Multiply Warped Products with a Semisymmetric Metric Connection

Yong Wang

doi:10.1155/2014/742371

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.

Highlights

The warped product B×bF of two pseudoRiemannian manifolds (B, gB) and (F, gF) with a smooth function b : B → (0, ∞) is the product manifold B × F with the metric tensor g = gB ⊕ b2gF
Let M = B×b1 F1×b2 F2 ⋅ ⋅ ⋅ ×bm Fm be a multiply twisted product and dim Fi > 1 and P ∈ Γ(TB); (M, ∇) is mixed Ricci-flat if and only if M can be expressed as a multiply warped product
Let M = B×b1 F1×b2 F2 ⋅ ⋅ ⋅ ×bm Fm be a multiply twisted product and dim Fi > 1 and P ∈ Γ(TFr); (M, ∇) is mixed Ricci-flat if and only if M can be expressed as a multiply warped product and br is only dependent on Fr

Summary

Introduction

When B = (c, d) with the negative definite metric gB = −dt and (Fi, gFi ) is a Riemannian manifold, we call M the multiply generalized Robertson-Walker space-time. In [5], Dobarro and Unal 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. 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].

Preliminaries

Special Multiply Warped Product with a Semisymmetric Connection

Generalized Kasner Space-Times with a Semisymmetric Metric Connection