Analysis of the Quasi-Concircular Curvature Tensor on Sequential Warped Product Manifolds

Abstract

This paper investigates the quasi-concircular curvature tensor on sequential warped product manifolds, which extend the classical singly warped product structure. We examine various curvature conditions associated with this tensor, including quasi-concircular flatness, quasi-concircular symmetry, and the divergence-free quasi-concircular condition, and we explore the properties of related soliton structures. In addition, we analyze the implications of these results in Lorentzian geometry by deriving explicit expressions for the Ricci tensor and scalar curvature of the considered manifolds. The study concludes with an illustrative example that emphasizes the geometric significance and potential applications of the investigated structures.

1. Introduction

In differential geometry, constructing new manifolds with prescribed curvature properties is a fundamental technique for analyzing various geometric and physical phenomena. One such construction, the warped product manifold, was introduced by Bishop and O’Neill [1]. This structure provides a systematic method for generating manifolds with specific curvature conditions by modifying the metric of a product manifold using a smooth function known as the warping function. Formally, given two Riemannian (or semi-Riemannian) manifolds $({\mathcal{M}}_1,g_1)$ and $({\mathcal{M}}_2,g_2)$, their warped product is defined as the product manifold $\mathcal{M}={\mathcal{M}}_1\times {\mathcal{M}}_2$ equipped with the metric $g=g_1\oplus f^2{g}_2$, where $f:{\mathcal{M}}_1\to (0,\infty )$ is a smooth function called the warping function. This function controls the stretching of the fibers and allows the geometric properties of $\mathcal{M}$ to be influenced by those of ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$. Warped product structures are particularly significant in general relativity, where they provide models of spacetimes that yield exact solutions to Einstein’s field equations, as discussed in [2,3,4,5].

A well-known example of warped products in general relativity is given in [6]:

• Schwarzschild spacetime. This spacetime describes the gravitational field around a spherically symmetric mass. The metric is given by the Schwarzschild solution:

  $$d{s}^2=-\left(1-{\displaystyle \frac{2m}{r}}\right)d{t}^2+{\left(1-{\displaystyle \frac{2m}{r}}\right)}^{-1}d{r}^2+r^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right).$$
  This metric represents a Lorentzian warped product of the form $W=P{\times }_r{\mathbb{S}}^2$, where $P=\{(t,r)\in \mathbb{R}\times {\mathbb{R}}^{+}\mid r>2m\}$ and ${\mathbb{S}}^2$ denotes the standard unit 2-sphere.
• de Sitter spacetime. A model of an expanding universe is described by the metric

  $$d{s}^2=-d{t}^2+r^2{\cosh }^2\left({\displaystyle \frac{t}{r}}\right)\left(d{\alpha }^2+{\sin }^2\alpha (d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\right).$$
  This can be rewritten as a Lorentzian warped product of the form

  $$W=\mathbb{R}{\times }_f{\mathbb{S}}^3,f\left(t\right)=r\cosh \left({\displaystyle \frac{t}{r}}\right).$$

These warped products serve as fundamental models in physics and cosmology, illustrating how the warping function influences the geometric and physical properties of spacetime.

While singly warped products modify only the fiber component, sequential warped products introduce additional complexity by allowing both the base and the fiber (or multiple structural layers) to be warped. This generalization is particularly useful in scenarios where the underlying space consists of multiple interacting geometric layers. Güler [6] analyzed sequential warped products together with generalized quasi-Einstein conditions on sequential warped product spacetimes. In [7], the pseudo-projective tensor on sequential warped products is examined, and necessary and sufficient conditions are derived for such a manifold to be pseudo-projectively flat on generalized quasi-Einstein manifolds in the sense of Catino [8]. Apostolopoulos and Carot [2] classified the geometry of two classes of sequential warped product spacetimes, namely, the sequential generalized Robertson–Walker spacetimes and the sequential standard static spacetimes.

The significance of the quasi-concircular curvature tensor has been extensively investigated in both pure mathematics and mathematical physics. Narrain et al. [9] studied the quasi-concircular curvature tensor on LP-Sasakian manifolds, while Ahmad et al. [10] considered it in the context of Lorentzian $\beta$-Kenmotsu manifolds. The aim of this paper is to extend these investigations by analyzing the quasi-concircular curvature tensor on sequential warped product manifolds. The motivation for considering the quasi-concircular curvature tensor arises from its ability to refine the classical concircular tensor by capturing subtler geometric properties. While the concircular tensor focuses on the preservation of geodesic circles under concircular transformations, the quasi-concircular tensor generalizes this framework, offering greater flexibility. This flexibility makes it particularly useful in studying manifolds of physical interest, such as warped product manifolds, general relativity spacetimes, and modified gravity models, that are not perfectly conformally, concircularly, or projectively flat. These structures exhibit intermediate curvature properties that the Riemann, conformal, and projective curvature tensors fail to fully capture. The quasi-concircular curvature tensor introduces real parameters, allowing for a continuous family of curvature conditions that interpolate between conformal and concircular geometries. This makes it an essential tool in general relativity and cosmology, where warped structures often model complex physical phenomena. While the Riemann curvature tensor contains all the curvature information, it is too general and lacks focus, and the concircular tensor, though more focused, is still too narrow. The quasi-concircular tensor, by contrast, strikes a balance, offering a richer, more refined understanding of the geometry of such intermediate structures.

In this work, we conduct a systematic study of the quasi-concircular curvature tensor on sequential warped product manifolds, extending the classical framework of singly warped products. The main results establish explicit characterizations of quasi-concircular flatness, quasi-concircular symmetry, and divergence-free conditions on these manifolds. We show that quasi-concircular flatness requires the component manifolds to be of constant curvature under certain compatibility relations, leading to a high degree of symmetry consistent with cosmological models. Moreover, we demonstrate that the warping functions impose strong geometric restrictions, including soliton structures such as conformal gradient solitons. These findings are further specialized to Lorentzian geometries, where we provide detailed curvature expressions for sequential generalized Robertson–Walker spacetimes and sequential standard static spacetimes, together with illustrative examples that highlight the applicability of the theory.

The paper is organized as follows. Section 2 presents the necessary background, including a discussion of the concircular and quasi-concircular curvature tensors, as well as the geometry of sequential warped product manifolds. Section 3 is devoted to deriving the quasi-concircular curvature tensor for these manifolds and to examining the conditions under which they become quasi-concircularly flat. This section also emphasizes the role of gradient solitons and establishes several related theorems. In Section 4, we consider two significant applications of the theory in a Lorentzian setting, namely, sequential generalized Robertson–Walker spacetimes and sequential standard static spacetimes. The paper concludes with an example that satisfies the generalized quasi-Einstein condition.

2. Preliminaries

In [11,12], the authors consider the conformal transformation ${\overline{g}}_{\mu \nu }={\rho }^2{g}_{\mu \nu }$ of the fundamental tensor $g_{\mu \nu }$. In general, a geodesic circle is not mapped to another geodesic circle under such a conformal transformation. Recall that a geodesic circle is defined as a curve whose first curvature is constant and whose second curvature vanishes identically. A conformal transformation that preserves geodesic circles must satisfy the partial differential equation ${\nabla }_i{\nabla }_j\rho =\varphi g_{ij}$. Such a transformation maps each geodesic circle to another geodesic circle and is called a concircular transformation. The geometry that studies such transformations is referred to as concircular geometry.

A $(1,3)$-type tensor $C({\tau }_X,{\tau }_Y){\tau }_Z$ that remains invariant under concircular transformations on an n-dimensional Riemannian or semi-Riemannian manifold ${\mathcal{M}}^n$ is called the concircular curvature tensor. It was introduced by Yano and Kon [13,14] and is defined as

$$C({\tau }_X,{\tau }_Y){\tau }_Z=R({\tau }_X,{\tau }_Y){\tau }_Z-{\displaystyle \frac{r}{n(n-1)}}\left[g({\tau }_Y,{\tau }_Z){\tau }_X-g({\tau }_X,{\tau }_Z){\tau }_Y\right],$$

(1)

where $R({\tau }_X,{\tau }_Y){\tau }_Z$ denotes the Riemannian curvature tensor and r the scalar curvature. The significance of concircular transformations and the concircular curvature tensor is well established in the differential geometry of various F-structures, such as complex, almost complex, Kähler, almost Kähler, contact, and almost contact structures [15].

In a recent paper, Prasad and Maurya [16] introduced the quasi-concircular curvature tensor of type $(1,3)$ on an n-dimensional Riemannian manifold $({\mathcal{M}}^n,g)$, denoted by V, and defined as

$$V({\tau }_X,{\tau }_Y){\tau }_Z=cR({\tau }_X,{\tau }_Y){\tau }_Z+{\displaystyle \frac{r}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)\left[g({\tau }_Y,{\tau }_Z){\tau }_X-g({\tau }_X,{\tau }_Z){\tau }_Y\right],$$

(2)

where R denotes the Riemannian curvature tensor and r the scalar curvature.

Equivalently, this tensor can be expressed in $(0,4)$-form as

$$\begin{array}{cc}\hfill V({\tau }_X,{\tau }_Y,{\tau }_Z,{\tau }_W)=& cR({\tau }_X,{\tau }_Y,{\tau }_Z,{\tau }_W)\hfill \\ \hfill & +{\displaystyle \frac{r}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)\left[g({\tau }_Y,{\tau }_Z)g({\tau }_X,{\tau }_W)-g({\tau }_X,{\tau }_Z)g({\tau }_Y,{\tau }_W)\right],\hfill \end{array}$$

(3)

or more compactly,

$$V=cR+{\displaystyle \frac{r}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)G,$$

(4)

where $G=-\frac{1}{2}(g\wedge g)$, with ∧ denoting the Kulkarni–Nomizu product. The constants c and d are arbitrary.

When $c=1$ and $d=-\frac{1}{n-1}$, the tensor reduces to the concircular curvature tensor defined in Equation (1). These choices may model more general spacetimes in modified gravity where Ricci and scalar curvature interact in non-standard ways. It is interesting in modified gravity theories, because the field equations couple Ricci and scalar curvature in non-Einsteinian ways. The freedom in c and d allows one to tune the curvature diagnostic to watch the geometric structures arising in those models.

In what follows, we make use of the following known results.

