On submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection

On submanifolds of a Riemannian manifold with a semi- symmetric non-metric connection

CIHAN OÈZGUÈR

Department of Mathematics, Faculty of Arts and Sciences, Balikesir University, 10145, Cagis, Balikesir, Turkey, e-mail:cozgur@balikesir.edu.tr

ABSTRACT

Under investigation were the submanifolds of a Riemannian manifold with a semi- symmetric non-metric connection.We have proved that the induced connection is also a semi-symmetric non-metric connection.The totaly geodesicness and the minimality of a submanifold of a Riemannian manifold with a semi-symmetric non-metric connection were also considered.We have obtained the Gauss, Codazzi and Ricci equations with respect to a semi-symmetric, non-metric connection.The relation between the sectional curvatures of the Levi-Civita connection and the semi-symmetric non-metric connection is also obtained.

Keywords: Semi-symmetric non-metric connection, submanifold.

INTRODUCTION

Hayden (1932) introduced the notion of a semi-symmetric metric connection on a Riemannian manifold.Yano (1970) studied some properties of a Riemannian manifold endowed with a semi-symmetric metric connection.Imai (1972a &

1972b† found some properties of a Riemannian manifold and a hypersurface of a Riemannian manifold with a semi-symmetric metric connection.Nakao (1976) studied submanifolds of a Riemannian manifold with a semi-symmetric metric connection.Agashe and Cha¯e (1992 & 1994) introduced the notion of a semi- symmetric non-metric connection and studied some of its properties and submanifolds of a Riemannian manifold with semi-symmetric non-metric connections.Sengupta, De & Binh (2000) de®ned a new type of semi-symmetric non-metric connection.

In the present paper, we have studied submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection as de®ned in Sengupta, De & Binh (2000).The paper is organized as follows: in Section 2, we have given some properties of the semi-symmetric non-metric connection; in Section 3, some necessary information about a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection has been given and we have

proved that the induced connection is also a semi-symmetric non-metric connection.We have also considered the totaly geodesicness and the minimality of a submanifold of a Riemannian manifold with the semi-symmetric non- metric connection.In Section 4, we have obtained the Gauss, Codazzi and Ricci equations with respect to the semi-symmetric non-metric connection.The relation between the sectional curvatures of the Levi-Civita connection and the semi-symmetric non-metric connection has been also found.

PRELIMINARIES

Let ~M be an …n ‡ d†ÿdimensional Riemannian manifold with a Riemannian metric g;and let ~r be the Levi-Civita connection on ~M: Sengupta, De & Binh (2000) de®ned a linear connection on ~M by

~³

rX~Y ˆ ~~ rX~Y ‡ !… ~~ Y† ~X ÿ g… ~X; ~Y† ~U ‡ g… ~X; ~Y† ~E; …1†

where ~U is a vector ®eld associated with the 1-form ! de®ned by

!… ~X† ˆ g… ~X; ~U† …2†

and ~E is a vector ®eld associated with the 1-form  by

… ~X† ˆ g… ~X; ~E†: …3†

Using (1), the torsion tensor T of ~M with respect to the connection ~r is given by³

T… ~X; ~Y† ˆ ~r³X~Y ÿ ~~ r³Y~X ÿ ~~ 2X; ~Y3

ˆ !… ~Y† ~X ÿ !… ~X† ~Y: …4†

A linear connection ~r satisfying (4) is called a semi-symmetric connection.If ~³ rg 6ˆ 0³ then ~r is called a non-metric connection.Using (1), we have³

~³ rX~g

 

… ~Y; ~Z† ˆ ÿ… ~Y†g… ~X; ~Z† ÿ … ~Z†g… ~X; ~Y†: …5†

Hence the connection ~r is not a metric connection.Because of this reason,³ this connection is called a semi-symmetric non-metric connection (for more details see Sengupta, De & Binh, 2000).

