Multiply Warped Products with a Semisymmetric Metric Connection

Abstract

We study the Einstein multiply warped products with a semisymmetric metric connection and the multiply warped products with a semisymmetric metric connection with constant scalar curvature, and we apply our results to generalized Robertson-Walker space-times with a semisymmetric metric connection and generalized Kasner space-times with a semisymmetric metric connection and find some new examples of Einstein manifolds with a semisymmetric metric connection and manifolds with constant scalar curvature with a semisymmetric metric connection.

1. Introduction

The (singly) warped product of two pseudo-Riemannian manifolds and with a smooth function is the product manifold with the metric tensor . Here, is called the base manifold and is called the fiber manifold and is called the warping function. Generalized Robertson-Walker space-times and standard static space-times are two well-known warped product spaces. The concept of warped products was first introduced by Bishop and O’Neill (see [1]) to construct examples of Riemannian manifolds with negative curvature. In Riemannian geometry, warped product manifolds and their generic forms have been used to construct new examples with interesting curvature properties since then. In [2], Dobarro and Dozo had studied from the viewpoint of partial differential equations and variational methods the problem of showing when a Riemannian metric of constant scalar curvature can be produced on product manifolds by a warped product construction. In [3], Ehrlich et al. got explicit solutions to warping function to have a constant scalar curvature for generalized Robertson-Walker space-times. In [4], explicit solutions were also obtained for the warping function to make the space-time as Einstein when the fiber is also Einstein.

One can generalize singly warped products to multiply warped products. Briefly, a multiply warped product is a product manifold of form with the metric , where for each is smooth and is a pseudo-Riemannian manifold. In particular, when with the negative definite metric and is a Riemannian manifold, we call the multiply generalized Robertson-Walker space-time. In [5], Dobarro and Ünal studied Ricci-flat and Einstein-Lorentzian multiply warped products and considered the case of having constant scalar curvature for multiply warped products and applied their results to generalized Kasner space-times.

Singly warped products have a natural generalization. A twisted product is a product manifold of form , with a smooth function , and the metric tensor . In [6], they showed that mixed Ricci-flat twisted products could be expressed as warped products. As a consequence, any Einstein twisted products are warped products. In this paper, we define the multiply twisted products as generalizations of multiply warped products and twisted products. A multiply twisted product is a product manifold of form with the metric , where for each is smooth.

The definition of a semisymmetric metric connection was given by Hayden in [7]. In 1970, Yano [8] considered a semisymmetric metric connection and studied some of its properties. He proved that a Riemannian manifold admitting the semisymmetric metric connection has vanishing curvature tensor if and only if it is conformally flat. Motivated by the Yano result, in [9], Sular and Özgür studied warped product manifolds with a semisymmetric metric connection; they computed curvature of semisymmetric metric connection and considered Einstein warped product manifolds with a semisymmetric metric connection. In this paper, we considered multiply twisted products with a semisymmetric metric connection and computed the curvature of a semisymmetric metric connection. We showed that mixed Ricci-flat multiply twisted products with a semisymmetric metric connection can be expressed as multiply warped products which generalizes the result in [6]. We also studied the Einstein multiply warped products with a semisymmetric metric connection and multiply warped products with a semisymmetric metric connection with constant scalar curvature; we applied our results to generalized Robertson-Walker space-times with a semisymmetric metric connection and generalized Kasner space-times with a semisymmetric metric connection and we found some new examples of Einstein affine manifolds and affine manifolds with constant scalar curvature. We also classified generalized Einstein Robertson-Walker space-times with a semisymmetric metric connection and generalized Einstein Kasner space-times with a semisymmetric metric connection.

Semisymmetric metric connections have some physical applications. In [10, 11], they considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators, and Dirac operators of Chamseddine-Connes type, they computed the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the standard model Lagrangian, they found the Holst term from loop quantum gravity, a coupling of the Holst term to the scalar curvature, and a prediction for the value of the Barbero-Immirzi parameter. For connections whose torsion is not zero, they showed that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on spinor fields. For the physical consequences of the use of torsion connections in Lorentzian geometry, we refer to the classical review [12] and the more recent overview [13] and references therein.