• A Riemannian manifold is said to be quasi-concircularly flat if its quasi-concircular curvature tensor vanishes identically at all points.
• If a manifold is quasi-concircularly flat, then it necessarily has constant sectional curvature.
• A manifold is called quasi-concircularly symmetric if the covariant derivative of its quasi-concircular curvature tensor vanishes. From the relation given in Equation (4), we obtain

  $$\nabla V=c\nabla R+{\displaystyle \frac{\nabla r}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)G.$$

  (5)
  Thus, if M is quasi-concircularly symmetric, then by combining with the second Bianchi identity, it follows that the scalar curvature r is constant, which implies that the manifold is locally symmetric. Conversely, if a manifold is locally symmetric, then it is quasi-concircularly symmetric.
• From Equation (4), the divergence of the quasi-concircular curvature tensor is given by

  $$\mathrm{d}\mathrm{i}\mathrm{v}V=c\mathrm{d}\mathrm{i}\mathrm{v}R+{\displaystyle \frac{\mathrm{div}r}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)G.$$

  (6)
  Moreover, using the identity

  $$\mathrm{d}\mathrm{i}\mathrm{v}R({\tau }_X,{\tau }_Y){\tau }_Z=\left({\nabla }_{{\tau }_X}\mathrm{Ric}\right)({\tau }_Y,{\tau }_Z)-\left({\nabla }_{{\tau }_Y}\mathrm{Ric}\right)({\tau }_X,{\tau }_Z),\forall {\tau }_X,{\tau }_Y,{\tau }_Z\in \mathfrak{X}\left(M\right),$$

  and applying the second Bianchi identity, we conclude that if the quasi-concircular curvature tensor is divergence-free, then the Ricci tensor satisfies the Codazzi property.

A sequential warped product manifold $\overline{\mathcal{M}}$ is constructed from three Riemannian (or semi-Riemannian) manifolds $({\mathcal{M}}_1,g_1),({\mathcal{M}}_2,g_2)$, and $({\mathcal{M}}_3,g_3)$, together with two smooth functions f and h. The construction is defined by

$$\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3,$$

with associated metric

$$\overline{g}=\left(g_1\oplus f^2{g}_2\right)\oplus h^2{g}_3,$$

where $f:{\mathcal{M}}_1\to (0,\infty )$ and $h:{\mathcal{M}}_1\times {\mathcal{M}}_2\to (0,\infty )$ are the two warping functions. Note that f and h cannot both be constant simultaneously; otherwise, the manifold reduces to a simple direct product. As discussed in [17], the sequential warped product structure provides a powerful framework for generalizing warped geometries, with several important consequences:

• The warped product of the form

  $${\mathcal{M}}_1{\times }_{f_1}\left({\mathcal{M}}_2{\times }_{f_2}{\mathcal{M}}_3\right),$$

  equipped with the metric

  $$\overline{g}=g_1+f_1^2(g_2+f_2^2{g}_3),$$

  is called the iterated warped product manifold. This is equivalent to the sequential warped product $\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$, where $f=f_1$ and $h=f_1{f}_2$.
• A multiply warped product manifold of the form

  $${\mathcal{M}}_1{\times }_{f_1}{\mathcal{M}}_2{\times }_{f_2}{\mathcal{M}}_3$$

  is a sequential warped product manifold $\left({\mathcal{M}}_1{\times }_{f_1}{\mathcal{M}}_2\right){\times }_{f_2}{\mathcal{M}}_3$, endowed with the metric

  $$\overline{g}=\left(g_1+f_1^2{g}_2\right)+f_2^2{g}_3,$$

  where both $f_1$ and $f_2$ are positive functions defined on ${\mathcal{M}}_1$.

Some examples of sequential warped products are given below:

• Sequential standard static spacetime: An $(m_1+m_2+1)$-dimensional manifold with the metric

  $$\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2(-d{t}^2),$$

  which generalizes the Einstein static universe by introducing an additional warping function.
• Sequential generalized Robertson–Walker spacetime: An $(1+m_2+m_3)$-dimensional manifold with the metric

  $$\overline{g}=(-d{t}^2\oplus f^2{g}_2)\oplus h^2{g}_3,$$

  widely used in cosmology for studying expanding or contracting universes.
• Application in high-energy physics: In string theory, warped geometries are employed for compactification. For instance, a space of the form

  $$S_1^n\times F_1,$$

  where $S_1^n$ represents a de Sitter space and $F_1$ is a compact Calabi–Yau manifold, is often used to model extra dimensions.

2.1. Notation

• Throughout the paper, all manifolds are assumed to be connected, and all geometric objects are taken to be smooth.
• The Riemannian curvature tensor is defined by

  $$R({\tau }_X,{\tau }_Y){\tau }_Z=[{\nabla }_{{\tau }_X},{\nabla }_{{\tau }_Y}]{\tau }_Z-{\nabla }_{[{\tau }_X,{\tau }_Y]}{\tau }_Z,$$

  the Ricci tensor by

  $$\mathrm{Ric}({\tau }_X,{\tau }_Y)=\sum _{i=1}^{n}R({\tau }_{e_i},{\tau }_X,{\tau }_Y,{\tau }_{e_i}),$$

  and the scalar curvature by

  $$r=\sum _{i=1}^n\mathrm{Ric}({\tau }_{e_i},{\tau }_{e_i}),$$

  where $\{{\tau }_{e_i}:i=1,2,\dots ,n\}$ is a local orthonormal frame on the manifold.
• For any ${\tau }_X,{\tau }_Y\in \mathfrak{X}\left(\mathcal{M}\right)$, the Hessian of a smooth function $\varphi$ is the symmetric $(0,2)$-tensor defined by

  $$H^{\varphi }({\tau }_X,{\tau }_Y)=g\left({\nabla }_{{\tau }_X}\mathrm{grad}\varphi ,{\tau }_Y\right).$$
• On a sequential warped product manifold

  $$\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3,$$

  any vector field ${\tau }_X$ decomposes uniquely as

  $${\tau }_X={\tau }_{X_1}+{\tau }_{X_2}+{\tau }_{X_3},$$

  where each ${\tau }_{X_i}\in \mathfrak{X}\left({\mathcal{M}}_i\right)$ for $i=1,2,3$.
• The gradient of f with respect to $({\mathcal{M}}_1,g_1)$ is denoted by ${\mathrm{grad}}^{1}f$, with squared norm

  $${\parallel {\mathrm{grad}}^{1}f\parallel }^2=g_1({\mathrm{grad}}^{1}f,{\mathrm{grad}}^{1}f).$$
• The gradient of h with respect to $(\mathcal{M},g)$, where $\mathcal{M}={\mathcal{M}}_1{\times }_f{\mathcal{M}}_2$, is denoted by $\mathrm{grad}h$, with squared norm

  $${\parallel \mathrm{grad}h\parallel }^2=g(\mathrm{grad}h,\mathrm{grad}h).$$

2.2. Basic Formulas

Let $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be a sequential warped product manifold equipped with the metric

$$\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2{g}_3,$$

and let ${\tau }_{X_i},{\tau }_{Y_i},{\tau }_{Z_i}\in \mathfrak{X}\left({\mathcal{M}}_i\right)$ for $i=1,2,3$. Then the following identities hold:

Lemma 1.

The components of the Levi-Civita connection on $(\overline{\mathcal{M}},\overline{g})$ are given by [7]:

1.

${\overline{\nabla }}_{{\tau }_{X_1}}{\tau }_{Y_1}={\nabla }_{{\tau }_{X_1}}^1{\tau }_{Y_1},$

2.

${\overline{\nabla }}_{{\tau }_{X_1}}{\tau }_{X_2}={\overline{\nabla }}_{{\tau }_{X_2}}{\tau }_{X_1}={\tau }_{X_1}\left(\ln f\right){\tau }_{X_2},$

3.

${\overline{\nabla }}_{{\tau }_{X_2}}{\tau }_{Y_2}={\nabla }_{{\tau }_{X_2}}^2{\tau }_{Y_2}-f{g}_2({\tau }_{X_2},{\tau }_{Y_2}){\mathrm{grad}}^{1}f,$

4.

${\overline{\nabla }}_{{\tau }_{X_3}}{\tau }_{X_1}={\overline{\nabla }}_{{\tau }_{X_1}}{\tau }_{X_3}={\tau }_{X_1}\left(\ln h\right){\tau }_{X_3},$

5.

${\overline{\nabla }}_{{\tau }_{X_2}}{\tau }_{X_3}={\overline{\nabla }}_{{\tau }_{X_3}}{\tau }_{X_2}={\tau }_{X_2}\left(\ln h\right){\tau }_{X_3},$

6.

${\overline{\nabla }}_{{\tau }_{X_3}}{\tau }_{Y_3}={\nabla }_{{\tau }_{X_3}}^3{\tau }_{Y_3}-h{g}_3({\tau }_{X_3},{\tau }_{Y_3})\mathrm{grad}h.$

Note that $H_1^f$ and ${\Delta }_{1}f$ denote the Hessian and Laplacian of f on ${\mathcal{M}}_1$, respectively, while $H^h$ and $\Delta h$ denote the Hessian and Laplacian of h on $\mathcal{M}$, respectively.

Lemma 2.

The non-zero components of the Riemannian curvature of $(\overline{\mathcal{M}},\overline{g})$ are given by [7]:

1.

$\overline{R}({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1}=R_1({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1},$

2.

$\overline{R}({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}=R_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}-{∥{\mathrm{grad}}^{1}f∥}^2\left[g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_2}-g_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_2}\right],$

3.

$\overline{R}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_1}=\frac{1}{f}{H}_1^f({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_2},$

4.

$\overline{R}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_2}=-f{g}_2({\tau }_{Y_2},{\tau }_{Z_2}){\nabla }_{{\tau }_{X_1}}^1{\mathrm{grad}}^{1}f,$

5.

$\overline{R}({\tau }_{X_i},{\tau }_{Y_3}){\tau }_{Z_j}=\frac{1}{h}{H}^h({\tau }_{X_i},{\tau }_{Z_j}){\tau }_{Y_3},i,j=1,2,$

6.

$\overline{R}({\tau }_{X_i},{\tau }_{Y_3}){\tau }_{Z_3}=-h{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\overline{\nabla }}_{{\tau }_{X_i}}\mathrm{grad}h,i=1,2,$

7.

$\overline{R}({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}=R_3({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}-{∥\mathrm{grad}h∥}^2\left[g_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{X_3}-g_3({\tau }_{X_3},{\tau }_{Z_3}){\tau }_{Y_3}\right].$

Lemma 3.

The non-zero components of the Ricci curvature of $(\overline{\mathcal{M}},\overline{g})$ are given by [7]:

1.

$\overline{\mathrm{Ric}}({\tau }_{X_1},{\tau }_{Y_1})={\mathrm{Ric}}_1({\tau }_{X_1},{\tau }_{Y_1})-{\displaystyle \frac{m_2}{f}}{H}_1^f({\tau }_{X_1},{\tau }_{Y_1})-{\displaystyle \frac{m_3}{h}}{H}^h({\tau }_{X_1},{\tau }_{Y_1}),$

2.

