Tangent Bundle of Hypersurface with Semi-Symmetric Metric Connection

Transcription

1 Int. J. ontemp. Math. Sciences Vol no Tangent Bundle of Hypersurface with Semi-Symmetric Metric onnection Ayşe ÇİÇEK GÖZÜTOK and Erdoğan ESİN Department of Mathematics Faculty of Science Gazi University Teknikokullar Ankara Turkey agozutok@gazi.edu.tr eresin@gazi.edu.tr Mathematics Subject lassification: 53B25; 5340 Keywords: Hypersurface semi-symmetric metric connection tangent bundle 1 Introduction Friedman A. and Schouten J. A. [1] introduced the idea of a semi-symmetric metric connection on a Riemann manifold. Later Hayden H. A. defined the semi-symmetric metric connection on a Riemann manifold [2]. In [4] Yano K. studied some properties of a semi-symmetric metric connection on a Riemann manifold. Imai T. considered a hypersurface with the semi-symmetric metric connection and obtained the Weingarten Gauss and odazzi-ricci equations with respect to semi-symmetric metric connection [5]. In [7] Nakao Z. expanded to submanifolds the study of Imai T. in [5]. Tani M. [3] developed the theory of hypersurfaces prolonged to tangent bundle with respect to the complete lift of metric tensor of the Riemann manifold. Yücesan A. in [10] studied semi-riemann submanifolds of a semi-riemann manifold with a semisymmetric metric connection. In this study we consider tangent bundle of a hypersurface with semi-symmetric metric connection especially by following [3] and [5]. In Section 2 we shall give the neccessary notions and results which will be used in the next sections. In Section 3 we show that the complete lift of semi-symmetric metric connection on hypersurface is semi-symmetric metric connection in tangent bundle of the hypersurface and we find some certain results concerning the tangent bundle. In the last section we obtain the structure equations with respect to semi-symmetric metric connection of tangent

2 280 A. ÇİÇEK GÖZÜTOK and E. ESİN bundle. 2 Preliminaries Let ˆ be a linear connection in an m dimensional Riemann manifold M. The torsion tensor ˆT of ˆ is given by ˆT ˆXŶ = ˆ ˆXŶ ˆ Ŷ ˆX [ ˆX Ŷ ] 2.1 for any vector fields ˆX and Ŷ in M. The connection ˆ is symmetric if its torsion tensor ˆT vansihes otherwise it is non-symmetric. If there is a Riemann metric ĝ in M Mĝ is called a Riemann manifold. If ĝ is a metric in M such that ˆ ĝ = 0 then the connection ˆ is a metric connection otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if and only if it is the Riemann connection [8]. Let be a metric connection in Mĝ which is non-symmetric. In [4 5] if torsion tensor T of defined by 2.1 satisfies T ˆXŶ =ŵŷ ˆX ŵ ˆXŶ 2.2 for a ŵ I 0 1 M then the connection is called semi-symmetric metric connection in Mĝ. A semi-symmetric metric connection in Mĝ is given by ˆXŶ = ˆ ˆXŶ +ŵŷ ˆX ĝ ˆXŶ 2.3 for arbitrary vector fields ˆX ve Ŷ in Mĝ where ˆ is a Riemann connection in Mĝ and is a vector field defined by ĝ ˆX =ŵ ˆX for any vector field ˆX in Mĝ. The tangent bundle of M is denoted by TM with the projection π M : TM M. I r s M is the space of tensor fields of type r s inm. According to [6] using the complete lift and vertical lift operations we have the following equalities:

