Black holes are among the most extraordinary phenomena in the universe. Despite significant progress in understanding black hole dynamics and solving Einstein’s field equations in complex scenarios, their deeper implications for space, time, causality, and information remain elusive. The identification of black hole solutions in the 1960s and the recent confirmation of their existence through gravitational wave detection and the direct imaging of the M87 black hole’s event horizon mark key milestones with profound philosophical implications. Defined as gravitationally collapsed objects, black holes are causally disconnected from the rest of the universe, with events inside them unable to influence the outside world. This raises the challenge of defining the boundary between the black hole’s interior and exterior. Since their discovery, black holes have driven the exploration of gravity theories and sparked profound questions about space, time, and the universe. These inquiries have led to theories seeking to unify the fundamental forces, with black holes, sitting at the intersection of classical and quantum effects, being of particular interest in the study of quantum gravity.

An essential feature for cataloging black holes is uniqueness. It has been shown (see [1], for example) that under modest conditions, such as asymptotic flatness and stationarity, 3+1 black holes are completely determined by their mass, angular momentum, and electromagnetic charge. The Kerr–Newman (KN) solution to the 3+1 Einstein–Maxwell equations represents the most general black hole, as it includes mass, angular momentum and electric charge. The previously discovered solutions—Schwarzschild [2] (mass only), Reissner–Nordström [3] (mass and charge), and Kerr [4] (mass and angular momentum) are limiting cases of the KN solution. Consequently, the properties of the KN solution are relevant for all 3+1 black holes. Notably, these black hole spacetimes contain a null surface, a topological two-sphere  [5] known as the event horizon. This surface serves as a one-way boundary that separates the interior of the black hole from events outside the horizon.

Regular black holes (RBHs) are a class of black holes that possess coordinate singularities (horizons) but lack essential singularities throughout the entire spacetime. Typically, the approach to defining an RBH [6], [7] involves ensuring that the spacetime has finite curvature invariants everywhere, particularly at the center of the black hole. This concept is related to Markov’s limiting curvature conjecture [8], [9], which posits that curvature invariants should be uniformly bounded by a certain universal value.

The study of RBHs can be traced back to the works of Sakharov and Gliner, who proposed that essential singularities could be avoided by replacing the vacuum with a vacuum-like medium characterized by a de Sitter metric. This concept was further developed by several researchers. The first RBH model was introduced by Bardeen [10], known as the Bardeen black hole. In this model, the mass of a Schwarzschild black hole is replaced with an -dependent function. Consequently, the essential singularity associated with the Kretschmann scalar is eliminated in the Bardeen black hole, and its core exhibits de Sitter characteristics, meaning the Ricci curvature is positive near the center of the black hole. In cylindrical symmetry, Lemos presented the first RBH solution in general relativity [11], and now known as the Lemos black hole. This black hole is an exact vacuum solution to the field equations with a negative cosmological constant, that is, an example of anti-de Sitter (AdS) spacetime.

In , Lemos made a significant contribution to the field of general relativity by developing a famous exact solution for a regular black hole spacetime. This exact solution to Einstein’s field equations provides a mathematical description of the gravitational field around black holes under specific conditions, including a negative cosmological constant. The line element describing this spacetime in cylindrical coordinates  with a metric signature of  is given by [11] where the non-zero components of the metric tensor  are given by: with  being the linear mass of the black hole,  is a positive constant, –, ,  and – defining the temporal, radial, angular and axial coordinates, respectively.

The study of black hole theory has garnered significant interest for a long time. While classical black hole theory, based on general relativity, neglects quantum effects at small scales, the theory itself continues to develop and improve gradually. In 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO) made a groundbreaking discovery by detecting gravitational waves from a binary black hole merger [12], [13], which confirmed the existence of binary black hole systems in nature [14], [15]. This observation has provided a powerful impetus for further research into the dynamics of binary black hole systems. These systems are highly dynamic, and it is essential to describe their behavior using numerical relativity to capture the complexity of their interactions. In addition to gravitational wave observations, the existence of black holes has been further confirmed by the Event Horizon Telescope (EHT) collaboration. In 2019, the EHT captured the first-ever image of the shadow of , a supermassive black hole at the center of the nearby galaxy Messier , offering direct visual evidence of black holes [16], [17], [18], [19], [20], [21], [22]. However, while these observational advancements have significantly expanded our understanding of black holes, many fundamental properties of black holes, such as Hawking radiation and their entropy, still remain to be tested and understood more fully.