This paper is arranged as follows. In Section 2, we compute curvature of multiply twisted products with a semisymmetric metric connection. In Section 3, we study the special multiply warped products with a semisymmetric metric connection. In Section 4, we study the generalized Robertson-Walker space-times with a semisymmetric metric connection. In Section 5, we consider the generalized Kasner space-times with a semisymmetric metric connection.

2. Preliminaries

Let be a Riemannian manifold with Riemannian metric . A linear connection on a Riemannian manifold is called a semisymmetric connection if the torsion tensor of the connection satisfies where is a one form associated with the vector field on defined by .   is called a semisymmetric metric connection if it satisfies . If is the Levi-Civita connection of , the semisymmetric metric connection is given by (see [8]). Let and be the curvature tensors of and , respectively. Then and are related by for any vector fields on [8]. By (3) and Proposition 2, we have a multiply twisted product which is a product manifold of form with the metric , where for each is smooth.

Here, is called the base manifold and is called the fiber manifold and is called the twisted function. Obviously, twisted products and multiply warped products are the special cases of multiply twisted products.

Proposition 1 (compare with [5, Proposition 2.2]). Let be a multiply twisted product and let and , and . Then(1);(2);(3);(4) if ;(5) if ,where denotes the semisymmetric metric connection on and denote the gradient vector fields on and , respectively.

Proof. Similar to Proposition 2.2 in [5] and Proposition 1 in [6], we have(1);(2);(3) if ;(4) if .So by (3), we get Proposition 1.

Similar to the proof of Proposition 1, we get the following.

Proposition 2 (compare with [5, Proposition 2.2]). Let be a multiply twisted product and let and , and for a fixed . Then(1);(2);(3);(4) if ;(5) if .

Remark 3. When for each is smooth and , we get Proposition 2.2 in [5] by Propositions 1 and 2.

Define the curvature, Ricci curvature, and scalar curvature as follows: where is an orthonormal base of with . The Hessian of is defined by .

Proposition 4 (compare with [5, Proposition 2.4]). Let be a multiply twisted product and let and , , , and . Then(1);(2);(3) if ;(4);(5) if ;(6) if or ;(7), if ;(8) if ;(9) if .

Proof. Similar to Proposition 2.4 in [5], we have(1);(2);(3) if ;(4);(5) if ;(6) if or ;(7), if ;(8) if ;(9) if .Then by (4), we get Proposition 4.

Similarly we may get the following.

Proposition 5 (compare with [5, Proposition 2.4]). Let be a multiply twisted product and let and , , , and for a fixed . Then(1);(2) if ;(3) if ;(4) if ;(5) if ;(6);(7) if ;(8) if or ;(9), if ;(10) if ;(11) if ;(12) if , where denotes the Kronecker symbol.

Remark 6. When for each is smooth and , we get Proposition 2.4 in [5] by Propositions 4 and 5.

By Propositions 4 and 5 and the definition of the Ricci curvature tensor, we have the following.

Proposition 7 (compare with [5, Proposition 2.5]). Let be a multiply twisted product and let and , , and . Then(1)        ;(2);(3) if ;(4)      +          +  if ,where is an orthonormal base of with and .

By and in Proposition 7, similar to the proof of Theorem 1 in [6], we have the following.

Corollary 8. Let be a multiply twisted product and and ; then is mixed Ricci-flat if and only if can be expressed as a multiply warped product. In particular, if is Einstein, then can be expressed as a multiply warped product.

Similar to Proposition 7, we have the following.

Proposition 9 (compare with [5, Proposition 2.5]). Let be a multiply twisted product and let and , , and for a fixed . Then(1)   +  , where for is an orthonormal basis of ;(2);(3);(4) if ;(5) +        if .