We denote by ~R³ the curvature tensor of ~M with respect to the semi-symmetric non-metric connection ~r: So we have³

R³~… ~X; ~Y† ~Z ˆ ~r³X~r~³Y~Z ÿ ~~ r³Y~r~³X~Z ÿ ~~ r³‰ Š ~X; ~~YZ

ˆ ~R… ~X; ~Y† ~Z ÿ s… ~Y; ~Z† ~X ‡ s… ~X; ~Z† ~Y‡

‡g… ~Y; ~Z† … ~8 X; ~E† ÿ … ~X; ~U†9

ÿ g… ~X; ~Z† … ~8 Y; ~E† ÿ … ~Y; ~U†9

;

…6†

where

R… ~~ X; ~Y† ~Z ˆ ~r_~

Xr~_~

YZ ÿ ~~ r_~

Yr~_~

XZ ÿ ~~ r _~

X; ~Y

‰ ŠZ~

is the curvature tensor of the manifold with respect to the Levi-Civita connection ~r and s is a …0; 2†ÿtensor ®eld de®ned by

s… ~X; ~Y†Z ˆ ~r_~

X!

 

Y ÿ !… ~~ X†!… ~Y† …7†

and

… ~X; ~Y† ˆ ~r_~

XY ‡ !… ~~ Y† ~X ÿ g… ~X; ~Y† ~U ‡ g… ~X; ~Y† ~E; …8†

(see Sengupta, De & Binh, 2000).The Riemannian Christo€el tensors of the connections ~r and ~³ rare de®ned by

R~³… ~X; ~Y; ~Z; ~W† ˆ g… ~R³… ~X; ~Y† ~Z; ~W†

and

R… ~~ X; ~Y; ~Z; ~W† ˆ g… ~R… ~X; ~Y† ~Z; ~W†;

respectively.

SUBMANIFOLDS

Let M be an nÿdimensional submanifold of an …n ‡ d†ÿdimensional Riemannian manifold ~M with the semi-symmetric non-metric connection ~r:³ Decomposing the vector ®elds ~Uand ~E on M uniquely into their tangent and normal components U^T; U^?and E^T; E^?;respectively, we have

U ˆ U~ ^T‡ U^?; …9†

E ˆ E~ ^T‡ E^?: …10†

The Gauss formula for a submanifold M of a Riemannian manifold ~M with respect to the Riemannian connection ~r is given by

r~

XY ˆ rXY ‡ h…X; Y†; …11†

where X; Yare vector ®elds tangent to M; and his the second fundamental form of M in ~M: If h ˆ 0;then M is called totally geodesic. H ˆ¹

ntraceh is called the mean curvature vector of the submanifold.If H ˆ 0 then M is called minimal.

For the second fundamental form h, the covariant derivative of h is de®ned by r

Xh

 

…Y; Z† ˆ r^?_X…h…Y; Z†† ÿ h…rXY; Z† ÿ h…Y; rXZ†

for any vector ®elds X; Y; Z tangent to M: Then rh is a normal bundle valued tensor ®eld of type 0; 3… †and is called the third fundamental form of M: r is called the van der Waerden-Bortolotti connection of M;i.e., ris the connection in TM 8 T^?M built with r and r^?Chen (1973).

Let r³ be the induced connection from the semisymmetric non-metric connection.We de®ne

~³

rXY ˆ r^{3 X}Y ‡ h³…X; Y†: …12†

Equation (12) is the Gauss equation with respect to the semi-symmetric non- metric connection ~r: Hence using (1), (11) and (12) we have³

r³ XY ‡ h³…X; Y† ˆ rXY ‡ h…X; Y† ‡ !…Y†X

ÿg…X; Y†U^Tÿ g…X; Y†U^?‡ g…X; Y†E^T‡ g…X; Y†E^?:

…13†

So comparing the tangential and normal parts of equation (13), we obtain

r³ XY ˆ rXY ‡ !…Y†X ÿ g…X; Y†U^T‡ g…X; Y†E^T …14†

and

h³…X; Y† ˆ h…X; Y† ÿ g…X; Y†U^?‡ g…X; Y†E^?: …15†

If h³ ˆ 0;then M is called totally geodesic with respect to the semi-symmetric non-metric connection (see Agashe & Cha¯e, 1994).

