ijk|h=−2y^l

F²

L˜_ijk|h+ 2

n+ 1{H_ij|hh^l_k+H_jk|hh^l_i+H_ik|hh^l_j}. (15)

Proof.

Dⁱ_jkl=Bⁱ_jkl− 2

n+ 1{Ejkδⁱ_l+Eklδⁱ_j+Eljδⁱ_k+Ejk,lyⁱ}. (16) Then

h^m_i Dⁱ_jkl|sy^s=h^m_i Bⁱ_jkl|sy^s− 2

n+ 1{Hjkh^m_l +H_klh^m_j +H_ljh^m_k }. (17)

By assumption we get

h^m_i B˜ⁱ_jkl= 2

n+ 1{Hjkh^m_l +H_klh^m_j +H_ljh^m_k}. (18) Taking a horizontal derivative of (18) yields

h^m_i B˜ⁱ_jkl|h= 2

n+ 1{H_jk|hh^m_l +H_kl|hh^m_j +H_lj|hh^m_k }. (19) Using

g_ipy^pBⁱ_jkl=−2Ljkl, (20) one can yields

h^m_i B˜ⁱ_jkl|h = (h^m_i Bⁱ_jkl)_|s|hy^s

= (B^m_jkl+2y^m

F² Ljkl)_|s|hy^s

= B˜^m_jkl|h+2y^m F²

L˜_jkl|h. (21)

By (19) and (21) we obtain (15).

Lemma 4.3. Let(M, F)be a generalized Douglas-Weyl space. Suppose thatF is a stretch metric. Then for any geodesicc(t)and any parallel vector fieldU(t) along c, the following function

E(t) =Ec˙(U(t), U(t)), (22) satisfying in the following equation

E(t) =H(0)t+E(0). (23) Proof. SinceF is a stretch metric, then we haveL_ijk|l=L_ijl|k. Contracting it withy^l yields ˜Lijk= 0. By considering Lemma 4.2, we have

B˜^l_ijk|h−B˜^l_ijh|k = 2

n+ 1{(H_jk|h−H_jh|k)h^l_i+ (H_ik|h−H_ih|k)h^l_j}

+ 2

n+ 1{H_ij|hh^l_k−H_ij|kh^l_h}. (24) Puttingj=l in (24), we get

H_ik|h−H_ih|k = 2

n+ 1{H_ik|h−H_ih|k}, (25) which yields

H_ik|h=H_ih|k. (26)

Contacting (26) with y^h

H_ik|hy^h= 0. (27)

Let

H(t) =H_c˙(U(t), U(t)). (28)

From the definition of H_y, we have

H(t) =E⁰(t). (29)

By (27) we have H⁰(t) = 0 which implies that H(t) =H(0).

Then by (29), we get the equation (23).

Remark 4.1. Let (M, F) be a Finsler space andc: [a, b]→M be a geodesic.

For a parallel vector fieldV(t) alongc, we haveg_c˙(V(t), V(t)) =constant.

Proof of Theorem 4.1: Take an arbitrary unit vector y ∈ T_xM and an arbitrary vectorv∈TxM. Letc(t) be the geodesic with ˙c(0) =y andV(t) the parallel vector field alongcwithV(0) =v. DefineE(t) and H(t) as in (22) and (28), respectively. Then by Lemma 4.3, we haveE(t) =tH(0) +E(t). Suppose that Ey is bounded, i.e., there is a constantN <∞such that

||E||x:= sup

y∈TxM0

sup

v∈TxM

Ey(v, v)

[g_y(v, v)]³² ≤N . (30) By Remark 4.1, we know thatT :=g_c˙(V(t), V(t)) = constant is positive con- stant. Thus

|E(t)| ≤N T³² <∞, andE(t) is a bounded function on [0,∞). This implies

Hy(v, v) =H(0) = 0.

ThereforeH= 0.

By the Theorem 4.1, every compact generalized Douglas-Weyl space with vanishing Landsberg curvature satisfiesH= 0. By a similar way, it follows that every compact Douglas space with vanishing stretch curvature satisfy inH= 0.

Proof of Theorem 1.2: By (18) we have h^m_i B˜ⁱ_jkl= 2

n+ 1{H_jkh^m_l +H_klh^m_j +H_ljh^m_k}. (31) Using (20) we get

h^m_i B˜ⁱ_jkl= ˜B^m_jkl+ 2 F²

L˜_jkly^m. (32) From assumption and the relations (31) and (32), we obtain

B˜^m_jkl= 2

n+ 1{H_jkh^m_l +H_klh^m_j +H_ljh^m_k }. (33) By puttingm=kin (33), we conclude thatH= 0.

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Akbar Tayebi

Faculty of Science, Department of Mathematics University of Qom

Qom, Iran

Email: [email protected] Hassan Sadeghi

Department of Mathematics, Faculty of Science University of Qom

Qom. Iran

Email: [email protected] Esmaeil Peyghan

Faculty of Science, Department of Mathematics Arak University

Arak 38156-8-8349, Iran Email: [email protected]