Symmetry is a cornerstone of modern physics, appearing in various forms, particularly as the symmetry of a theory and the symmetry of specific solutions within that theory. In gravitational physics, symmetries of solutions, referred to as isometries, are represented by Killing vectors. A symmetry in a physical theory determined by its action and initial or boundary conditions remains invariant under transformations that map one solution to another. These transformations can be either discrete or continuous, with the latter often described by Lie groups. Continuous symmetries are further classified as either global or local. Local symmetries in particular allow transformations that can depend on arbitrary functions of spacetime, which leads to the formulation of gauge theories. In such theories, different solutions can be connected through local symmetries, and physical theories assume that these solutions are equivalent, forming equivalence classes of gauge solutions. As such, gauge symmetries represent redundant degrees of freedom in the theory. Emmy Noether’s theorems provide a deep connection between symmetries and conservation laws. The first theorem links conserved charges (such as energy or momentum) to continuous symmetries, while the second constrains the form of field equations in gauge theories. Noether’s work has become foundational in theoretical physics, solidifying the idea that symmetries underpin the conservation of physical quantities and shaping the structure of modern physical theories. In the context of general relativity, the curvature tensor is an intrinsic geometric property of spacetime, quantifying its curvature. It describes the degree to which spacetime deviates from being flat, serving as a key element in understanding the gravitational field. The curvature tensor, along with the Christoffel symbols, can be used to calculate the curvature in Riemannian geometry. This tensor not only characterizes the “flatness” of space but also provides insight into the relative acceleration between two nearby points in spacetime. This behavior is captured by the geodesic deviation equation, which describes how the separation between nearby geodesics (trajectories of freely falling particles) changes over time, revealing the influence of spacetime curvature on the motion of objects.

Consider a smooth manifold  characterized by the metric  with signature  or , where  forms an open connected subset of  and , here,  ranges from  to . For , the semi-Riemannian manifold  is Lorentzian; for , it is Riemannian. A spacetime is defined as a -dimensional connected Lorentzian manifold. The manifold  is further characterized by the Riemann curvature tensor , the Ricci curvature tensor , the scalar curvature , the Levi-Civita connection , and additional tensors such as the conformal curvature tensor , the projective curvature tensor , the concircular curvature tensor , and the conharmonic curvature tensor , respectively. It may be mentioned that under conformal transformation, the angle between two intersecting curves remains invariant both in magnitude and orientation, and the conformal curvature tensor is the invariant of conformal transformation on . Again, projective transformation is a geodesic preserving map, whereas the concircular and conharmonic transformations are, respectively, geodesic circle and harmonic function-preserving maps, and the respective curvature tensors are the invariants of such mappings on . Throughout the paper by the manifold , we mean a semi-Riemannian manifold, unless stated otherwise.

In general relativity, the energy–momentum tensor of rank  provides insight into the nature of the sources present in spacetime. This tensor deeply influences the geometry of the underlying space, with its Ricci curvature, scalar curvature, and the metric tensor shaping the spacetime geometry. Together, these components describe the curvature and overall structure of the space, which is crucial for understanding its geometric properties. It is important to note that the scalar curvature represents the trace of the Ricci curvature, while the Ricci curvature itself is the trace of the Riemann curvature tensor. Therefore, we assume that the geometric quantity “curvature” and its characteristics reveal various physical properties in space. For instance, a Brinkmann wave is determined to be a pp-wave by the condition  [23], [24], [25], and these spacetimes with pseudosymmetric Weyl tensors are of Petrov type D [26]. Refs. [27] demonstrate that a quasi-Einstein spacetime corresponds to a perfect fluid solution and vice versa. Specifically, we identify the perfect fluid Friedmann–Lemaître–Robertson–Walker (FLRW) [28] spacetime as quasi-Einstein, highlighting its unique geometric properties and its role in cosmological models describing a homogeneous and isotropic universe.