From equation (12), we have

T³…X; Y† ˆ r³ XY ÿ r³ YX ÿ X; Y‰ Š ˆ !…Y†X ÿ !…X†Y; …16†

where T³ is the torsion tensor of Mwith respect to r³and X; Y are vector ®elds tangent to M: Moreover using equation (14), we have

r³ Xg

 

…Y; Z† ˆ r³ Xg…Y; Z† ÿ g…r³ XY; Z† ÿ g…Y; r³ XZ†

ˆ ÿ…Y†g…X; Z† ÿ …Z†g…X; Y†;

…17†

for all vector ®elds X; Y; Z tangent to M:In view of equations (1), (14), (16) and (17), we can state the following theorem:

Theorem 1. The induced connection r³ on a submanifold of a Riemannian manifold admitting the semi-symmetric non-metric connection in the sense of Sengupta, De & Binh (2000) is also a semi-symmetric non-metric connection.

Let Ef 1; E2; :::; Engbe an orthonormal basis of the tangent space of M:We de®ne the mean curvature vector H³ of M with respect to the semi-symmetric non-metric connection r³ by

H³ ˆ1 n

Xⁿ

iˆ1h³…Ei; Ei†;

(see Agashe & Cha¯e, 1994).So from equation (15) we ®nd

H³ ˆ H ÿ U^?‡ E^?:

If H³ ˆ 0 then M is called minimal with respect to the semi-symmetric metric connection (see Agashe & Cha¯e, 1994).

So we have the following result:

Theorem 2. Let M be an nÿdimensional submanifold of an …n ‡ d†ÿdimensional Riemannian manifold ~M with the semi-symmetric non-metric connection ~r in the³ sense of Sengupta, De & Binh (2000).Then

i) M is totally geodesic with respect to the Levi-Civita connection and with respect to the semi-symmetric non-metric connection if and only if the vector ®elds ~U and ~E are tangent to M or U^?ˆ E^?:

ii) The mean curvature normal of M and that of M with respect to the semi- symmetric non-metric connection coincide if and only if the vector ®elds ~U and ~E are tangent to M or U^? ˆ E^?: Hence M is minimal with respect to the Levi-Civita connection and with respect to the semi-symmetric non- metric connection if and only if the vector ®elds ~U and ~E are tangent to M or U^? ˆ E^?:

Let  be a normal vector ®eld on M: From (1), we have

~³

rX ˆ ~r_X ‡ !…†X: …18†

It is well-known that

r~

X ˆ ÿAX ‡ r^?_X; …19†

which is the Weingarten formula for a submanifold of a Riemannian manifold, where Ais the shape operator of M in the direction of : So from (19), equation (18) can be written as

~³

rX ˆ ÿA_X ‡ r^?_X ‡ !…†X: …20†

Now we de®ne a …1; 1†ÿtensor ®eld Aon M by

A³ _ ˆ Aÿ ÿ !…†1

I: …21†

Then equation (20) turns into

~³

r_X ˆ ÿ A³ X ‡ r^?_X: …22†

Equation (22) is the Weingarten's formula with respect to the semi-symmetric non-metric connection ~r: Since A³ is symmetric, it is easy to see that

g A^{3 }X; Y

ˆ g X; A ^{3 }Y

and

g A³ ; A³ 

h i

 X; Y

ˆ g A2 ; A3

ÿ X; Y1

; …23†

where A³ ; A³

h i

ˆ A³ A³ ÿ A³ A³ _ and A2 ; A3

ˆ AAÿ AA and ;  are unit normal vector ®elds on M:

So we can state the following theorem:

Theorem 3. Let M be an nÿdimensional submanifold of an …n ‡ d†ÿdimensional Riemannian manifold ~M with the semi-symmetric non-metric connection ~r in the³ sense of Sengupta, De & Binh (2000).Then the shape operators with respect to the semi-symmetric non-metric connection are simultaneously diagonalizable if and only if the shape operators with respect to the Levi-Civita connection are simultaneously diagonalizable.