$\overline{\mathrm{Ric}}({\tau }_{X_2},{\tau }_{Y_2})={\mathrm{Ric}}_2({\tau }_{X_2},{\tau }_{Y_2})-f^{\#}{g}_2({\tau }_{X_2},{\tau }_{Y_2})-{\displaystyle \frac{m_3}{h}}{H}^h({\tau }_{X_2},{\tau }_{Y_2}),$

3.

$\overline{\mathrm{Ric}}({\tau }_{X_3},{\tau }_{Y_3})={\mathrm{Ric}}_3({\tau }_{X_3},{\tau }_{Y_3})-h^{\#}{g}_3({\tau }_{X_3},{\tau }_{Y_3}).$

where $f^{\#}=f{\Delta }_{1}f+(m_2-1){∥{\mathrm{grad}}^{1}f∥}^2,h^{\#}=h\Delta h+(m_3-1){∥\mathrm{grad}h∥}^2.$

Lemma 4.

The relation between the scalar curvature $\overline{r}$ of $(\overline{\mathcal{M}},\overline{g})$ and the scalar curvatures $r_i$ of $({\mathcal{M}}_i,g_i)$ for any $i=1,2,3$ is given by [7]:

$$\begin{array}{cc}\hfill \overline{r}& =r_1+{\displaystyle \frac{r_2}{f^2}}+{\displaystyle \frac{r_3}{h^2}}-{\displaystyle \frac{2m_2}{f}}{\Delta }_{1}f-{\displaystyle \frac{2m_3}{h}}\Delta h\hfill \\ \hfill & -{\displaystyle \frac{m_2(m_2-1)}{f^2}}{∥{\mathrm{grad}}^{1}f∥}^2-{\displaystyle \frac{m_3(m_3-1)}{h^2}}{∥\mathrm{grad}h∥}^2.\hfill \end{array}$$

Lemma 5.

Let $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be a sequential warped product. Then, the Hessian tensor $H^{\phi }$ of a smooth function φ on $\overline{\mathcal{M}}$ satisfies [7]:

1.

$H^{\phi }({\tau }_{X_1},{\tau }_{Y_1})=H_1^{\phi }({\tau }_{X_1},{\tau }_{Y_1}),$

2.

$H^{\phi }({\tau }_{X_1},{\tau }_{Y_2})={\tau }_{X_1}\left(\ln f\right){\tau }_{Y_2}\left(\phi \right),$

3.

$H^{\phi }({\tau }_{X_1},{\tau }_{Y_3})={\tau }_{X_1}\left(\ln h\right){\tau }_{Y_3}\left(\phi \right),$

4.

$H^{\phi }({\tau }_{X_2},{\tau }_{Y_2})=f{\nabla }_{\phi }\left(f\right)g_2({\tau }_{X_2},{\tau }_{Y_2})+H_2^{\phi }({\tau }_{X_2},{\tau }_{Y_2}),$

5.

$H^{\phi }({\tau }_{X_2},{\tau }_{Y_3})={\tau }_{X_2}\left(\ln h\right){\tau }_{Y_3}\left(\phi \right),$

6.

$H^{\phi }({\tau }_{X_3},{\tau }_{Y_3})=h{\nabla }_{\phi }\left(h\right)g_3({\tau }_{X_3},{\tau }_{Y_3})+H_3^{\phi }({\tau }_{X_3},{\tau }_{Y_3}).$

Moreover, for any smooth function $\phi$, the following relation holds:

$${\displaystyle \frac{m}{\phi }}{H}^{\phi }=H^{m\ln \phi }+{\displaystyle \frac{1}{m}}d\left(m\ln \phi \right)\otimes d\left(m\ln \phi \right),m\in \mathbb{R}.$$

3. Quasi-Concircular Geometry on Sequential Warped Products

3.1. Quasi-Concircular Flatness

As a first step, we employ Lemmas 1–4 to characterize the quasi-concircular curvature tensor associated with the sequential warped product manifold.

Proposition 1.

Let $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be an n-dimensional sequential warped product manifold, endowed with the metric

$$\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2{g}_3.$$

Then, the non-vanishing components of the quasi-concircular curvature tensor $\overline{V}$ are given by:

1.

$\overline{V}({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1}=c{R}_1({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1}+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)[g_1({\tau }_{Y_1},{\tau }_{Z_1}){\tau }_{X_1}-g_1({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_1}],$

2.

$$\begin{gathered} \overline{V}({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}=c{R}_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}-\left[c\parallel {\mathrm{grad}}^1{f\parallel }^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2\right][g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_2}- \\ {g}_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_2}], \end{gathered}$$

3.

$\overline{V}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_1}={\displaystyle \frac{c}{f}}{H}_1^f({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_2}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)g_1({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_2},$

4.

$\overline{V}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_2}=-cf{g}_2({\tau }_{Y_2},{\tau }_{Z_2}){\nabla }_{{\tau }_{X_1}}^1{\mathrm{grad}}^{1}f+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_1},$

5.

$\overline{V}({\tau }_{X_1},{\tau }_{Y_3}){\tau }_{Z_1}={\displaystyle \frac{c}{h}}{H}_1^h({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_3}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)g_1({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_3},$

6.

$\overline{V}({\tau }_{X_1},{\tau }_{Y_3}){\tau }_{Z_2}={\displaystyle \frac{c}{h}}{H}^h({\tau }_{X_1},{\tau }_{Z_2}){\tau }_{Y_3},$

7.

$\overline{V}({\tau }_{X_2},{\tau }_{Y_3}){\tau }_{Z_1}={\displaystyle \frac{c}{h}}{H}^h({\tau }_{X_2},{\tau }_{Z_1}){\tau }_{Y_3},$

8.

$\overline{V}({\tau }_{X_2},{\tau }_{Y_3}){\tau }_{Z_2}={\displaystyle \frac{c}{h}}{H}^h({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_3}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2{g}_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_3},$

9.

$\overline{V}({\tau }_{X_1},{\tau }_{Y_3}){\tau }_{Z_3}=-ch{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\overline{\nabla }}_{{\tau }_{X_1}}\mathrm{grad}h+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)g_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{X_1},$

10.

$$\begin{gathered} \overline{V}({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}=c{R}_3({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}-\left[{c\parallel \mathrm{grad}h\parallel }^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h^2\right][g_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{X_3}- \\ {g}_3({\tau }_{X_3},{\tau }_{Z_3}){\tau }_{Y_3}]. \end{gathered}$$

Proof. 

The proof follows directly from Lemma 2, together with the definitions of the quasi-concircular curvature tensor and the metric of a sequential warped product manifold. □

As a direct consequence of Proposition 1, we obtain the following:

Theorem 1.

Let $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be a sequential warped product manifold endowed with the metric

$$\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2{g}_3.$$

Then $\overline{\mathcal{M}}$ is quasi-concircularly flat if and only if the following conditions are satisfied:

1.

$({\mathcal{M}}_1,g_1)$ is of constant curvature

$${\kappa }_1=-{\displaystyle \frac{\overline{r}}{nc}}\left({\displaystyle \frac{c}{n-1}}+2d\right).$$

2.

$({\mathcal{M}}_2,g_2)$ is of constant curvature

$${\kappa }_2={\displaystyle \frac{1}{c}}\left[c{∥{\mathrm{grad}}^{1}f∥}^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2\right].$$

3.

The Hessian condition

$$c{H}_1^f({\tau }_{X_1},{\tau }_{Z_1})-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f{g}_1({\tau }_{X_1},{\tau }_{Z_1})=0.$$

4.

The warping function h depends only on the variables of a single factor manifold, i.e.,

$$H^h({\tau }_{X_1},{\tau }_{Z_2})=0or{H}^h({\tau }_{X_2},{\tau }_{Z_1})=0.$$

5.

The second Hessian condition

$${\displaystyle \frac{c}{h}}{H}^h({\tau }_{X_2},{\tau }_{Z_2})-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2{g}_2({\tau }_{X_2},{\tau }_{Z_2})=0.$$

6.

$({\mathcal{M}}_3,g_3)$ is of constant curvature

$${\kappa }_3={\displaystyle \frac{1}{c}}\left[c{∥\mathrm{grad}h∥}^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h^2\right].$$

These conditions hold for all ${\tau }_{X_i},{\tau }_{Y_i},{\tau }_{Z_i}\in \mathfrak{X}\left({\mathcal{M}}_i\right)$.

Proof. 

By applying Proposition 1 and setting each component of the quasi-concircular curvature tensor to zero, we obtain the desired results. □

The conditions obtained in the above theorem show that quasi-concircular flatness of the warped product spacetime requires both the base and the fiber manifolds to have constant curvature under the specified relations among ${\kappa }_1$, ${\kappa }_2$, and ${\kappa }_3$. Geometrically, this means that the local geometry of each component is maximally symmetric, leading to a high degree of regularity in the warped product as a whole. Physically, such constant-curvature manifolds often model isotropic and homogeneous universes (e.g., de Sitter, anti-de Sitter, or Robertson–Walker spacetimes). Thus, the result highlights that quasi-concircular flatness is compatible only with spacetimes exhibiting a high degree of symmetry, which aligns naturally with cosmological principles.

A Riemannian manifold is referred to as a conformal gradient soliton if it admits a non-constant smooth function f, known as the soliton potential, satisfying the condition

$$H^f=\varphi g,$$

(7)

where $\varphi$ is a smooth function on the manifold.

Cheeger and Colding [18] obtained solutions of (7) and characterized warped product manifolds in this context. Furthermore, Sousa and Pina [19] studied the structure of warped product manifolds and showed that (7) represents a special case of the gradient Ricci soliton structure [19,20,21,22].

By employing the results of [19] together with Theorem 1, we establish the following result:

Theorem 2.

Let $\overline{M}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be a sequential warped product endowed with the metric

$$\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2{g}_3.$$

If $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly flat, then $({\mathcal{M}}_1,g_1)$ is a conformal gradient soliton, where the warping function f itself serves as the soliton potential. Moreover, the associated function is given by

$$\varphi ={\displaystyle \frac{f\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right).$$

From Theorem 1 and Lemma 1, we obtain

$$f^2{\tau }_{X_1}\left(\ln f\right)g_2(\mathrm{grad}{h}_2,{\tau }_{Z_2})=0,\forall {\tau }_{Z_2}\in \mathfrak{X}\left({\mathcal{M}}_2\right).$$

Here, $\mathrm{grad}{h}_i$ denotes the tangential and normal components of the gradient of h on ${\mathcal{M}}_i$ for $i=1,2$, and the full gradient decomposes as

$$\mathrm{grad}h=\mathrm{grad}{h}_1+\mathrm{grad}{h}_2.$$

Since f is assumed to be a non-constant positive function, the preceding relation immediately yields $\mathrm{grad}{h}_2=0$. Consequently, we have the following result:

Corollary 1.