3 Tangent bundle of hypersurface 281 [ ˆX Ŷ ] = [X Y ] 2.4 ŵ V ˆX ˆX V = ŵ ŵ ˆX ˆX = ŵ ˆF ˆX = ˆF ˆX ĝ ˆX V Ŷ = ĝ ˆX Ŷ V = ĝ ˆXŶ V ĝ ˆX V Ŷ = ĝ ˆXŶ ˆ ˆXŶ = ˆ ˆXŶ V = ˆT ˆX Ŷ = ˆR ˆX Ŷ Ẑ = ˆ ˆXŶ V ˆ ˆXŶ ˆT ˆX Ŷ ˆR ˆX Ŷ Ẑ for any ˆXŶ Ẑ I1 0 M ŵ I0 1 M ˆF I 1 1 M ĝ I 0 2 M ˆT I 1 2 M and ˆR I 1 3M. Let S be an m 1 dimensional manifold imbedded differentially as a submanifold in Mĝ and denote by ı : S M its imbedding [3 7]. The differential mapping dı is a mapping from TS into TM which is called the tangent map of ı where TS and TM are the tangent bundles of S and M respectively. The tangent map dı is denoted by B. The tangent map of B is denoted by B : T TS T TM. The hypersurface S is also a Riemann manifold with the induced metric g defined by gx Y =ĝbxby for arbitrary X Y I 1 0 S. Thus is a Riemann connection with the induced connection on S g from ˆ defined by [3] ˆ BX BY = B X Y +hx Y N 2.5 for any X Y I 1 0S where N is unit normal vector field on S g and h is the second fundamental tensor field of S g. Also the following equality hx Y =ghxy for any X Y I 1 0S where H I 1 1S. If h vanishes then S is called totally geodesic with respect to and if h is proportional to g then S is called totally umbilical with respect to [3].

4 282 A. ÇİÇEK GÖZÜTOK and E. ESİN 3 Tangent Bundle of Hypersurface with Semi- Symmmetric Metric onnection In [5] is a semi-symmetric metric connection induced on the hypersurface S from which satisfies the equation BX BY = B X Y +mx Y N 3.1 for X Y I 1 0 S where m is a tensor field type of 0 2 in S. Defining M = H ηi we obtain the equality mx Y =gmxy 3.2 for any X Y I 1 0 S where I is the unit tensor field of type 1 1 in S. If m vanishes then S is called totally geodesic with respect to and if m is proportional to g then S is called totally umbilical with respect to. Theorem 1 The connection induced on a hypersurfaces of a Riemann manifold with a semi-symmetric metric connection with respect to the unit normal is also a semi-symmetric metric one [5]. Then we have X Y = X Y + wy X gx Y P for arbitrary X Y I 1 0S. Here P is a vector field in S determined by = BP + ηn where η is a function in S and w is a 1 form in S determined by wx =ŵbx. For the Riemann metric ĝ in M the complete lift ĝ of ĝ is the pseudo-riemann metric in TM. Therefore if we denote the induced metric on TS from ĝ by g then gx Y =ĝ BX BY for arbitrary X Y I 1 0S. Thus the complete lift ˆ of the Riemann connection ˆ in Mĝ is the Riemann connection in the pseudo-riemann manifold TMĝ. Similarly the complete lift of the induced connection on S g is also the Riemann connection in TS g. Theorem 2 If ˆT is torsion tensor of ˆ in Mĝ then ˆT is torsion tensor of ˆ in TMĝ [6].

5 Tangent bundle of hypersurface 283 Now the main theorem of this study follows. Theorem 3 Let be a semi-symmetric metric connection with respect to ˆ Riemann connection in Mĝ. Then is also a semi-symmetric metric connection with respect to ˆ Riemann connection in TMĝ. Proof. Firstly we shall show that ŵ V BX =ŵbx V and ŵ BX = ŵbx. In [3] using 3.10 we get ŵ V BX = ŵ V BX =# ŵ V ˆX ŵ BX = ŵ BX =# ŵ ˆX =# =# ˆX V ŵ =ŵbx V ŵ ˆX =ŵbx for arbitrary X Y I 1 0 S. Here we denote the operation of restriction to π 1 M ıs by #. Also we denote the vertical and complete lift operations on π 1 M ıs by V and respectively. Now taking the complete lift of both sides of the Equation 2.3 and using the Equation 1 we get BX BY = BX BY = ˆ BX BY +ŵby BX ĝ BXBY ˆ BX BY +ŵby BX V +ŵby V BX ĝ BXBY V ĝ BXBY V BX BY = ˆ BX BY +ŵ BY BX V +ŵ V BY BX ĝ BX BY V ĝ BX V BY. Then we have BX BY BY BX [X Y ] = ŵ BY BX V +ŵ V BY BX ŵ BX BY V ŵ V BX BY. Therefore from the Equation 2.1 and Theorem 2 we obtain T BX BY = ŵ BY BX V +ŵ V BY BX ŵ BX BY V ŵ V BX BY. 3.3