By a similar proof of Theorem 3.3 in Agashe & Cha¯e (1994), we have the following theorem:

Theorem 4. Principal directions of the unit normal vector ®eld  with respect to the Levi-Civita connection and the semi-symmetric non-metric connection in the sense of Sengupta, De & Binh (2000) coincide, and the principal curvatures are equal if and only if  is orthogonal to U^?:

GAUSS, CODAZZI AND RICCI EQUATIONS WITH RESPECT TO SEMI-SYMMETRIC NON-METRIC CONNECTION

We denote the curvature tensor of a submanifold M of a Riemannian manifold M with respect to the induced semi-symmetric non-metric connection r~ ³ and the induced Riemannian connection r by

R³…X; Y†Z ˆ r³ Xr³ YZ ÿ r³ Yr³ XZ ÿ r³ ‰X;YŠZ …24†

and

R…X; Y†Z ˆ rXrYZ ÿ rYrXZ ÿ r‰X;YŠZ;

respectively, where X; Y; Z are tangent vector ®elds on M:

From equations (12) and (20) we get

~³

rXr~³YZ ˆ r³ Xr³ YZ ‡ h³…X; r³ YZ† ÿ A

h3 …Y;Z†X

‡r^?_Xh³…Y; Z† ‡ !…³…Y; Z††X; …25†

~³

rYr~³XZ ˆ r³ Yr³

XZ ‡ h³…Y; r³ XZ† ÿ A

h³…X;Z†Y

‡r^?_Yh³…X; Z† ‡ !…³…X; Z††Y

…26†

and

~³

r‰X;YŠZ ˆ r³ ‰X;YŠZ ‡ h³…‰X; YŠ; Z†: …27†

Hence in view of (24), from (25)-(27), we have

R³~…X; Y†Z ˆ R³…X; Y†Z ‡ h³X; r³ YZ

ÿ h³Y; r³ XZ

ÿ h³… X; Y‰ Š; Z†

ÿAh³…Y;Z†X ‡ A

h3

…X;Z†Y ‡ r^?_Xh³…Y; Z† ÿ r^?_Yh³…X; Z†

‡! h³…Y; Z†

X ÿ ! h³…X; Z† Y:

…28†

Since, g…AX; Y† ˆ g…h…X; Y†; †; using (15) we ®nd

R³~…X; Y; Z; W† ˆ R³…X; Y; Z; W† ÿ g…h…Y; Z†; h…X; W†† ‡ g…h…X; Z†; h…Y; W††

‡g…Y; Z† !…h…X; W†† ÿ …h…X; W††‰ Š

‡g…X; Z† …h…Y; W†† ÿ !…h…Y; W††‰ Š

‡!…h…Y; Z††g…X; W† ÿ !…h…X; Z††g…Y; W†

‡! E… † g…Y; Z†g…X; W† ÿ g…X; Z†g…Y; W†^?‰ Š

‡! U… † g…X; Z†g…Y; W† ÿ g…Y; Z†g…X; W†^? ‰ Š;

…29†

where W is a tangent vector ®eld on M:

From (28), the normal component of ~R³…X; Y†Z is given by

h³X; r³ YZ

ÿ h³Y; r³ XZ

ÿ h³… X; Y‰ Š; Z† ‡ r^?_Xh³…Y; Z† ÿ r^?_Yh³…X; Z†

ˆ r³Xh³

…Y; Z† ÿ r³Yh³ …X; Z†

‡! Y… † h³…X; Z† ÿ ! X… † h³…Y; Z† ˆ ~R³…X; Y†Z

 ?

;

…30†

where

³ rXh³

 

…Y; Z† ˆ r^?_Xh³…Y; Z† ÿ h³r³ XY; Z

ÿ h³Y; r³ XZ :

³

r is the connection in TM 8 T built with r³ and r^?: It can be called the van der Waerden-Bortolotti connection with respect to the semi-symmetric non-metric connection.Equation (30) is the equation of Codazzi with respect to the semi- symmetric non-metric connection.

From equations (22) and (12), we get

~³

rXr~³Y ˆ ÿ r^{3 X} A 3 Y

 

ÿ h³ X; A 3 Y

 

ÿ A³ _r?