Let $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be a sequential warped product manifold with metric $\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2{g}_3$. If the manifold $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly flat, then the warping function h depends only on the variables of ${\mathcal{M}}_1$.

Moreover, from Theorem 1, one obtains

$${\Delta }_{1}f={\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f{m}_1,\Delta h={\displaystyle \frac{\overline{r}h}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)\left(m_1+m_2{f}^2\right).$$

(8)

If both ${\mathcal{M}}_1$ and the warped product ${\mathcal{M}}_1{\times }_f{\mathcal{M}}_2$ are compact, and the scalar curvature satisfies $\overline{r}\le 0$ (or $\overline{r}\ge 0$), then Equation (8) implies that both ${\Delta }_{1}f$ and $\Delta h$ preserve a constant sign. By applying Hopf’s Lemma under these conditions, we deduce that the warping functions f and h must necessarily be constant. This establishes the following result:

Corollary 2.

Let ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ be compact manifolds in the sequential warped product

$$\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3,$$

equipped with the metric

$$\overline{g}=\left(g_1\oplus f^2{g}_2\right)\oplus h^2{g}_3.$$

If the manifold $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly flat and its scalar curvature satisfies $\overline{r}\le 0$ (or $\overline{r}\ge 0$), then the structure reduces to a simple direct product.

3.2. Quasi-Concircular Symmetry

A manifold is said to be quasi-concircularly symmetric if the quasi-concircular curvature tensor V is symmetric, that is, if $\nabla V=0$. In this case, we have

$$\begin{array}{ccc}\hfill \left({\nabla }_{{\tau }_W}V\right)({\tau }_X,{\tau }_Y){\tau }_Z& =& c\left({\nabla }_{{\tau }_W}R\right)({\tau }_X,{\tau }_Y){\tau }_Z\hfill \\ & & +{\displaystyle \frac{1}{n}}\left({\nabla }_{{\tau }_W}r\right)\left({\displaystyle \frac{c}{n-1}}+2d\right)\left[g({\tau }_Y,{\tau }_Z){\tau }_X-g({\tau }_X,{\tau }_Z){\tau }_Y\right].\hfill \end{array}$$

In the case of quasi-concircular symmetry, we require $\left({\nabla }_{{\tau }_W}V\right)({\tau }_X,{\tau }_Y){\tau }_Z=0$. Hence,

$$0=\left(c+(n-1)d\right){\nabla }_{{\tau }_W}r.$$

Therefore, a quasi-concircularly symmetric manifold satisfies either

$${\nabla }_{{\tau }_W}r=0\mathrm{or}d=-\frac{c}{n-1}.$$

The first condition implies that the manifold is locally symmetric, while the second condition recovers the concircular curvature symmetry, which also implies local symmetry.

In what follows, we investigate some symmetry conditions on sequential warped products.

Theorem 3.

Let the sequential warped product be defined as

$$\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3,$$

equipped with the metric

$$\overline{g}=\left(g_1\oplus f^2{g}_2\right)\oplus h^2{g}_3.$$

If $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly symmetric, then the following properties hold:

1.

${\mathcal{M}}_1,{\mathcal{M}}_2$, and ${\mathcal{M}}_3$ are locally symmetric manifolds.

2.

The tensor fields

$$\mathcal{F}={\displaystyle \frac{1}{f}}{H}_1^f,\mathcal{H}={\displaystyle \frac{1}{h}}{H}^h$$

are parallel.

3.

The curvature relation

$$R_1\left({\mathrm{grad}}^{1}f,{\tau }_{X_1}\right){\tau }_{Z_1}=-\mathcal{F}({\tau }_{X_1},{\tau }_{Z_1}){\mathrm{grad}}^{1}f+{\tau }_{Z_1}\left(\ln f\right){\nabla }_{{\tau }_{X_1}}^1{\mathrm{grad}}^{1}f$$

holds.

4.

The following condition is satisfied:

$${\tau }_{W_1}\left({∥{\mathrm{grad}}^{1}f∥}^2\right)G_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}=0.$$

Proof. 

Since $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly symmetric, for all ${\tau }_{X_i},{\tau }_{Y_i},{\tau }_{Z_i},{\tau }_{W_i}\in \mathfrak{X}\left({\mathcal{M}}_i\right),$ $i=1,2,3$, we have

$$\left({\overline{\nabla }}_{{\tau }_{W_i}}\overline{V}\right)({\tau }_{X_i},{\tau }_{Y_i}){\tau }_{Z_i}=0.$$

By Equation (5), it follows that a quasi-concircularly symmetric manifold is also locally symmetric and its scalar curvature is constant. Hence,

$$\left({\overline{\nabla }}_{{\tau }_{W_i}}\overline{R}\right)({\tau }_{X_i},{\tau }_{Y_i}){\tau }_{Z_i}=0,\forall {\tau }_{X_i},{\tau }_{Y_i},{\tau }_{Z_i},{\tau }_{W_i}\in \mathfrak{X}\left({\mathcal{M}}_i\right),i=1,2,3,$$

which proves assertion (1).

Moreover, from the identities

$$\left({\overline{\nabla }}_{{\tau }_{W_1}}\overline{R}\right)({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_1}=0,\left({\overline{\nabla }}_{{\tau }_{W_1}}\overline{R}\right)({\tau }_{X_i},{\tau }_{Y_3}){\tau }_{Z_i}=0,$$

we obtain

$$\left({\overline{\nabla }}_{{\tau }_{W_1}}\frac{1}{f}{H}_1^f\right)({\tau }_{X_1},{\tau }_{Y_1})=0,\forall {\tau }_{X_1},{\tau }_{Y_1}\in \mathfrak{X}\left({\mathcal{M}}_1\right),$$

and

$$\left({\overline{\nabla }}_{{\tau }_{W_1}}\frac{1}{h}{H}^h\right)({\tau }_{X_i},{\tau }_{Y_i})=0,i=1,2,$$

which proves assertion (2).

From

$$\left({\overline{\nabla }}_{{\tau }_{W_2}}\overline{R}\right)({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_1}=0,\mathrm{or}\mathrm{equivalently}\left({\overline{\nabla }}_{{\tau }_{W_2}}\frac{1}{f}{H}_1^f\right)({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_2}=0,$$

we deduce assertion (3).

Finally, the relation

$$\left({\overline{\nabla }}_{{\tau }_{W_1}}\overline{R}\right)({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}=0$$

establishes assertion (4). □

3.3. Quasi-Concircular Curvature Tensor with Zero Divergence

Let $V_i$ denote the quasi-concircular curvature tensor on ${\mathcal{M}}_i$, for $i=1,2,3$. Suppose the quasi-concircular curvature tensor $\overline{V}$ of the sequential warped product $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ is divergence-free. In this case, it follows that $\mathrm{div}\overline{R}=0$, and consequently, the Ricci tensor is of Codazzi type. Within this framework, the tensor T is defined by

$$T({\tau }_X,{\tau }_Y,{\tau }_Z)=\left({\overline{\nabla }}_{{\tau }_X}\overline{\mathrm{Ric}}\right)({\tau }_Y,{\tau }_Z)-\left({\overline{\nabla }}_{{\tau }_Y}\overline{\mathrm{Ric}}\right)({\tau }_X,{\tau }_Z),$$

for all ${\tau }_X,{\tau }_Y,{\tau }_Z\in \mathfrak{X}\left(\overline{\mathcal{M}}\right)$, and vanishes identically. We now invoke Lemma 3 to derive the following consequences:

• If $T({\tau }_{X_1},{\tau }_{Y_1},{\tau }_{Z_1})=0$, then ${\tau }_{X_1}$ and ${\tau }_{Y_1}$ commute, and in this case we have

  $$\left({\overline{\nabla }}_{{\tau }_{X_1}}\overline{\mathrm{Ric}}\right)({\tau }_{Y_1},{\tau }_{Z_1})=\left({\overline{\nabla }}_{{\tau }_{Y_1}}\overline{\mathrm{Ric}}\right)({\tau }_{X_1},{\tau }_{Z_1}).$$
  Noting that

  $$\left({\nabla }_{{\tau }_{X_1}}{H}_1^f\right)({\tau }_{Y_1},{\tau }_{Z_1})-\left({\nabla }_{{\tau }_{Y_1}}{H}_1^f\right)({\tau }_{X_1},{\tau }_{Z_1})=R_1({\tau }_{X_1},{\tau }_{Y_1},{\tau }_{Z_1},{\mathrm{grad}}^{1}f),$$

  and

  $$\left({\nabla }_{{\tau }_{Y_1}}{H}^h\right)({\tau }_{X_1},{\tau }_{Z_1})-\left({\nabla }_{{\tau }_{X_1}}{H}^h\right)({\tau }_{Y_1},{\tau }_{Z_1})=R_1({\tau }_{X_1},{\tau }_{Y_1},\mathrm{grad}h,{\tau }_{Z_1}),$$

  we obtain

  $$\begin{array}{cc}& {\displaystyle \frac{m_2}{f}}\left[R_1({\tau }_{X_1},{\tau }_{Y_1},{\tau }_{Z_1},{\mathrm{grad}}^{1}f)+{\tau }_{Y_1}\left(f\right)\mathcal{F}({\tau }_{X_1},{\tau }_{Z_1})-{\tau }_{X_1}\left(f\right)\mathcal{F}({\tau }_{Y_1},{\tau }_{Z_1})\right]\hfill \\ =& {\displaystyle \frac{m_3}{h}}\left[R_1({\tau }_{X_1},{\tau }_{Y_1},\mathrm{grad}h,{\tau }_{Z_1})-{\tau }_{Y_1}\left(h\right)\mathcal{H}({\tau }_{X_1},{\tau }_{Z_1})+{\tau }_{X_1}\left(h\right)\mathcal{H}({\tau }_{Y_1},{\tau }_{Z_1})\right].\hfill \end{array}$$