On the other hand, the geometry of a semi-Riemannian space is fundamentally shaped by its curvature, intricately linked to the covariant derivatives of various tensors, up to different orders. The condition , indicating a locally symmetric space [29], implies that at each point in the manifold , the geometry exhibits symmetry where all local geodesic symmetries are isometric. This property is foundational in differential geometry, highlighting the uniformity of geometric properties across nearby points. Ruse [30], [31], [32], [33] expanded upon this concept by introducing “kappa” spaces, which relax this stringent condition to explore broader classes of geometric structures. Initially emerging from investigations into preserving curvature under parallel transport, “kappa” spaces allow for scaling and were later recognized as a significant class of structures. Walker [34] called them recurrent manifolds, and subsequent research has introduced various families of manifolds that extend and refine the concept of recurrence. For those who are interested in exploring these manifold families in detail, articles such as [35] provide comprehensive discussions. Shirokov [36] previously investigated spaces characterized by covariantly constant tensors with additional insights available in related literature [37].

The integrability conditions arising from the equations  are succinctly expressed by the formula . These conditions have been extensively studied in the works of Cartan [38] and Shirokov [36], where they were analyzed in depth. When , the manifold  is classified as semisymmetric. We note that a semi-Riemannian manifold is semisymmetric if and only if under parallel transport of any plane  at any point  around any infinitesimal coordinate parallelogram centered at , its sectional curvature  is invariant up to second order (see [39], [40], [41]). The specific conditions under which  are thoroughly explored in Sinyukov’s work [42], where  denotes any smooth function defined on the subset , with  representing the Tachibana tensor. Geometrically, the Tachibana tensor  measures the change in sectional curvatures for any plane under the action of an infinitesimal rotation without leaving that location [41]. These conditions are referred to as generalized symmetric and are explained in detail in [43], [44], with more concise equations provided in [45], [46], [47].

Adamów and Deszcz [48] introduced a second order-symmetry extending beyond semisymmetry, stemming from their profound exploration of symmetrical properties within Einstein’s field equations and detailed examination of totally umbilical hypersurfaces. This development greatly enhances our comprehension of geometric structures and their symmetries in the fields of differential geometry and mathematical physics. We note that all -dimensional Thurston geometries are either spaces of constant curvature or Deszcz pseudosymmetric spaces of constant type, and also all -dimensional d’Atri spaces are Deszcz pseudosymmetric spaces of constant type. The class of pseudosymmetric spaces is mostly anisotropic spaces. Pseudosymmetry also exhibits in the design of higher symmetry crystals of racemic compounds and also in the chirality in molecules, in nature, and in the cosmos. The concept of pseudosymmetry is exemplified in numerous spacetimes, such as Robertson–Walker spacetimes [49], [50], Schwarzschild [2], and the Reissner–Nordström black hole [3], among others, serving as models of pseudosymmetric manifolds. The concepts of local symmetry, semisymmetry, and pseudosymmetry are skillfully expounded upon by Haesen and Verstraelen in [40], [41], [51]. Deszcz’s notion of pseudosymmetry has been a focal point in general relativity and cosmology for the past forty years, drawing significant attention for its wide-ranging applications. Numerous spacetimes have been identified as exemplifying pseudosymmetry, as documented in [49], [50], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], continuing to play a crucial role in advancing our understanding of geometric and physical properties within theoretical frameworks. Furthermore, Mikeš et al. [43], [45], [46], [47], [62] conducted comprehensive investigations into geodesic mappings across various symmetric Riemannian manifolds. It is noteworthy that the concepts of Deszcz and Chaki pseudosymmetries may overlap, as discussed in [63], contributing to a nuanced understanding of the geometric properties and structures in spacetime analysis.

In 1982, mathematician Richard Hamilton developed the Ricci flow process as a method to study how Riemannian metrics change over time on compact three-dimensional manifolds with positive Ricci curvature [64]. The evolution of solutions to the Ricci flow in a self-similar manner gives rise to what are known as Ricci solitons. These solitons represent an extension of the concept of Einstein metrics and have been the subject of extensive study [23], [65]. Over time, the concept of Ricci solitons has been expanded to encompass various generalizations, including almost Ricci solitons, -Ricci solitons, and almost -Ricci solitons.