YX ‡ r^?_Xr^?_Y; …31†

~³

rYr~³X ˆ ÿ r³ Y A 3 X

 

ÿ h³ Y; A 3 X

 

ÿ A³ _r?

XY ‡ r^?_Yr^?_X …32†

and

~³

r‰X;YŠ ˆ ÿ³_‰X; YŠ ‡ r^?_‰X;YŠ: …33†

So using (31)-(33), we have

R³~…X; Y; ; † ˆ R^?…X; Y; ; † ÿ g h³ X; A 3 Y

 

; 

 

‡ g h³ Y; A 3 X

 

; 

 

;

where ;  are unit normal vector ®elds on M: Hence in view of (15) and (21) the last equation turns into

R³~…X; Y; ; † ˆ R^?…X; Y; ; † ÿ g h X; Aÿ Y1

ÿ ; 1

‡ g h Y; Aÿ X1

ÿ ; 1

; which is equivalent to

R³~…X; Y; ; † ˆ R^?…X; Y; ; † ‡ g Aÿ Aÿ AA1

ÿ X; Y1

ˆ R^?…X; Y; ; † ‡ g A2 ; A3

ÿ X; Y1

:

…34†

Equation (34) is the equation of Ricci with respect to the semi-symmetric non-metric connection.

Now assume that ~M is a space of constant curvature c with the semi- symmetric non-metric connection ~r: Then

R³~…X; Y†Z ˆ c g…Y; Z†X ÿ g…X; Z†Y… † ÿ s…Y; Z†X ‡ s…X; Z†Y

‡g…Y; Z† …X; ~ÿ E† ÿ …X; ~U†1 ÿg…X; Z† …Y; ~ÿ E† ÿ …Y; ~U†1 :

…35†

Hence

R³~…X; Y†Z

 ?

ˆ g…Y; Z† …X; ~ E†^?ÿ …X; ~U†^? ÿg…X; Z† …Y; ~ E†^?ÿ …Y; ~U†^?

;

which gives us

R³~…X; Y†Z

 ?

ˆ g…Y; Z† h…X; E8 ^T† ‡ r^?_X

E^?ÿ h…X; U^T† ÿ r^?_XU^?

‡ !…X† ÿ …X†… † U… ^?ÿ E^?†g ÿ g…X; Z† h…Y; E8 ^T† ‡ r^?_Y

E^?ÿ h…Y; U^T† ÿ r^?_YU^?

‡ !…Y† ÿ …Y†… † U… ^?ÿ E^?†g:

So the Ricci equation becomes

³ rXh³

 

…Y; Z† ÿ r³Yh³

…X; Z† ‡ !…Y† h³…X; Z† ÿ !…X† h³…Y; Z†

ˆ g…Y; Z† h…X; E8 ^T† ‡ r^?_X

E^?ÿ h…X; U^T† ÿ r^?_XU^?

‡ !…X† ÿ …X†… † U… ^?ÿ E^?†g ÿg…X; Z† h…Y; E8 ^T† ‡ r^?_Y

E^?ÿ h…Y; U^T† ÿ r^?_YU^?

‡ !…Y† ÿ …Y†… † U… ^?ÿ E^?†g:

Since ~M is a space of constant curvature c with the semi-symmetric non- metric connection, from (35), we have ~R³…X; Y; ; † ˆ 0: Therefore using (34) and (23) we obtain

R^?…X; Y; ; † ˆ g A2 ; A3

ÿ X; Y1

ˆ g Ah³ ;³i

 X; Y :

Hence using (23), we can state the following theorem:

Theorem 5. Let M be an nÿdimensional submanifold of an …n ‡ d†ÿdimensional space of constant curvature ~M…c† with the semi-symmetric non-metric connection ~r³ in the sense of Sengupta, De & Binh (2000).Then the normal connection r^?is ¯at if and only if all second fundamental tensors with respect to the semi-symmetric non- metric connection and the Levi-Civita connection are simultaneously diagonalizable.