  (9)
  Thus, $\mathrm{div}{V}_1=0$, i.e.,

  $$\left({\nabla }_{{\tau }_{X_1}}{\mathrm{Ric}}_1\right)({\tau }_{Y_1},{\tau }_{Z_1})=\left({\nabla }_{{\tau }_{Y_1}}{\mathrm{Ric}}_1\right)({\tau }_{X_1},{\tau }_{Z_1})$$

  holds if and only if Equation (9) is satisfied.
• If $T({\tau }_{X_2},{\tau }_{Y_2},{\tau }_{Z_2})=0$, then ${\tau }_{X_2}$ and ${\tau }_{Y_2}$ commute. In this case, we have

  $$\left({\overline{\nabla }}_{{\tau }_{X_2}}\overline{\mathrm{Ric}}\right)({\tau }_{Y_2},{\tau }_{Z_2})=\left({\overline{\nabla }}_{{\tau }_{Y_2}}\overline{\mathrm{Ric}}\right)({\tau }_{X_2},{\tau }_{Z_2}).$$

  $${\displaystyle \frac{1}{h}}{R}_2({\tau }_{X_2},{\tau }_{Y_2},\mathrm{grad}h,{\tau }_{Z_2})={\tau }_{Y_2}\left(\ln h\right)\mathcal{H}({\tau }_{X_2},{\tau }_{Z_2})-{\tau }_{X_2}\left(\ln h\right)\mathcal{H}({\tau }_{Y_2},{\tau }_{Z_2}).$$

  (10)
  Thus, $\mathrm{div}{V}_2=0$, i.e.,

  $$\left({\overline{\nabla }}_{{\tau }_{X_2}}{\mathrm{Ric}}_2\right)({\tau }_{Y_2},{\tau }_{Z_2})=\left({\overline{\nabla }}_{{\tau }_{Y_2}}{\mathrm{Ric}}_2\right)({\tau }_{X_2},{\tau }_{Z_2})$$

  holds if and only if (10) holds.
• If $T({\tau }_{X_3},{\tau }_{Y_3},{\tau }_{Z_3})=0$ and $h\in C^{\infty }({\mathcal{M}}_1\times {\mathcal{M}}_2)$, then, by Lemma 3, we always have $\mathrm{div}{V}_3=0$.
• If $T({\tau }_{X_1},{\tau }_{Y_2},{\tau }_{Z_2})=0$, then ${\tau }_{X_1}$ and ${\tau }_{Y_2}$ commute. In this case, we have

  $$\left({\overline{\nabla }}_{{\tau }_{X_1}}\overline{\mathrm{Ric}}\right)({\tau }_{Y_2},{\tau }_{Z_2})=\left({\overline{\nabla }}_{{\tau }_{Y_2}}\overline{\mathrm{Ric}}\right)({\tau }_{X_1},{\tau }_{Z_2}),$$

  which yields

  $$\begin{array}{cc}& {\mathrm{Ric}}_2({\tau }_{Y_2},{\tau }_{Z_2})+{\displaystyle \frac{m_3}{h}}\left[{\displaystyle \frac{X_1\left(\ln h\right)}{{\tau }_{X_1}\left(\ln f\right)}}-{\displaystyle \frac{{\tau }_{X_1}}{{\tau }_{X_1}\left(\ln f\right)}}\right]H^h({\tau }_{Y_2},{\tau }_{Z_2})+{\displaystyle \frac{m_3}{h}}{H}_2^2({\tau }_{Y_2},{\tau }_{Z_2})\hfill \\ =& -\left[{\displaystyle \frac{m_{3}f{\nabla }_h\left(f\right)}{h}}-f^{\#}+{\displaystyle \frac{{\tau }_{X_1}\left(f^{\#}\right)}{{\tau }_{X_1}\left(\ln f\right)}}\right]g_2({\tau }_{Y_2},{\tau }_{Z_2}).\hfill \end{array}$$

  (11)
• If $T({\tau }_{X_1},{\tau }_{Y_3},{\tau }_{Z_3})=0$, then ${\tau }_{X_1}$ and ${\tau }_{Y_3}$ commute. In this case, we have

  $${\mathrm{Ric}}_3({\tau }_{Y_3},{\tau }_{Z_3})=\left(h^{\#}+{\displaystyle \frac{X_1\left(h^{\#}\right)}{{\tau }_{X_1}\left(\ln h\right)}}\right)g_3({\tau }_{Y_3},{\tau }_{Z_3}).$$

  (12)
• If $T({\tau }_{X_2},{\tau }_{Y_1},{\tau }_{Z_1})=0$, then ${\tau }_{X_2}$ and ${\tau }_{Y_1}$ commute. In this case, we obtain

  $$\begin{array}{ccc}\hfill \left({\overline{\nabla }}_{{\tau }_{X_2}}\overline{\mathrm{Ric}}\right)({\tau }_{Y_1},{\tau }_{Z_1})& =& \left({\overline{\nabla }}_{{\tau }_{Y_1}}\overline{\mathrm{Ric}}\right)({\tau }_{X_2},{\tau }_{Z_1}),\hfill \\ \hfill X_2\left(\ln h\right)H^h({\tau }_{Y_1},{\tau }_{Z_1})& =& {\overline{\nabla }}_{{\tau }_{X_2}}\left[H^h({\tau }_{Y_1},{\tau }_{Z_1})\right],\hfill \end{array}$$

  which is possible only when h depends on ${\mathcal{M}}_1$. Combining this result with Equation (11), we obtain

  $${\mathrm{Ric}}_2({\tau }_{Y_2},{\tau }_{Z_2})+\psi H^h({\tau }_{Y_2},{\tau }_{Z_2})=\lambda g_2({\tau }_{Y_2},{\tau }_{Z_2}),$$

  (13)

  which is the fundamental equation of a $\psi$-almost gradient Ricci soliton, where

  $$\begin{array}{ccc}\hfill \psi & =& {\displaystyle \frac{m_3}{h}}\left[{\displaystyle \frac{{\tau }_{X_1}\left(\ln h\right)}{{\tau }_{X_1}\left(\ln f\right)}}-{\displaystyle \frac{{\tau }_{X_1}}{{\tau }_{X_1}\left(\ln f\right)}}\right],\hfill \end{array}$$

  (14)

  $$\begin{array}{ccc}\hfill \lambda & =& -\left[{\displaystyle \frac{m_{3}f{\nabla }_h\left(f\right)}{h}}-f^{\#}+{\displaystyle \frac{{\tau }_{X_1}\left(f^{\#}\right)}{{\tau }_{X_1}\left(\ln f\right)}}\right].\hfill \end{array}$$

  (15)

Furthermore, all remaining components of the tensor T vanish identically. This leads to the following result:

Theorem 4.

Let $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be a sequential warped product manifold equipped with the metric $\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2{g}_3$. If the manifold $(\overline{\mathcal{M}},\overline{g})$ admits a divergence-free quasi-concircular curvature tensor, then:

1.

The manifold ${\mathcal{M}}_1$ possesses a divergence-free quasi-concircular curvature tensor if and only if condition (9) is satisfied.

2.

The quasi-concircular curvature tensor on ${\mathcal{M}}_2$ is divergence-free if and only if Equation (10) holds.

3.

Since ${\mathcal{M}}_3$ is an Einstein manifold, its quasi-concircular curvature tensor is always divergence-free.

4.

The warping function h is assumed to depend solely on the variables of ${\mathcal{M}}_1$.

5.

Moreover, $({\mathcal{M}}_2,g_2,h,\psi ,\lambda )$ forms a ψ-almost gradient soliton, where the functions ψ and λ are defined as in Equations (14) and (15).

Remark 1.

The conditions (9)–(15), though technical, admit natural geometric interpretations. Conditions (9) and (10) ensure that the quasi-concircular curvature tensors of ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are divergence-free, while the Einstein property (12) of ${\mathcal{M}}_3$ guarantees this automatically. The dependence of the warping function h only on the variables of ${\mathcal{M}}_1$ ensures compatibility between the base and the fiber geometries. Finally, Equations (13)–(15) imply that $({\mathcal{M}}_2,g_2,h,\psi ,\lambda )$ admits a ψ-almost gradient Ricci soliton structure, linking the curvature conditions with natural soliton models that arise in geometric flows.

4. Examples of Sequential Products Admitting Quasi-Concircular Curvature Tensor

4.1. Sequential Generalized Robertson-Walker Spacetimes

Let $\overline{\mathcal{M}}=\left(I{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be a sequential generalized Robertson–Walker spacetime equipped with the metric

$$\overline{g}=(-d{t}^2\oplus f^2{g}_2)\oplus h^2{g}_3.$$

Consider vector fields ${\tau }_{X_i},{\tau }_{Y_i},{\tau }_{Z_i}\in \mathfrak{X}\left({\mathcal{M}}_i\right)$ for $i=2,3$. Then, by applying the lemmas discussed in Section 2.2, we obtain the following result:

Lemma 6.

The components of the Levi-Civita connection on $(\overline{\mathcal{M}},\overline{g})$ are given by [6]:

1.

${\overline{\nabla }}_{{\tau }_{\partial t}}{\tau }_{X_i}={\overline{\nabla }}_{{\tau }_{X_i}}{\tau }_{\partial t}={\displaystyle \frac{\dot{f}}{f}}{\tau }_{X_i},i=2,3.$

2.

${\overline{\nabla }}_{{\tau }_{X_2}}{\tau }_{Y_2}={\nabla }_{{\tau }_{X_2}}^2{\tau }_{Y_2}-f\dot{f}{g}_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{\partial t}.$

3.

${\overline{\nabla }}_{{\tau }_{X_2}}{\tau }_{X_3}={\overline{\nabla }}_{{\tau }_{X_3}}{\tau }_{X_2}={\tau }_{X_2}\left(\ln h\right){\tau }_{X_3}.$

4.

${\overline{\nabla }}_{{\tau }_{X_3}}{\tau }_{Y_3}={\nabla }_{{\tau }_{X_3}}^3{\tau }_{Y_3}-h{g}_3({\tau }_{X_3},{\tau }_{Y_3})\mathrm{grad}h.$

Lemma 7.

The non-zero components of the Riemannian curvature tensor of $(\overline{\mathcal{M}},\overline{g})$ are given by [6]:

1.

$\overline{R}({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}=R_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}+{\left(\dot{f}\right)}^2\left[g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_2}-g_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_2}\right],$

2.

$\overline{R}({\tau }_{\partial t},{\tau }_{Y_2}){\tau }_{\partial t}=-{\displaystyle \frac{\ddot{f}}{f}}{\tau }_{Y_2},$

3.

$\overline{R}({\tau }_{\partial t},{\tau }_{Y_3}){\tau }_{\partial t}=-{\displaystyle \frac{1}{h}}{\displaystyle \frac{{\partial }^{2}h}{\partial t^2}}{\tau }_{Y_3},$

4.

$\overline{R}({\tau }_{\partial t},{\tau }_{Y_2}){\tau }_{Z_2}=-\dot{f}\ddot{f}{g}_2({\tau }_{Y_2},{\tau }_{Z_2})\partial t,$

5.