A semi-Riemannian manifold  is defined as a Ricci soliton if its Ricci curvature  and metric tensor  satisfy the equation £where  is a constant and £ represents the Lie derivative in the direction of the soliton vector field . The nature of the Ricci soliton depends on : it is expanding if , steady if , and shrinking if . If  is a non-constant smooth function, then  is called an almost Ricci soliton.

If the soliton vector field  of a Ricci soliton is a Killing field, the Ricci soliton becomes an Einstein manifold. Furthermore,  is termed as -Ricci soliton [66] if there exists a non-zero -form  on  and constants  and  such that £When  and  are allowed to vary as smooth functions on , the -Ricci soliton is referred to as an almost -Ricci soliton [67].

In 1988, Hamilton [64] introduced the concept of Yamabe flow in conjunction with Ricci flow. More recently, Güler and Crăşmăreanu [68] developed a combined geometric flow called Ricci–Yamabe flow, which combines aspects of both Ricci and Yamabe flows. The self-similar solutions of the Ricci–Yamabe flow are known as Ricci–Yamabe solitons, while the self-similar solutions of the Yamabe flow are termed Yamabe solitons. In a semi-Riemannian manifold , if the Ricci curvature  and the metric tensor  satisfy the equation £where , , and  are constants,  is the scalar curvature, and  is the soliton vector field, then  is called a Ricci–Yamabe soliton [69]. This concept of Ricci–Yamabe solitons enriches the study of geometric flows by providing a unified framework that incorporates both Ricci and Yamabe solitons, offering deeper insights into the behavior and properties of manifolds under these combined flows. The development of Ricci–Yamabe flow represents a significant advancement in geometric analysis, building on Hamilton’s foundational work and extending it to explore new directions and applications in the study of semi-Riemannian manifolds. In the context of semi-Riemannian manifolds, specific configurations of constants ,  and  lead to distinct geometric structures known as solitons. For instance, setting , and  corresponds to a Yamabe soliton, while , and  corresponds to a Ricci soliton. These solitons represent critical solutions under the respective geometric flows. Moreover, allowing , , and  to vary as non-constant smooth functions characterizes an almost Ricci–Yamabe soliton [69]. This generalization broadens the scope to include more flexible and varied geometric behaviors within the manifold.

Additionally, introducing a non-zero 1-form  that satisfies £where  is a constant, defines an -Ricci–Yamabe soliton [69]. Extending these constants to smooth functions allows for the exploration of almost -Ricci–Yamabe solitons, showcasing richer geometric structures and their implications in differential geometry. Over the past three decades, numerous research papers have emerged on Ricci solitons, Yamabe solitons, and their various generalizations (see, for example, [70] and references therein). These studies have propelled the topic to the forefront of research in modern differential geometry.