Now assume that X and Y are orthogonal unit tangent vector ®elds on M:

Then in view of (29) we can write

R³~…X; Y; Y; X† ˆ R³…X; Y; Y; X† ÿ g…h…Y; Y†; h…X; X††

‡g…h…X; Y†; h…Y; X†† ‡ !…h…X; X†† ÿ …h…X; X††‰ Š

‡!…h…Y; Y†† ‡ ! E… † ÿ ! U^{? … ?}†:

So we get

K~³…† ˆ K³…† ÿ g…h…Y; Y†; h…X; X††

‡g…h…X; Y†; h…Y; X†† ‡ !…h…X; X†† ÿ …h…X; X††‰ Š

‡!…h…Y; Y†† ‡ ! E… † ÿ ! U^{? … ?}†:

…36†

Let M be an nÿdimensional submanifold of an …n ‡ d†ÿdimensional Riemannian manifold ~M with the semi-symmetric non-metric connection ~r³in the sense of Sengupta, De & Binh (2000) and  be a subspace of the tangent space spanned by the orthonormal base X; Yf g: Denote by ~K³…† and K³…† the sectional curvatures of ~M and M at a point p 2 ~M; respectively with respect to the semi-symmetric non-metric connection ~r in the sense of Sengupta, De &³ Binh (2000).Let be a geodesic in ~M which lies in M;and T be a unit tangent vector ®eld of in M: Then from (15) we have

h…T; T† ˆ 0;

h³…T; T† ˆ ÿU^?‡ E^?: …37†

Let  be the subspace of the tangent space spanned by X; T;and ~U and ~E be vector ®elds tangent to M: Then from (37), we have h³…T; T† ˆ 0: Hence using (36), we obtain

K~³…† ˆ K³…† ‡ g…h…X; T†; h…X; T††:

Let X be a unit tangent vector ®eld on M which is parallel along in M:SorTX ˆ 0:

Hence we have the following theorem:

Theorem 6. Let M be an nÿdimensional submanifold of an …n ‡ d†ÿdimensional Riemannian manifold ~M with the semi-symmetric non-metric connection ~r in the³ sense of Sengupta, De & Binh (2000), and be a geodesic in ~M which lies in M;

and T be a unit tangent vector ®eld of in M;  be the subspace of the tangent space spanned by X; T: If the vector ®elds ~U and ~Eare tangent to M; then

i) K³~…†  K³…† along .

ii) If Xis a unit tangent vector ®eld on M which is parallel along in M; then the equality case of (i) holds if and only if Xis parallel along in ~M:

REFERENCES

Agashe, N. S. & Cha¯e, M. R. 1994. On submanifolds of a Riemannian manifold with a semi- symmetric non-metric connection.Tensor (New Series) 55: 120-130.

Agashe, N. S. & Cha¯e, M. R. 1992. A semi-symmetric nonmetric connection on a Riemannian manifold.Indian Journal of Pure and Applied Mathematics 23: 399-409.

Chen, B. Y. 1973. Geometry of submanifolds.Pure and Applied Mathematics No.22.Marcel Dekker, Inc., New York.

Friedmann, A. & Schouten, J. A. 1924. UÈber die Geometrie der halbsymmetrischen UÈbertragungen.

Mathematische Zeitschrift 21: 211-223.

Hayden, H. A. 1932. Subspaces of a space with torsion.Proceedings of the London Mathematical Society 34: 27-50.

Imai, T. 1972a. Notes on semi-symmetric metric connections.Tensor (New Series) 24: 293- 296.

Imai, T. 1972b. Hypersurfaces of a Riemannian manifold with semi-symmetric metric connection.

Tensor (New Series), 23: 300-306.

Nakao, Z. 1976. Submanifolds of a Riemannian manifold with semisymmetric metric connections.

Proceedings American Mathematical Society 54: 261-266.

Sengupta, J., De, U. C. & Binh, T. Q. 2000. On a type of semi-symmetric non-metric connection on a Riemannian manifold.Indian Journal of Pure and Applied Mathematics 31: 1659-1670.

Yano, K. 1970. On semi-symmetric metric connection.Revue Roumaine de MatheÂmatiques Pures et AppliqueÂes 15: 1579-1586.

Submitted : 19/11/2009 Revised : 13/5/2010 Accepted : 31/5/2010

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