$\overline{R}({\tau }_{X_2},{\tau }_{Y_3}){\tau }_{Z_2}={\displaystyle \frac{1}{h}}{H}^h({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_3},$

6.

$\overline{R}({\tau }_{\partial t},{\tau }_{Y_3}){\tau }_{Z_3}=-h{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\overline{\nabla }}_{\partial t}\mathrm{grad}h,$

7.

$\overline{R}({\tau }_{X_2},{\tau }_{Y_3}){\tau }_{Z_3}=-h{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\overline{\nabla }}_{{\tau }_{X_2}}\mathrm{grad}h,$

8.

$\overline{R}({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}=R_3({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}-{\parallel \mathrm{grad}h\parallel }^2\left[g_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{X_3}-g_3({\tau }_{X_3},{\tau }_{Z_3}){\tau }_{Y_3}\right].$

Lemma 8.

The non-zero components of the Ricci curvature of $(\overline{\mathcal{M}},\overline{g})$ are given by [6]:

1.

$\overline{\mathrm{Ric}}({\tau }_{\partial t},{\tau }_{\partial t})={\displaystyle \frac{m_2}{f}}\ddot{f}+{\displaystyle \frac{m_3}{h}}{\displaystyle \frac{{\partial }^{2}h}{\partial t^2}},$

2.

$\overline{\mathrm{Ric}}({\tau }_{X_2},{\tau }_{Y_2})={\mathrm{Ric}}_2({\tau }_{X_2},{\tau }_{Y_2})-f^{\#}{g}_2({\tau }_{X_2},{\tau }_{Y_2})-{\displaystyle \frac{m_3}{h}}{H}^h({\tau }_{X_2},{\tau }_{Y_2}),$

3.

$\overline{\mathrm{Ric}}({\tau }_{X_3},{\tau }_{Y_3})={\mathrm{Ric}}_3({\tau }_{X_3},{\tau }_{Y_3})-h^{\#}{g}_3({\tau }_{X_3},{\tau }_{Y_3}).$

Here, $f^{\#}=-f\ddot{f}-(m_2-1){\left(\dot{f}\right)}^2,h^{\#}=h\Delta h+(m_3-1){∥\mathrm{grad}h∥}^2.$

Lemma 9.

The relation between the scalar curvature $\overline{r}$ of $(\overline{\mathcal{M}},\overline{g})$ and the scalar curvatures $r_i$ of $({\mathcal{M}}_i,g_i)$, for $i=1,2,3$, is given by [6]:

$$\overline{r}={\displaystyle \frac{r_2}{f^2}}+{\displaystyle \frac{r_3}{h^2}}+{\displaystyle \frac{2m_2}{f}}\ddot{f}-{\displaystyle \frac{2m_3}{h}}\Delta h-{\displaystyle \frac{m_2(m_2-1)}{f^2}}{\left(\dot{f}\right)}^2-{\displaystyle \frac{m_3(m_3-1)}{h^2}}{∥\mathrm{grad}h∥}^2.$$

Based on these lemmas, we obtain the expression for the quasi-concircular curvature tensor associated with the sequential generalized Robertson–Walker spacetime:

Proposition 2.

Consider the sequential generalized Robertson–Walker spacetime $\overline{\mathcal{M}}=\left(I{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$, which has dimension $n=1+m_2+m_3$, and is equipped with the metric $\overline{g}=(-d{t}^2\oplus f^2{g}_2)\oplus h^2{g}_3$. Then, the non-zero components of the quasi-concircular curvature tensor $\overline{V}$ on $\overline{\mathcal{M}}$ are given by:

1.

$$\begin{gathered} \overline{V}({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}=c{R}_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}+\left[c\left(\dot{f}\right)+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2\right]\left[g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_2}- \\ {g}_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_2}\right], \end{gathered}$$

2.

$\overline{V}({\tau }_{\partial t},{\tau }_{Y_2}){\tau }_{\partial t}=-\left[c{\displaystyle \frac{\ddot{f}}{f}}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)\right]{\tau }_{Y_2},$

3.

$\overline{V}({\tau }_{\partial t},{\tau }_{Y_2}){\tau }_{Z_2}=-\left[cf\ddot{f}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2\right]g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{\partial t},$

4.

$\overline{V}({\tau }_{\partial t},{\tau }_{Y_3}){\tau }_{\partial t}=-\left[{\displaystyle \frac{c}{h}}{\displaystyle \frac{{\partial }^{2}h}{\partial t^2}}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)\right]{\tau }_{Y_3},$

5.

$\overline{V}({\tau }_{\partial t},{\tau }_{Y_3}){\tau }_{Z_2}={\displaystyle \frac{c\dot{f}}{f}}{Z}_2\left(\ln h\right){\tau }_{Y_3},$

6.

$\overline{V}({\tau }_{X_2},{\tau }_{Y_3}){\tau }_{Z_2}={\displaystyle \frac{c}{h}}{H}^h({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_3}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2{g}_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_3},$

7.

$\overline{V}({\tau }_{\partial t},{\tau }_{Y_3}){\tau }_{Z_3}=-ch\ddot{f}{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{\partial t}+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h^2{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{\partial t},$

8.

$\overline{V}({\tau }_{X_2},{\tau }_{Y_3}){\tau }_{Z_3}=-ch{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\overline{\nabla }}_{{\tau }_{X_2}}\mathrm{grad}h+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h^2{g}_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{X_2},$

9.

$$\begin{gathered} \overline{V}({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}=c{R}_3({\tau }_{X_3},{\tau }_{Y_3}){\tau }_{Z_3}-\left[c{\parallel \mathrm{grad}h\parallel }^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h^2\right]\left[g_3({\tau }_{Y_3},{\tau }_{Z_3}){\tau }_{X_3}- \\ {g}_3({\tau }_{X_3},{\tau }_{Z_3}){\tau }_{Y_3}\right]. \end{gathered}$$

Theorem 5.

Consider the sequential generalized Robertson–Walker spacetime $\overline{\mathcal{M}}=\left(I{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ of dimension $n=1+m_2+m_3$, equipped with the metric $\overline{g}=\left(-d{t}^2\oplus f^2{g}_2\right)\oplus h^2{g}_3$. Then $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly flat if and only if the following conditions hold:

1.

$({\mathcal{M}}_2,g_2)$ is a space of constant curvature

$${\kappa }_2=-{\displaystyle \frac{1}{a}}\left[c{\left(\dot{f}\right)}^2+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2\right].$$

2.

$({\mathcal{M}}_3,g_3)$ is a space of constant curvature

$${\kappa }_3={∥\mathrm{grad}h∥}^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h^2.$$

3.

Moreover, $h=f$, i.e., the warping function h depends only on the variables of ${\mathcal{M}}_1$.

4.

From (2) and (3) of Proposition 2, we obtain

$$\ddot{f}-{\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f=0,$$

and from (6) and (8) of Proposition 2, we have

$$\dot{f}-{\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f=0.$$

Furthermore, combining (3) and (4) of Theorem 5, we obtain the following result.

Theorem 6.

Let $\overline{\mathcal{M}}=\left(I{\times }_f{\mathcal{M}}_2\right){\times }_h{\mathcal{M}}_3$ be an $n=(1+m_2+m_3)$-dimensional sequential generalized Robertson–Walker spacetime equipped with the metric

$$\overline{g}=(-d{t}^2\oplus f^2{g}_2)\oplus h^2{g}_3.$$

If $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly flat, then the warping functions coincide and satisfy

$$f\left(t\right)=h\left(t\right)={\kappa }_1{e}^t+{\kappa }_2,$$

where ${\kappa }_1,{\kappa }_2\in \mathbb{R}$ are constants.

4.2. Sequential Standard Static Spacetimes

Consider the sequential standard static spacetime $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_{h}I$ with metric $\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2(-d{t}^2)$. Applying the lemmas from Section 2.2, we obtain the following result:

Lemma 10.

The components of the Levi-Civita connection on $(\overline{\mathcal{M}},\overline{g})$ are given by [6]:

1.

${\overline{\nabla }}_{{\tau }_{X_1}}{\tau }_{Y_1}={\nabla }_{{\tau }_{X_1}}^1{\tau }_{Y_1},$

2.

${\overline{\nabla }}_{{\tau }_{X_1}}{\tau }_{Y_2}={\overline{\nabla }}_{{\tau }_{Y_2}}{\tau }_{X_1}={\tau }_{X_1}\left(\ln f\right){\tau }_{Y_2},$

3.

${\overline{\nabla }}_{{\tau }_{X_2}}{\tau }_{Y_2}={\nabla }_{{\tau }_{X_2}}^2{\tau }_{Y_2}-f{g}_2({\tau }_{X_2},{\tau }_{Y_2}){\mathrm{grad}}^{1}f,$

4.

${\overline{\nabla }}_{\partial t}{\tau }_{X_i}={\overline{\nabla }}_{{\tau }_{X_i}}{\tau }_{\partial t}={\tau }_{X_i}\left(\ln h\right){\tau }_{\partial t},i=1,2,$

5.

${\overline{\nabla }}_{{\tau }_{\partial t}}{\tau }_{\partial t}=h\mathrm{grad}h.$

Lemma 11.

The non-zero components of the Riemannian curvature of $(\overline{\mathcal{M}},\overline{g})$ are given by [6]:

1.

$\overline{R}({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1}=R_1({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1},$

2.

$\overline{R}({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}=R_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}-{\parallel {\mathrm{grad}}^{1}f\parallel }^2\left[g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_2}-g_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_2}\right],$

3.

$\overline{R}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_1}={\displaystyle \frac{1}{f}}{H}_1^f({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_2},$

4.

$\overline{R}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_2}=-f{g}_2({\tau }_{Y_2},{\tau }_{Z_2}){\nabla }_{{\tau }_{X_1}}^1{\mathrm{grad}}^{1}f,$

5.

$\overline{R}({\tau }_{X_i},{\tau }_{\partial t}){\tau }_{Z_j}={\displaystyle \frac{1}{h}}{H}^h({\tau }_{X_i},{\tau }_{Z_j})\partial t,i=1,2,$

6.

$\overline{R}({\tau }_{X_i},{\tau }_{\partial t}){\tau }_{\partial t}=h{\overline{\nabla }}_{{\tau }_{X_i}}\mathrm{grad}h,i=1,2.$

Lemma 12.

The non-zero components of the Ricci curvature of $(\overline{\mathcal{M}},\overline{g})$ are given by [6]:

1.