Remark 10. When for each is smooth and , we get Proposition 2.5 in [5] by Propositions 7 and 9.

Corollary 11. Let be a multiply twisted product and and ; then is mixed Ricci-flat if and only if can be expressed as a multiply warped product and is only dependent on . In particular, if is Einstein, then can be expressed as a multiply warped product.

Proof. By and in Proposition 9, we have that is mixed Ricci-flat if and only if . By , similar to the proof of Corollary 8, we get that can be expressed as a multiply warped product. When , . When , by , then depends only on .

By Proposition 7 and the definition of the scalar curvature, we have the following.

Proposition 12 (compare with [5, Proposition 2.6]). Let be a multiply twisted product and ; then the scalar curvature has the following expression:

By Proposition 9 and the definition of the scalar curvature, we have the following.

Proposition 13 (compare with [5, Proposition 2.6]). Let be a multiply twisted product and ; then the scalar curvature has the following expression:

Remark 14. When for each is smooth and , we get Proposition 2.6 in [5] by Propositions 12 and 13.

3. Special Multiply Warped Product with a Semisymmetric Connection

Let be a multiply warped product with the metric tensor and is an open interval in and .

Theorem 15. Let be a multiply warped product with the metric tensor and . Then is Einstein with the Einstein constant if and only if the following conditions are satisfied for any :(1) is Einstein with the Einstein constant , ;(2);(3).

Proof. By Proposition 7, we have By (8) and the Einstein condition, we get Theorem 15.

Theorem 16. Let be a multiply warped product with the metric tensor and with and . Then is Einstein with the Einstein constant if and only if the following conditions are satisfied for any :(1) is Einstein with the Einstein constant , ;(2) is a constant and , where are constants;(3), for ;(4).

Proof. By Proposition 9 and , we have that is a constant. By Proposition 9, then By variables separation, we have When , then , so By variables separation, we have that is Einstein with the Einstein constant and When and is a constant, then So we prove the above theorem.

Remark 17. Comparing with Theorem 3.3 in [5], in Theorems 15 and 16, the unit vector field emerges. For Theorem 3.3 in [5], equals zero. So Theorem 3.3 in [5] is not the special case of Theorems 15 and 16. Then equations in our theorems are different from the equations in theorems in [5].

When is a multiply warped product and , by Proposition 12, we have The following result just follows from the method of separation of variables and the fact that each is function defined on .

Proposition 18. Let be a multiply warped product and . If has constant scalar curvature , then each has constant scalar curvature .

When , by Proposition 13, we have

Similarly we have the following.

Proposition 19 (compare with [5, Proposition 3.5]). Let be a multiply warped product and . If has constant scalar curvature , then (1)each has constant scalar curvature ;(2)moreover, if , are also constants, then is a constant.

Remark 20. When in Proposition 19, we get Proposition 3.5 in [5].

4. Generalized Robertson-Walker Space-Times with a Semisymmetric Metric Connection

In this section, we study with the metric tensor . As a corollary of Theorem 15, we obtain the following.

Corollary 21. Let with the metric tensor and . Then is Einstein with the Einstein constant if and only if the following conditions are satisfied: (1) is Einstein with the Einstein constant ;(2);(3).

Remark 22. In Theorem 5.1 in [9], they got the Einstein condition of with a semisymmetric metric connection, but they did not consider conditions (2) and (3).

Corollary 23. Let with the metric tensor and and . Then is Einstein with the Einstein constant if and only if .

By Corollary 21 (2) and (3), we get the following.

Corollary 24. Let with the metric tensor and and . Then is Einstein with the Einstein constant if and only if the following conditions are satisfied: (1) is Einstein with the Einstein constant ;(2);(3).

By Corollary 23 and elementary methods for ordinary differential equations, we get the following.

Theorem 25. Let with the metric tensor and and . Then is Einstein with the Einstein constant if and only if (1), (2), (3).

Let ,  ,  ; then, . When , by Corollary 24, we have the following three cases.