6 284 A. ÇİÇEK GÖZÜTOK and E. ESİN By computing ĝ BX BY BZ +ĝ BY BX BZ ˆ BX BY +ŵ BY BX V = ĝ +ŵ V BY BX ĝ BX BY V ĝ BX V BY BZ BY ˆ BX BZ +ŵ BZ BX V +ĝ +ŵ V BZ BX ĝ BX BZ V ĝ BX V BZ = ĝ ˆ BY BX BZ +ĝ BZ ˆ BZ BX = BX ĝ BY BZ we get BX BY ĝ BZ = The Equation 1 and the Equation 3.4 imply the desired result. orollary 4 Let be a semi-symmetric metric connection with respect to Riemann connection in S g. Then is also semi-symmetric metric connection with respect to Riemann connection in TS g. Proof. We have BX BY = ˆ BX BY +ŵby BX ĝ BXBY BX BY = ˆ BX BY +ŵby BX V +ŵby V BX ĝ BXBY V ĝ BXBY V BX BY = ˆ BX BY +ŵ BY BX V +ŵ V BY BX ĝ BX BY V ĝ BX V BY for any X Y I 1 0 S. Hence from the Equation 2.5 and the Equation 3.1 we obtain B X Y + mx Y N = B X Y +hx Y N +ŵ BY BX V +ŵ V BY BX ĝ BX BY BP V + η V N V ĝ BX V BY BP + η V N + η N V

7 Tangent bundle of hypersurface 285 Moreover we get B X Y + m V X Y N + m X Y N V = B X Y + h V X Y N + h X Y N V +ŵ BY BX V +ŵ V BY BX ĝ BX BY BP V η V ĝ BX BY N V ĝ BX V BY BP η V ĝ BX V BY N η ĝ BX V BY N V. B X Y = B X Y +ŵ BY BX V +ŵ V BY BX ĝ BX BY BP V ĝ BX V BY BP 3.5 and m V X Y N + m X Y N V = h V X Y η V ĝ BX V BY N h X Y η V ĝ BX BY η ĝ BX V BY N V. From the Equation 1 it follows that X Y = X Y + w Y X V + w V Y X gx Y P V gx V Y P and finally we obtain X Y = X Y + w Y X V + w V Y X gx Y P V gx V Y P. Thus we have that is X Y Y X [X Y ] = w Y X V + w V Y X w X Y V w V X Y T X Y =w Y X V + w V Y X w X Y V w V X Y. 3.7 Similarly g X Y Z + g Y X Y = X g Y Z

8 286 A. ÇİÇEK GÖZÜTOK and E. ESİN we obtain X g Y Z = The Equation 3.7 and the Equation 3.8 complete. The semi-symmetric metric connection on TS g can be given by X Y = X Y + w Y X V + w V Y X gx Y P V gx V Y P and taking the complete lift of both sides of the Equation 3.1 we obtain BX BY = B From the Equation 1 it follows that X Y + m V X Y N + m X Y N V. m V X Y = h V X Y η V ĝ BX V BY m X Y = h X Y η V ĝ BX BY η ĝ BX V BY. According to [3] TS is totally umbilical if and only if there exist differentiable functions λ and μ such that m V XỸ = λ g XỸ m XỸ = μ g XỸ for any XỸ I1 0TS. If both λ and μ vanish then TS is totally geodesic. It is trivial to prove the following theorems by using the Equation 1. Theorem 5 TS is totally umbilical with respect to the semi-symmetric metric connection if and only if it is totally umbilical or totally geodesic with respect to the Riemann connection. Theorem 6 TS is totally umbilical with respect to the semi-symmetric metric connection if and only if S is totally umbilical with respect to the semisymmetric metric connection. Theorem 7 TS is totally geodesic with respect to the semi-symmetric metric connection if and only if it is totally geodesic with respect to the Riemann connection and the vector field is tangent to S. Theorem 8 TS is totally geodesic with respect to the semi-symmetric metric connection if and only if S is totally geodesic with respect to the semisymmetric metric connection.