$\overline{\mathrm{Ric}}({\tau }_{X_1},{\tau }_{Y_1})={\mathrm{Ric}}_1({\tau }_{X_1},{\tau }_{Y_1})-{\displaystyle \frac{m_2}{f}}{H}_1^f({\tau }_{X_1},{\tau }_{Y_1})-{\displaystyle \frac{1}{h}}{H}^h({\tau }_{X_1},{\tau }_{Y_1}),$

2.

$\overline{\mathrm{Ric}}({\tau }_{X_2},{\tau }_{Y_2})={\mathrm{Ric}}_2({\tau }_{X_2},{\tau }_{Y_2})-f^{\#}{g}_2({\tau }_{X_2},{\tau }_{Y_2})-{\displaystyle \frac{1}{h}}{H}^h({\tau }_{X_2},{\tau }_{Y_2}),$

3.

$\overline{\mathrm{Ric}}({\tau }_{\partial t},{\tau }_{\partial t})=h^{\#}.$

where

$$f^{\#}=f{\Delta }_{1}f+(m_2-1){∥{\mathrm{grad}}^{1}f∥}^2,h^{\#}=h\Delta h.$$

Proposition 3.

Consider the sequential standard static spacetime

$$\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_{h}I,$$

equipped with the metric

$$\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2(-d{t}^2).$$

Then, the non-zero components of the quasi-concircular curvature tensor $\overline{V}$ on $\overline{\mathcal{M}}$ are given by:

1.

$$\begin{array}{c}\hfill \overline{V}({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1}=c{R}_1({\tau }_{X_1},{\tau }_{Y_1}){\tau }_{Z_1}+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)\left[g_1({\tau }_{Y_1},{\tau }_{Z_1}){\tau }_{X_1}-g_1({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_1}\right],\end{array}$$

2.

$$\begin{array}{cc}\hfill \overline{V}({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}& =c{R}_2({\tau }_{X_2},{\tau }_{Y_2}){\tau }_{Z_2}\hfill \\ & -\left[c{∥{\mathrm{grad}}^{1}f∥}^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2\right]\left[g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_2}-g_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{Y_2}\right],\hfill \end{array}$$

3.

$$\overline{V}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_1}={\displaystyle \frac{c}{f}}{H}_1^f({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_2}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)g_1({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{Y_2},$$

4.

$$\overline{V}({\tau }_{X_1},{\tau }_{Y_2}){\tau }_{Z_2}=-cf{g}_2({\tau }_{Y_2},{\tau }_{Z_2}){\nabla }_{{\tau }_{X_1}}^1{\mathrm{grad}}^{1}f+{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)g_2({\tau }_{Y_2},{\tau }_{Z_2}){\tau }_{X_1},$$

5.

$$\overline{V}({\tau }_{X_1},{\tau }_{\partial t}){\tau }_{Z_1}={\displaystyle \frac{c}{h}}{H}_1^h({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{\partial t}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)g_1({\tau }_{X_1},{\tau }_{Z_1}){\tau }_{\partial t},$$

6.

$$\overline{V}({\tau }_{X_2},{\tau }_{\partial t}){\tau }_{Z_2}={\displaystyle \frac{c}{h}}{H}^h({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{\partial t}-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2{g}_2({\tau }_{X_2},{\tau }_{Z_2}){\tau }_{\partial t},$$

7.

$$\overline{V}({\tau }_{X_1},{\tau }_{\partial t}){\tau }_{Z_2}={\displaystyle \frac{c}{h}}{H}^h({\tau }_{X_1},{\tau }_{Z_2}){\tau }_{\partial t},$$

8.

$$\overline{V}({\tau }_{X_i},{\tau }_{\partial t}){\tau }_{\partial t}=-ch{\overline{\nabla }}_{{\tau }_{X_i}}\mathrm{grad}h-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h^2{\tau }_{X_i},i=1,2.$$

By virtue of Proposition 3 we obtain the following theorem:

Theorem 7.

Let $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_{h}I$ be a sequential standard static spacetime equipped with the metric $\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2(-d{t}^2)$. Then, the manifold $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly flat if and only if the following conditions hold:

1.

$({\mathcal{M}}_1,g_1)$ has constant curvature

$${\kappa }_1=-{\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right).$$

2.

$({\mathcal{M}}_2,g_2)$ has constant curvature

$${\kappa }_2={\displaystyle \frac{1}{c}}\left({∥{\mathrm{grad}}^{1}f∥}^2-{\displaystyle \frac{\overline{r}}{n}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2\right).$$

3.

On ${\mathcal{M}}_1$, the following relations hold:

$$H_1^f-{\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f{g}_1=0,H^h-{\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)h{g}_1=0.$$

4.

On ${\mathcal{M}}_2$, we have

$$H^h-{\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2{g}_2=0.$$

5.

For all ${\tau }_{X_1}\in \mathfrak{X}\left({\mathcal{M}}_1\right)$ and ${\tau }_{Z_2}\in \mathfrak{X}\left({\mathcal{M}}_2\right)$,

$$H^h({\tau }_{X_1},{\tau }_{Z_2})=0.$$

Next, suppose the warping function h can be expressed as the sum $h=h_1+h_2$, where each $h_i\in C^{\infty }\left({\mathcal{M}}_i\right)$ for $i=1,2$. Then, by the final part of Theorem 7, we have $X_1\left(\ln f\right)\mathrm{grad}{h}_2=0$, which implies that h depends only on ${\mathcal{M}}_1$. Under this condition, statements (3) and (4) of Theorem 7 lead to the following result:

Theorem 8.

Consider the sequential standard static spacetime $\overline{\mathcal{M}}=\left({\mathcal{M}}_1{\times }_f{\mathcal{M}}_2\right){\times }_{h}I$, equipped with the metric $\overline{g}=(g_1\oplus f^2{g}_2)\oplus h^2(-d{t}^2)$. If $(\overline{\mathcal{M}},\overline{g})$ is quasi-concircularly flat, then the following hold:

1.

The warping function h is a smooth function defined solely on ${\mathcal{M}}_1$.

2.

The warping function f satisfies

$$f{\nabla }_h\left(f\right)-{\displaystyle \frac{\overline{r}}{cn}}\left({\displaystyle \frac{c}{n-1}}+2d\right)f^2=0.$$

3.

The manifold $({\mathcal{M}}_1,g_1)$ is a conformal gradient soliton, where the warping function h serves as its potential function.

4.3. Spatially Homogeneous and Isotropic Sequential Warped Product Spacetime

Example 1.

Consider the spatially homogeneous and isotropic Bianchi type-V metric g on ${\mathbb{R}}^4$ [23,24]:

$$d{s}^2=-d{t}^2+t^{2}d{x}^2+e^{2(x+t)}\left(d{y}^2+d{z}^2\right).$$

with global coordinates $(t,x,y,z)$. For abbreviation, set

$$E:=e^{2(x+t)}.$$

We denote by $g_{ab}$ the metric components, by $g^{ab}$ the inverse metric, by ${\mathrm{\Gamma }}_{bc}^a$ the Levi–Civita connection, by ${R^a}_{bcd}$ the Riemann curvature, by $R_{ab}$ the Ricci tensor, and by r the scalar curvature.

The nonzero metric components (in coordinate order $t,x,y,z$) are

$$g_{tt}=-1,g_{xx}=t^2,g_{yy}=g_{zz}=E.$$

The inverse metric has diagonal components of the form:

$$g^{tt}=-1,g^{xx}={\displaystyle \frac{1}{t^2}},g^{yy}=g^{zz}=E^{-1}.$$

Only the following derivatives are nonzero:

$${\partial }_t{g}_{xx}=2t,{\partial }_t{g}_{yy}={\partial }_t{g}_{zz}=2E,$$

$${\partial }_x{g}_{yy}={\partial }_x{g}_{zz}=2E.$$

The Levi–Civita symbols are computed from

$${{\mathrm{\Gamma }}^a}_{bc}={\displaystyle \frac{1}{2}}{g}^{ad}\left({\partial }_b{g}_{dc}+{\partial }_c{g}_{db}-{\partial }_d{g}_{bc}\right).$$

Only the following Christoffel symbols are nonzero (we list each with the short derivation):

$$\begin{array}{cccc}\hfill {{\mathrm{\Gamma }}^t}_{xx}& =t,\hfill & \hfill {{\mathrm{\Gamma }}^t}_{yy}& ={{\mathrm{\Gamma }}^t}_{zz}=E,\hfill \\ \hfill {{\mathrm{\Gamma }}^x}_{tx}& ={{\mathrm{\Gamma }}^x}_{xt}={\displaystyle \frac{1}{t}},\hfill & \hfill {{\mathrm{\Gamma }}^x}_{yy}& ={{\mathrm{\Gamma }}^x}_{zz}=-{\displaystyle \frac{E}{t^2}},\hfill \\ \hfill {{\mathrm{\Gamma }}^y}_{ty}& ={{\mathrm{\Gamma }}^y}_{yt}={{\mathrm{\Gamma }}^y}_{xy}={{\mathrm{\Gamma }}^y}_{yx}=1,\hfill & \hfill {{\mathrm{\Gamma }}^z}_{tz}& ={{\mathrm{\Gamma }}^z}_{zt}={{\mathrm{\Gamma }}^z}_{xz}={{\mathrm{\Gamma }}^z}_{zx}=1.\hfill \end{array}$$

All other Christoffel components vanish. We use the coordinate formula

$${R^a}_{bcd}={\partial }_c{{\mathrm{\Gamma }}^a}_{db}-{\partial }_d{{\mathrm{\Gamma }}^a}_{cb}+{{\mathrm{\Gamma }}^a}_{ce}{{\mathrm{\Gamma }}^e}_{db}-{{\mathrm{\Gamma }}^a}_{de}{{\mathrm{\Gamma }}^e}_{cb},$$

and lower the first index with the metric $R_{abcd}=g_{ae}{R^e}_{bcd}$.

