Sequential warped products: Curvature and conformal vector fields

Abstract

In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein's equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consider the geometry of two classes of sequential warped product space-time models which are sequential generalized Robertson-Walker spacetimes and sequential standard static spacetimes.

Sequential Warped Products and Their Applications

“…Note that, H f 1 and ∆ 1 f denote the Hessian and Laplacian of f on M 1 respectively, while H h and ∆h denote the Hessian and Laplacian of h on M , respectively. In calculations of the next proposition (and its analogues in Subsections 4.1 and 4.2), there is a difference of one minus sign with the results in paper [11]. The reason for this is that we adhere to the Riemann curvature tensor and contruction rules given in Notations 1-(iii).…”

“…Then, we have: Lemma 2.1. [11] The components of the Levi-Civita connection on ( M , ḡ) are given by:…”

“…These type of metrics also play an important role to find the solutions of the Einstein field equations so they have been previously studied by many authors, such as Dobarro and Ünal [13] and Allison [2]. Moreover, first in [11], these two structures have been studied on the sequential warped product structure and then in [11] and [16] the necessary and sufficient conditions are given for the existence of Einstein and quasi Einstein manifolds on the sequential warped products, respectively.…”

“…Ricci flat Riemannian manifold admitting a complex structure) F and the de Sitter spacetime S n 1 , the warped product S n 1 × F has the same feature, [4]. Thus, a new concept of warped product metric has been introduced in which its base or fiber or both also have the warped product structure and it is called sequential warped product, [20,11]. The formal definition of this notion is given as follows:…”

Gray’s decomposition on doubly warped product manifolds and applications

“…The maps π i : M 1 × M 2 → M i are the natural projections M onto M i whereas * denotes the pull-back operator on tensors. In particular, if for example f 2 = 1, then M = M 1 × f 1 M 2 is called a singly warped product manifold (see [15,35] for doubly warped products and [4,14,16,27,33,34] for singly warped products).…”