Case (i)   (). We have . By Corollary 24, then When , we get . By (16), , so . In this case . When , then , , and are linearly independent, so and . Then , by , so ; then . Thus .

Case (ii)  (). One has . By Corollary 24, then The coefficient of is , so . The coefficient of is , so ; in this case we have no solutions.

Case (iii)  (). One has , where . By Corollary 24, then Considering the coefficients of and , we get Adding the above two equalities, then and . There is a contradiction with and in this case we have no solutions. So we obtain the following theorem.

Theorem 26. Let with the metric tensor and and . Then is Einstein with the Einstein constant if and only if and and is Einstein with the Einstein constant .

By (14) and (15), we have the following.

Corollary 27. Let with the metric tensor and . Then has constant scalar curvature if and only if has constant scalar curvature and

Corollary 28. Let with the metric tensor and and , where and are constants. Then has constant scalar curvature if and only if has constant scalar curvature and

In (20), we make the change of variable and have the following equation:

Theorem 29. Let with the metric tensor and and . Then has constant scalar curvature if and only if has constant scalar curvature (1) and ,;(2);(3);(4).

Proof. If , then we have a simple differential equation as follows: If , putting , it follows that . The above solutions (1)–(3) follow directly from elementary methods for ordinary differential equations. When , then , and we get solution (4).

Theorem 30. Let with the metric tensor and and and . If has constant scalar curvature if and only if

(1) ,  ;

(2) ;

(3) ,  .

Proof. In this case, (22) is changed into the simpler form Putting , then satisfies the equation ; by the elementary methods for ordinary differential equations we prove the above theorem.

When and , putting , then satisfies the following equation:

5. Generalized Kasner Space-Times with a Semisymmetric Metric Connection

In this section, we consider the scalar and Ricci curvature of generalized Kasner space-times with a semisymmetric metric connection. We recall the definition of generalized Kasner space-times [5].

Definition 31. A generalized Kasner space-time is a Lorentzian multiply warped product of the form with the metric , where is smooth and , for any and also .

We introduce the following parameters and for generalized Kasner space-times. By Theorem 15 and direct computations, we get the following.

Proposition 32. Let be a generalized Kasner space-time and . Then is Einstein with the Einstein constant if and only if the following conditions are satisfied for any :(1) is Einstein with the Einstein constant , ;(2);(3).

By (14), we obtain the following.

Proposition 33. Let be a generalized Kasner space-time and . Then has constant scalar curvature if and only if each has constant scalar curvature and

Next, we first give a classification of four-dimensional generalized Kasner space-times with a semisymmetric metric connection and then consider Ricci tensors and scalar curvatures of them.

Definition 34. Let with the metric . Consider the following:(i) is said to be of type (I) if and ;(ii) is said to be of type (II) if and and ;(iii) is said to be of type (III) if and , and .

By Theorems 26 and 29, we have given a classification of type (I) Einstein spaces and type (I) spaces with the constant scalar curvature.

5.1. Classification of Einstein Type (II) Generalized Kasner Space-Times with a Semisymmetric Metric Connection

Let be an Einstein type (II) generalized Kasner space-time and . Then , . By Theorem 15, we have

where is a constant. Consider the following two cases.

Case (i) (). In this case, ,  . Then by (27a)–(5.2iii), we have

Case (i)(a) (). One has ; by (28a), . By (28b), ; this is a contradiction.

Case (i)(b) (). One has .

Case (i)(b)(1) (). By (28b) and (28c), and then which does not satisfy the first equation in (29); this a contradiction.

Case (i)(b)(2) (). By (28b) and (28c), we have , so is a constant. By (28b), , so ; this is a contradiction. In a word, we have no solutions when .

Case (ii) (). One has . Putting , then . Hence,(1), , (2), , (3), . We make (27a), (5.2ii), (5.2iii) into

When , type (II) spaces turn into type (I) spaces, so we assume . By (30b) and (30c), then

Case (ii)(1).Consider,  , where .