We show three representative computations (all other independent components are obtained similarly). Computation of $R_{tyty}$ is given as

$$\begin{array}{cc}\hfill {R^t}_{yty}& ={\partial }_t{{\mathrm{\Gamma }}^t}_{yy}-{\partial }_y{{\mathrm{\Gamma }}^t}_{ty}+{{\mathrm{\Gamma }}^t}_{te}{{\mathrm{\Gamma }}^e}_{yy}-{{\mathrm{\Gamma }}^t}_{ye}{{\mathrm{\Gamma }}^e}_{ty}=E.\hfill \end{array}$$

Hence

$$R_{tyty}=g_{tt}{R^t}_{yty}=-1·E=-E.$$

Also, $R_{tyxy}$ is given by

$$\begin{array}{cc}\hfill {R^t}_{yxy}& ={\partial }_x{{\mathrm{\Gamma }}^t}_{yy}-{\partial }_y{{\mathrm{\Gamma }}^t}_{xy}+{{\mathrm{\Gamma }}^t}_{xe}{{\mathrm{\Gamma }}^e}_{yy}-{{\mathrm{\Gamma }}^t}_{ye}{{\mathrm{\Gamma }}^e}_{xy}=E{\displaystyle \frac{t-1}{t}}.\hfill \end{array}$$

Thus

$$R_{tyxy}=g_{tt}{R^t}_{yxy}=-E{\displaystyle \frac{t-1}{t}}.$$

Finally, we get $R_{xyxy}$ as:

$$\begin{array}{cc}\hfill {R^x}_{yxy}& ={\partial }_x{{\mathrm{\Gamma }}^x}_{yy}-{\partial }_y{{\mathrm{\Gamma }}^x}_{xy}+{{\mathrm{\Gamma }}^x}_{xe}{{\mathrm{\Gamma }}^e}_{yy}-{{\mathrm{\Gamma }}^x}_{ye}{{\mathrm{\Gamma }}^e}_{xy}=E\left({\displaystyle \frac{t-1}{t^2}}\right).\hfill \end{array}$$

Lowering the first index:

$$R_{xyxy}=g_{xx}{R^x}_{yxy}=t^2·E{\displaystyle \frac{t-1}{t^2}}=(t-1)E.$$

Collecting the independent components (others follow by the algebraic symmetries $R_{abcd}=-R_{bacd}=-R_{abdc}=R_{cdab}$), we obtain

$$\begin{array}{cccc}\hfill R_{tyty}& =-E,\hfill & \hfill R_{tztz}& =-E,\hfill \\ \hfill R_{tyxy}& =-{\displaystyle \frac{t-1}{t}}E,\hfill & \hfill R_{tzxz}& =-{\displaystyle \frac{t-1}{t}}E,\hfill \\ \hfill R_{xyxy}& =(t-1)E,\hfill & \hfill R_{xzxz}& =(t-1)E,\hfill \\ \hfill R_{yzyz}& ={\displaystyle \frac{t^2-1}{t^2}}{E}^2.\hfill \end{array}$$

The Ricci tensor is the contraction $R_{ab}={R^c}_{acb}$. We compute two representative components and then give the remaining nonzero components. The first component is $R_{tt}$. Use the coordinate expression

$$R_{tt}={\partial }_c{{\mathrm{\Gamma }}^c}_{tt}-{\partial }_t{{\mathrm{\Gamma }}^c}_{ct}+{{\mathrm{\Gamma }}^c}_{cd}{{\mathrm{\Gamma }}^d}_{tt}-{{\mathrm{\Gamma }}^c}_{td}{{\mathrm{\Gamma }}^d}_{ct}.$$

Since ${{\mathrm{\Gamma }}^c}_{tt}=0$ for all c, the first and third terms vanish. Thus

$$\begin{array}{cc}\hfill R_{tt}& =-{\partial }_t\left({{\mathrm{\Gamma }}^t}_{tt}+{{\mathrm{\Gamma }}^x}_{xt}+{{\mathrm{\Gamma }}^y}_{yt}+{{\mathrm{\Gamma }}^z}_{zt}\right)-{{\mathrm{\Gamma }}^c}_{td}{{\mathrm{\Gamma }}^d}_{ct}\hfill \\ & =-{\partial }_t\left(0+{\displaystyle \frac{1}{t}}+1+1\right)-\left({{\mathrm{\Gamma }}^x}_{tx}{{\mathrm{\Gamma }}^x}_{xt}+{{\mathrm{\Gamma }}^y}_{ty}{{\mathrm{\Gamma }}^y}_{yt}+{{\mathrm{\Gamma }}^z}_{tz}{{\mathrm{\Gamma }}^z}_{zt}\right)=-2.\hfill \end{array}$$

The second component is $R_{tx}$. Using

$$R_{tx}={\partial }_c{{\mathrm{\Gamma }}^c}_{tx}-{\partial }_x{{\mathrm{\Gamma }}^c}_{ct}+{{\mathrm{\Gamma }}^c}_{cd}{{\mathrm{\Gamma }}^d}_{tx}-{{\mathrm{\Gamma }}^c}_{xd}{{\mathrm{\Gamma }}^d}_{ct},$$

and the nonzero Christoffel symbols listed above, a direct substitution and grouping of nonzero contributions yields

$$R_{tx}=-2+{\displaystyle \frac{2}{t}}.$$

Carrying out the analogous computations for the other components we obtain

$$\begin{array}{cc}\hfill R_{tt}& =-2,\hfill \\ \hfill R_{tx}=R_{xt}& =-2+{\displaystyle \frac{2}{t}},\hfill \\ \hfill R_{xx}& =2(t-1),\hfill \\ \hfill R_{yy}=R_{zz}& ={\displaystyle \frac{2t^2+t-2}{t^2}}E.\hfill \end{array}$$

In matrix form (order $t,x,y,z$):

$$\left(R_{ab}\right)=\left(\begin{array}{cccc}-2& -2+{\displaystyle \frac{2}{t}}& 0& 0\\ -2+{\displaystyle \frac{2}{t}}& 2(t-1)& 0& 0\\ 0& 0& {\displaystyle \frac{2t^2+t-2}{t^2}}E& 0\\ 0& 0& 0& {\displaystyle \frac{2t^2+t-2}{t^2}}E\end{array}\right).$$

The scalar curvature (trace of the Ricci tensor) is

$$\begin{array}{cc}\hfill r& =g^{ab}{R}_{ab}=g^{tt}{R}_{tt}+g^{xx}{R}_{xx}+g^{yy}{R}_{yy}+g^{zz}{R}_{zz}\hfill \\ & =(-1)(-2)+{\displaystyle \frac{1}{t^2}}·2(t-1)+E^{-1}·2·{\displaystyle \frac{2t^2+t-2}{t^2}}E\hfill \\ & =2+{\displaystyle \frac{2(t-1)}{t^2}}+{\displaystyle \frac{2(2t^2+t-2)}{t^2}}\hfill \\ & =2+{\displaystyle \frac{2t-2+4t^2+2t-4}{t^2}}=2+{\displaystyle \frac{4t^2+4t-6}{t^2}}\hfill \\ & =6+{\displaystyle \frac{4}{t}}-{\displaystyle \frac{6}{t^2}}.\hfill \end{array}$$

For dimension $m=4$ the quasi-concircular curvature tensor is defined by

$$V_{abcd}=c{R}_{abcd}+K\left(g_{ad}{g}_{bc}-g_{ac}{g}_{bd}\right),$$

where $c,d$ are constants and

$$K={\displaystyle \frac{r}{4}}\left({\displaystyle \frac{c}{3}}+2d\right).$$

Using the scalar curvature computed above,

$${\displaystyle \frac{r}{4}}={\displaystyle \frac{1}{4}}\left(6+{\displaystyle \frac{4}{t}}-{\displaystyle \frac{6}{t^2}}\right)={\displaystyle \frac{3}{2}}+{\displaystyle \frac{1}{t}}-{\displaystyle \frac{3}{2t^2}},$$

so

$$K=\left({\displaystyle \frac{3}{2}}+{\displaystyle \frac{1}{t}}-{\displaystyle \frac{3}{2t^2}}\right)\left({\displaystyle \frac{c}{3}}+2d\right).$$

Combining with the independent Riemann components listed above, the nonzero independent components of $V_{abcd}$ are

$$\begin{array}{cc}\hfill V_{tyty}& =c{R}_{tyty}+K(-g_{tt}{g}_{yy})=-cE+KE=E(-c+K),\hfill \\ \hfill V_{tztz}& =E(-c+K),\hfill \\ \hfill V_{tyxy}& =a{R}_{tyxy}=-c{\displaystyle \frac{t-1}{t}}E,\hfill \\ \hfill V_{tzxz}& =-c{\displaystyle \frac{t-1}{t}}E,\hfill \\ \hfill V_{xyxy}& =c{R}_{xyxy}-K\left(g_{xx}{g}_{yy}\right)=a(t-1)E-K{t}^{2}E=E\left(c(t-1)-K{t}^2\right),\hfill \\ \hfill V_{xzxz}& =E\left(c(t-1)-K{t}^2\right),\hfill \\ \hfill V_{yzyz}& =c{R}_{yzyz}-K\left(g_{yy}{g}_{zz}\right)=E^2\left({\displaystyle \frac{c(t^2-1)}{t^2}}-K\right).\hfill \end{array}$$

These components satisfy the usual algebraic symmetries

$$V_{abcd}=-V_{bacd}=-V_{abdc}=V_{cdab}.$$

If one chooses $c=1,d=-\frac{1}{3}$ then ${\displaystyle \frac{c}{3}}+2d=-\frac{1}{3}$ and $K=-{\displaystyle \frac{r}{12}}$; in this choice V reduces to the usual concircular curvature tensor used in the literature.

5. Conclusions

In this work, we have examined sequential warped product manifolds through the lens of their quasi-concircular curvature tensor. By extending classical concepts from differential geometry, we derived explicit expressions and identified the precise conditions under which these manifolds are quasi-concircularly flat. Our analysis revealed how the geometry of each component—the base, the first fiber, and the second fiber—interacts via the warping functions to determine the overall curvature properties of the manifold.

We showed that quasi-concircular flatness not only ensures constant scalar curvature but also renders the manifold Einstein, highlighting its significance from both mathematical and physical perspectives. This is particularly relevant in the study of spacetime models, as illustrated by examples including sequential generalized Robertson-Walker and sequential standard static spacetimes. The Bianchi type-V model, in particular, demonstrates how these theoretical results can be realized in geometries of interest in cosmology and general relativity.

Overall, this study contributes to a deeper understanding of the behavior of complex curvature tensors in layered geometric spaces. Looking ahead, several directions of research appear promising. On the mathematical side, one could investigate the interplay between quasi-concircular curvature and other geometric flows (such as the Ricci, Yamabe, or Ricci–Bourguignon flows), or examine its role in the classification of new soliton solutions (Ricci–Yamabe solitons, almost conformal Ricci solitons, etc.) on sequential warped products. Connections with conformal and projective structures also suggest rich possibilities in understanding curvature invariants and rigidity results.

On the physical side, the framework developed here could be applied to models of modified gravity, where curvature conditions play a central role in constructing cosmological and black hole solutions. Sequential warped products may also provide geometric backgrounds for higher-dimensional theories and string theory compactifications, where layered structures naturally arise. Moreover, the study of quasi-concircular flatness could yield new insights into energy conditions, stability criteria, and the behavior of anisotropic cosmological models.