The geometry of the LBH spacetime, however, remains underexplored in the existing literature, with limited documentation regarding its detailed geometric properties. This gap in understanding makes the LBH spacetime an intriguing subject for further investigation, particularly concerning its curvature-related characteristics. The curvature properties of a spacetime provide essential insights into its underlying geometric structure and its behavior under gravitational interactions, which are crucial for understanding the dynamical and physical features of black holes. In the context of differential geometry, the study of curvature involves examining tensors that describe how spacetime is curved or deviates from flatness. These tensors, such as the Riemann curvature tensor, Ricci tensor, and Ricci scalar, serve as fundamental objects in characterizing the intrinsic curvature of spacetime. This paper specifically aims to elucidate the curvature properties inherent to the LBH spacetime, using differential geometric methods to uncover the underlying structure. Through rigorous analysis, we show that LBH spacetime does not exhibit semisymmetric properties, which are often associated with spacetimes that possess certain symmetries in their curvature tensor, such as the tensor being symmetric in its lower indices. Instead, it is shown that LBH spacetime qualifies as pseudosymmetric, which represents a more general class of spacetimes in which the curvature tensor does not necessarily adhere to the constraints of semisymmetry but still exhibits certain symmetry-like characteristics under specific transformations. Pseudosymmetry, in differential geometry, refers to the scenario where the curvature tensor of a spacetime satisfies certain conditions similar to those of semisymmetric spaces, but with an important distinction: the curvature tensor is not entirely symmetric under index exchange. Specifically, pseudosymmetric spaces exhibit a form of “partially” symmetric behavior in the curvature tensor, leading to distinctive geometric features, including unique propagation behaviors for gravitational waves or other perturbations. The curvature tensor  of a manifold  is determined by all its sectional curvatures and  is of constant curvature, (i.e., space form) if sectional curvatures are constant, i.e., sectional curvatures are independent with respect to the sections by the planes as well as points. Again, the quantity  represents the measures of the changes of the sectional curvatures under parallel transport around any infinitesimal coordinate parallelograms. For a space form, the Tachibana tensor vanishes, and hence it measures the changes of the sectional curvatures under infinitesimal rotations of the planes with respect to other planes. Let , where  is of non-constant curvature. If , then a plane  () is said to be curvature dependent with regard to another plane  () at a point p ,  being the tangent space at any point . Furthermore, if  is curvature dependent on , then the scalar is called the Deszcz sectional curvature or double sectional curvature [39], [40], [41], [51] of the plane  with respect to  at . A semi-Riemannian manifold is pseudosymmetric in terms of Deszcz sectional curvature if at every point ,  is independent of the planes  and . We note that under a geodesic transformation, a semisymmetric space transforms to a pseudosymmetric space, and also, pseudosymmetry remains invariant under such a transformation. The classification of pseudosymmetric spacetimes has been divulged by Haesen and Verstraelen [71]. In the case of LBH spacetime, the pseudosymmetric nature manifests through various curvature conditions that describe how the spacetime’s curvature behaves differently compared to standard symmetric or semisymmetric spaces. These conditions might involve the interplay between the components of the Riemann curvature tensor, the Ricci tensor, and the metric, revealing nontrivial relationships between spacetime geometry and the distribution of matter and energy. Further analysis of the LBH spacetime’s pseudosymmetric curvature conditions reveals a rich and complex structure, which may have profound implications for understanding the nature of gravitational fields in higher-dimensional spacetimes. The examination of such curvature conditions can provide deeper insights into the dynamical stability of the black hole, the propagation of gravitational waves in this background, and the broader impact of higher-dimensional gravitational theories. By exploring these curvature-related features, this paper aims to contribute to a more comprehensive understanding of the LBH spacetime and its geometric properties within the broader framework of differential geometry and general relativity. In conclusion, the pseudosymmetric curvature properties of the LBH spacetime present a fascinating area of study in differential geometry. The investigation into these properties not only deepens our understanding of the LBH black hole solutions but also enriches the broader field of geometric analysis in gravitational theories. It is noteworthy to mention that LBH spacetime admits pseudosymmetric Weyl tensor  which is invariant under conformal transformation of the metric. The commutator tensor  shows a linear dependence on the tensors  and , which is relevant to applicable is conformal field theory (CFT). This spacetime is neither classified as a Roter-type manifold nor as a generalized Roter-type manifold. Moreover, the Ricci tensor of the LBH spacetime exhibits cyclic parallelism and Codazzi type properties. Analysis using the Tachibana tensor associated with the energy–momentum tensor  demonstrates that the spacetime satisfies various types of pseudosymmetric structures.

The curvature properties of black holes play a fundamental role in understanding the structure of spacetime and its singularities. The curvature of a spacetime is infinite at the singularity. In D gravity, LBH provides a simplified framework to study these properties in AdS spacetime, which is crucial for holography and lower-dimensional quantum gravity. In contrast, D regular black holes, such as the BBH, offer insights into singularity resolution through nonlinear electrodynamics. Comparing their curvature properties sheds light on how dimensionality and source fields shape spacetime geometry, paving the way for theoretical advancements in regular black hole models and AdS/CFT dualities.

The comparison of curvature properties between the LBH and BBH spacetimes serves as a powerful tool to understand the influence of dimensionality and source terms on spacetime geometry. In LBH spacetime with a negative cosmological constant, curvature singularities emerge prominently, reflecting the simplicity of D gravity and its limited degrees of freedom. Conversely, the BBH in D provides an elegant example of a regular black hole, avoiding singularities through nonlinear electrodynamics.

This comparison has several applications in theoretical physics. First, it informs the development of lower-dimensional models of black holes that mimic features of their 4D counterparts. For example, by analyzing the curvature regularization in the BBH spacetime, one can propose modifications to the LBH, such as coupling to nonlinear fields, to achieve a regular black hole solution in 3D. