α -metrically chebyshev conformal motions in Riemannian spaces

α-METRICALLY CHEBYSHEV CONFORMAL MOTIONS IN RIEMANNIAN SPACES

S. Aynur Uysal1 §, R. Ozlem Laleoglu2 1_{Department of Mathematics}

Dogus University

Kadikoy, Istanbul, 34722, TURKEY e-mail: auysal@dogus.edu.tr 2_{Department of Mathematics} Faculty of Sciences Porto University Porto, PORTUGAL e-mail: rojbinl@gmail.com

Abstract: In this paper we introduced α-metrically Chebyshev nets in a Riemannian manifold with semi-symmetric metric connection. We also studied conformal motions with trajectories defined by the components of these nets in Riemannian spaces with Riemannian connection and with semi-symmetric metric connection.

AMS Subject Classification: 53A30, 53A60, 53B20

Key Words: Riemannian space, Chebyshev net, conformal motion, connec-tion

1. Introduction

A linear semi-symmetric connection on a manifold was defined by Friedmann and Schouten in 1924, [1]. In 1932, Hayden investigated the metric connections with torsion on a Riemannian manifold, [2]. Afterwards in 1970, Yano enlarged the idea of semi-symmetric connection, introduced by Friedmann, Schouten and Hayden, to the semi-symmetric metric connection, by adding the metric to the space and putting the metric condition on the connection, [9].

Received: February 22, 2008 _{2008, Academic Publications Ltd.}c

Infinitesimal conformal motions of geometric objects has been widely stud-ied in Riemannian and semi-Riemannian spaces. In 1998 and 1999 Pujar gave the conditions of being isometric to a sphere for a Riemannian manifold with semi-symmetric metric connection, without putting any condition on the sec-tional curvature tensor, [7], [6].

Chebyshev nets in Riemannian and in generalized spaces are studied by many authors. Definition of Chebyshev nets in Weyl spaces was given by Zla-tanov, in 1988, [10]. Generalized metrically Chebyshev and equidistant nets in Weyl spaces were studied by ¨Ozde˘ger, Uysal and ¨Unal, [5]. In this paper we introduced the notion of α-metrically Chebyshev nets in a Riemannian space with semi-symmetric metric connection and investigated conformal motions de-fined with respect to components of these nets, i.e. we dede-fined the trajectories of the conformal motion as the components of α-metrically Chebyshev nets.

2. α-Metrically Chebyshev Conformal Motions in a Riemannian Space

We consider an n-dimensional (n ≥ 2), Riemannian manifold Rn of class C∞ with the fundamental positive definite metric

ds2 = gijdxidxj

covered by any system of coordinate neighborhoods (xh_{), where h, i, j = 1, 2, ...,} n. Here and throughout the paper we use the summation convention.

Let υ α

i_{(α = 1, 2, ..., n) be the contravariant components of the n-independent} vector fields υ

α in Rn, which are normalized by the condition gijυα i_υ

α

j _{= 1.} Fol-lowing [4], we define the covector fieldsυα by the following equalities

υ α

i α_υ

j = δij, υ_αi βυi= δβα (α, β = 1, 2, ..., n). Let δ = (υ

1, υ2, ..., υn) be a net belonging to Rn and ∇ be the Riemannian con-nection of Rn. If for a fixed α, the condition

∇jυαi− ∇i α

υj = 0 (1)

holds, then the net δ is called α-metrically Chebyshev (α-MC in short) net. If the condition (1) holds for each α (α = 1, 2, ..., n) in this case the net is named strongly-metrically Chebyshev, [3].

defined by,

′_xh_{= x}h_{+ υ}h_{(x)dt, (2)}

where υh _{a contravariant vector field and dt is an infinitesimal. If the} transfor-mation (2) does not change the angle of two directions it is called an infinitesi-mal conforinfinitesi-mal transformation or a conforinfinitesi-mal motion, and the vector field υ is called the conformal killing vector field.

An infinitesimal conformal motion (2) is characterized by

L_υgji = 2∇(jυi) = ∇jυi+ ∇iυj = 2ρgji, (3) where υi = gjiυj_{, and υ}j _{are the components of υ, L}

υ denotes the Lie derivative with respect to υ and ρ is the smooth function on Rn given by

ρ= 1 n∇iυ

i ₌ 1

n(divυ). (4)

A vector field υ, that satisfies (3), defines a homothetic motion or a motion (isometry) if ρ is constant or zero, respectively, [8].

If a vector υh _{satisfies (3), it generates a one-parameter group of conformal} motions.

Now we consider the conformal motions defined with respect to the α. com-ponent of α-MC nets, and give the definition of α-MC conformal motions,

Definition 1. If the trajectory of a conformal motion are the α. com-ponent of an α-MC net δ = (υ

1, υ2, ..., υn), then the conformal motion is called a α-MC conformal motion, and the vector field υ

α is called α-MC conformal killing vector field. υ

α defines a α-MC homothetic motion or a α-MC motion if ρ, which is defined by the equality (4), is constant or zero, respectively.

Then the Lie derivative of the fundamental metric tensor gji with respect to α-MC conformal killing vector fields υ

α is, L_υ α gji= 2∇j α υi = 2ρgji. (5)

Thus we arrive to the following theorems,

Theorem 2. The covariant derivative of an α-MC conformal killing vector field αυj with respect to any coordinate function xi is

∇jυαi = ρgji, (6)

where ρ is the function which arises from the α-MC conformal motion. Proof. The proof follows directly from the equality (5).

Theorem 3. Let υ

α be a α-MC conformal killing vector field in Rn. Then

under the infinitesimal conformal point transformation′_xh _{= x}h_+υ α

h_{dtwe have}

the following assertions;

(i) υ

α defines a motion if and only if ∇j α υi = 0. (ii) If υ

α defines a conformal motion then ρ = 1 ndivυ_α. (iii) υ

α defines a homothetic motion if and only if ∇k∇j α υi = 0.

Proof. (i) Proof follows directly from the equality (5).

(ii) Let us assume that υ

α defines a conformal motion, then the equality ∇j

α

υi= ρgji holds. Contracting this equation with gij we have ρ= 1

ndivυα.

(iii) By taking the covariant derivative of the equation (6) with respect to the coordinate function xk _{we have}

∇k∇jυαi= gji ∇kρ.

Then from the above equality we have ∇kρ = 0, i.e. ρ is constant, if and only if ∇k∇jαυi = 0.

Now we give the well known Green Theorem which is used to prove the some of the proceeding theorem and theorems;

Green Theorem. In a compact orientable Riemannian space Rn, we have Z Rn ∇iυidσ= 0 or equivalently Z Rn divυ dσ = 0

for an arbitrary vector field υh_{(x), where dσ is the volume element dσ = √gdx}1_∧ dx2_{∧ ... ∧ dx}n_.

Theorem 4. In a compact orientable Riemannian space Rn, for a α-MC

conformal killing vector field υ

α following equality holds: Z Rn Rjiυ α j_υ α i − n(n − 1)ρ2 dσ = 0, (7)

where ρ is the smooth function arising from the α-MC conformal motion.

υ α

h _{in Rn and calculate the divergence of the vector field} υ α i (∇iυ α j ) − υ_αj(∇i)υ α i_, then we get ∇j[υ α i_(∇iυ α j_{) − υ} α j_(∇i)υ α i_{] = R} jiυ α j_υ α i_{+ (∇jυ} α i_)(∇iυ α j_{) − (∇iυ} α i₎2_{. (8)} Substituting (4) in (6) we have ∇jυαi = 1 n(∇aυα a_)g ji, (9)

by means of which (8) reduces to ∇j[υ α i (∇iυ α j ) − υ_αj(∇i)υ α i_{] = R} jiυ α j_υ α i −n − 1_n (∇iυ α i₎2_.

Integrating the both sides of the above equality and using the Green The-orem we have Z Rn Rjiυ α j_υ α i −n − 1_n (∇iυ α i₎2_{dσ= 0 . (10)} Finally with the aid of (4), (10) reduces to

Z Rn Rjiυ α j_υ α i − n(n − 1)ρ2dσ= 0. (11)

Corollary 5. In a compact orientable Riemannian space Rn, with the

Ricci curvature Rijυ α

i_υ α

j _{≦ 0, a conformal α-MC conformal killing vector field}

has zero divergence:

∇jυ α j _{= 0,} i.e. υ α is harmonic. In case Rijυα i_υ α

j _{< 0 a α-MC conformal killing vector field}

does not exist other than the zero vector field. Consequently Rndoes not admit

a local one-parameter group of conformal motions.

3. α-Metrically Chebyshev Conformal Motions in a Riemannian Space with Semi-Symmetric Metric Connection

Let M be a n-dimensioanal (n ≥ 2) differentiable manifold of class C∞ _{and D} be a linear connection on M . If the torsion tensor T satisfies

for a one-form π and any vector fields X, Y , then the connection is called semi-symmetric connection. If in addition the covariant derivative of the Riemannian metric tensor g with respect to the connection D vanishes, then D defines a semi-symmetric metric connection [3]. The manifold M , equipped with the semi-symmetric metric connection D will be denoted by (M, D).

The semi-symmetric metric connection D has the components Γh_ji=n h ji o + δh_jπi_{− g}jiπh, (12) wheren h ji o

are the Christoffel symbols of the second type. If Kh

kji, Kji and K are the curvature tensor, Ricci tensor and the scalar curvature of M with respect to D respectively, similar to those with respect to Riemannian connection ∇, then from (12), we have

K_kjih = R_kjih − αjiδh_k+ αkiδ_jh_{− gji}Ah_k+ gkiAh_j (13) where Rh_kji denotes the Riemannian curvature tensor, Ah_j = ghiαji and

αji = ∇jρi− ρjρi+ 1 2gjiρ k_ρ k (14) From (13) we have Kji= Rji− (n − 2)αji− αgji, (15) K _{= R − 2(n − 1)α} (16)

where Rji and R denote the Ricci tensor and the scalar curvature with respect to Riemannian connection and k = gji_{Kji, Kji= K}t

tji and α= gji_α

ji = ∇iρi+n − 1 2 ρkρ

k_{. (17)}

Let the smooth function ρ given by (4), arising from the infinitesimal con-formal transformation and satisfying (2), induce the semi-symmetric metric connection given by (12), i.e. π = ρ. We define the covariant and contravariant components of ρ as ρi _{= ∇}iρ and ρj _{= g}ij_{ρi, [7].}

Lemma 1. For any vector field υ, the following relation is satisfied between the semi-symmetric metric connection D and the Riemannian connection ∇,

D_(jυ_{i)= ∇(j}υ_{i)− υ(j}ρ_i)+ gjiρhυh. (18) Now we define the α-MC nets in an (M, D) manifold.

Let δ = (υ

α υ satisfies Dj α υi− Di α υj = 0, (19)

then δ is called a α-metrically Chebyshev net with respect to the semi-symmetric metric connection D in M . If the equality (19) holds for each α then the net is called strongly-metrically Chebyshev.

In (18), if we specialize the vector field υ as the covariant components of an α-MC net and use the condition of being an α-MC net, we have the following lemma.

Lemma 2. Let δ = (υ

1, υ2, ..., υn) be a α-MC net in (M, D). Then the

following relation between the semi-symmetric metric connection D and the Riemannian connection ∇ holds,

Dj α

υi= ∇(j α

υ_i)−υα_(jρ_i)+ gjiρh αυh. (20) Transvecting the equality (20) with gji _{we obtain the following equality for} the divergence of the function ρ;

div υ

α= nρ + (n − 1)ρ i α_υi. Hence we can state the following lemma;

Lemma 3. If the infinitesimal point transformation ′_xh _{= x}h _{+ υ} α

h_(x),

defined by means of the components of the α-metrically Chebyshev net δ =

(υ

1, υ2, ..., υn), defines a conformal motion then

div υ α= divυα+ (n − 1)Lυ α ρ, (21) where div υ α = g ij_D i α υj, div υ α = nρ and Lυ α ρ= ρi α_υ i.

Theorem 6. Let M be a compact, orientable Riemannian manifold with semi-symmetric metric connection D. Then for the α-MC conformal killing vector field αυ we have the following integral equality;

Z M div υ α dσ= (n − 1) Z MLυα ρ dσ.

Proof. The proof follows from Lemma 3 and the Green Theorem.

Theorem 7. In a compact orientable Riemannian space M , given by the semi-symmetric metric connection D, the following equality holds

Z M Kjiυ α j_υ α i_{+ (Diυ} α j_)(Djυ α i ) − (Diυ α i₎2_{dσ = (n − 1)} Z MLw ρ dσ (22)

for the α-MC conformal killing vector field υ α h_{, where w}j _{= υ} α i_(Diυ α j_)−υ α j_(Diυ α i_).

Proof. Let us begin with considering a vector field υh _{in M , and calculate} the divergence of the vector field

wj = υi(Diυj_{) − υ}j(Diυi), then we get

Dj[υi(Diυj) − υj(Diυi)] = Kijυjυi+ (Diυj)(Djυi) − (Diυi)2,

where Kji’s are the components of the Ricci curvature tensor with respect to the connection D. Integrating both sides over M and using Theorem 2.1, we arrive to the equality

Z M Kjiυ α j_υ α i_{+ (D} iυ α j_)(D jυ α i ) − (Diυ α i₎2_{dσ = (n − 1)}Z MLw ρ dσ .  Corollary 8. Changing the indices j and i in (22) and subtracting it from the former one we derive the integral equality

Z M K_[ji]υ α j_υ α i_{dσ= 0,} which is equivalent to Z M rotρ dσ= 0. References

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