By (31),

Case (ii)(1)(a) (). One has

Case (ii)(1)(a)(1) (). One has and and and . By (30b) and , we get and . But , and this is a contradiction.

Case (ii)(1)(a)(2) (). If , then , and , so and . By (30c), we get and which is a contradiction.

If , by (30b) and , we get or . When , then ; this is a contradiction. There is a similar contradiction for .

Case (ii)(1)(a)(3) (). One has ; by (30a), . By (30b), ; this is a contradiction.

Case (ii)(1)(b) (). One has

Case (ii)(1)(b)(1) (). One has and , and and . By (30b), then and , so and and satisfies (30c). In this case, we get .

Case (ii)(1)(b)(2) (). If , then , and and , so and . By (30c), we get and satisfies (30b) and (30c). In this case, .

If , by (30b) and , then and . By (30c), then and satisfies (30b) and (30c). In this case, ,  .

Case (ii)(1)(c) (). If , then are linearly independent; by (32), then So , and this is a contradiction.

If , then by (32), so and we get a contradiction.

Case (ii)(1)(d) (). When  , we have similar discussions. When , we have . By , then and , so and . But ; then . This is a contradiction.

Case (ii)(2) (,  ). By (31), we have where .

Case (ii)(2)(a) (). One has and . By (30c), , then and ; this is a contradiction with .

Case (ii)(2)(b) (). By (37), we have If , then and . If , then and by (30b), and . By (30c), so and we have a contradiction by (39).

If , then , so and . Then by (30b), we have which contradicts with .

If , then and and and . By and (30b), we have a contradiction. In a word, we have no solutions in case (ii)(2).

Case (ii)(3). One has  ,  , where . By (31), we have If , then and so . This is a contradiction.

If , then Then . This is a contradiction. By the above discussions, we get the following theorem.

Theorem 35. Let be a generalized Kasner space-time and and . Then is Einstein with the Einstein constant if and only if is Einstein with the Einstein constant , and one of the following conditions is satisfied:(1);(2);(3).

5.2. Type (II) Generalized Kasner Space-Times with a Semisymmetric Metric Connection with Constant Scalar Curvature

By Proposition 33, then has constant scalar curvature and If , when , then and . If , then If , putting , we get

5.3. Type (III) Generalized Kasner Space-Times with a Semisymmetric Metric Connection with Constant Scalar Curvature

By Proposition 33, then If , then , and we get .

If , then , so when , there are no solutions, when , is a constant, and when , .

If , then ; putting , then So, we get (1), , (2),  ,(3),  . So we get the following theorem.

Theorem 36. Let be a generalized Kasner space-time and , and . Then is a constant if and only if one of the following cases holds. (1)One has ,  .(2)One has ; when , there are no solutions, when , is a constant, and when , .(3)If (a), , (b), , (c), .

5.4. Einstein Type (III) Generalized Kasner Space-Times with a Semisymmetric Metric Connection

By Proposition 32, we have

If , by (48a), and by (48b), ; this is a contradiction.

If , adding (48b), (48c), and (48d), we get . By (48a), and . But by (48b), then , and this is a contradiction.

Consider . If , we get type (I), so we may let . By (48b) and (48c), we have and , so and or . When , is a constant; by (48a), , this is a contradiction. When , , and , so . In this case, we get when for some ,  . We get the following theorem.

Theorem 37. Let be a generalized Kasner space-time for for some and , and . Then is Einstein with the Einstein constant if and only if .

Remark 38. Comparing with Proposition 4.3, Proposition 4.11, Section 5 in [5], in Proposition 32, Proposition 33, and Theorems 35–37, the unit vector field emerges. For Proposition 4.3, Proposition 4.11, Section 5 in [5], equals zero. So equations in our theorems are different from the equations in theorems in [5].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by NCET-13-0721 and NSFC 11271062. The author would like to thank referees for their careful reading and helpful comments.