9 Tangent bundle of hypersurface The Structure Equations of Tangent Bundle with Semi-symmetric Metric onnection Theorem 9 In [5] the structure equations of S are given by: BX N = BMX g R X Y Z W = ĝ R BXBY BZBW +g MX myz MY mx ZW ĝ R BXBY NBZ = g Y MX X MY + M[X Y ]Z for X Y Z I 1 0S. Theorem 10 If ˆR is the curvature tensor field of the Riemann connection ˆ in Mĝ then the complete lift ˆR of ˆR is the curvature tensor field of the Riemann connection ˆ in TMĝ. Similarly the complete lift R of R is the curvature tensor field of the Riemann connection in TS g where R is the curvature tensor field of the induced connection on S g [6]. Let R be a curvature tensor field of the semi-symmetric connection in Mĝ. Then the curvature tensor field of the semi-symmetric connection is R in TMĝ. Similarly the complete lift R of R is the curvature tensor field of the semi-symmetric metric connection in TS g where R is the curvature tensor field of the induced connection on S g. Theorem 11 The Weingarten equation of TS is obtained as: V BXN = BM V X BX N = BM X for any X I 1 0 S. Proof. Using the Equation 1 Theorem 9 and by virtue of the Section 2 in [3] we get BX N V = BX N V = BMX V = B MX V = BM V X BX N = BX N = BMX = BMX = BM X. Theorem 12 The Gauss equation of TS is obtained as: g R X Y Z W = ĝ R BX BY BZ BW + g M X m V Y Z + M V X m Y Z W g M Y m V X Z + M V Y m X Z W for any X Y Z I 1 0S.

10 288 A. ÇİÇEK GÖZÜTOK and E. ESİN Proof. Using the Equation 1 Theorem 9 and by virtue of the Section 2 and the Equation 6.4 in [3] we get g R X Y Z W = g R X Y Z W ĝ R BXBY BZBW = +g MX myz MY mx ZW ĝ R BXBY BZ BW = + g MX myz V +MX V myz W g MY mx Z V +MY V myz W = ĝ R BX BY BZ BW + g M X m V Y Z + M V X m Y Z W g M Y m V X Z + M V Y m X Z W. Theorem 13 The odazzi-ricci equation of TS is obtained as: R BX BY N V = B Y M V X X M V Y + M V [X Y ] R BX BY N = B Y M X X M Y + M [X Y ] R N V N BX = 0 for any X Y Z I 1 0 S. Proof. Using the Equation 1 Theorem 9 and by virtue of the Section 2 and the Equation 6.4 in [3] we get R BX BY N V = V R BXBY N = B Y MX X MY M[X Y ] V = B Y MX X MY M[X Y ] = B Y M V X X M V Y + M V [X Y ] V

11 Tangent bundle of hypersurface 289 R BX BY N = R BXBY N and = B Y MX X MY M[X Y ] = B Y MX X MY M[X Y ] = B Y M X X M Y + M [X Y ] R N V N BX = R NN BX =0. References [1] Friedman A. Schouten J. A. Über die Geometrie der Halbsymmetrischen Übertragungen Math. Z [2] Hayden H. A. Subspace of a Space with TorsionProceedings of the London Mathematical Society II Series [3] Tani M. Prolongations of Hypersurfaces to Tangent Bundles Kodai Math. Semp. Rep [4] Yano K. On Semi-Symmetric Metric onnections Rev. Roum. Math. Pures et Appl [5] Imai T. Hypersurfaces of A Riemannian Manifold with Semi-Symmetric Metric onnection Tensor N.S [6] Yano K. IshiharaS. Tangent and otangent Bundles Marcel Dekker Inc. NewYork [7] Nakao Z. Submanifolds of a Reimann Manifold with Semisymmetric Metric onnections Proc. Amer. Math. Soc [8] hen Bang-Yen Geometry of Submanifolds Marcel Dekker Inc. NewYork [9] De Leon M Rodrigues P. R. Methods of Differential Geometry in Analytical Mechanics Elsevier Science Publishers B.V [10] Yücesan A. On semi-riemann submanifolds of a semi-riemann manifold with a semi-symmetric metric connection Kuwait J. Sci. Eng. 351A Received: August 2011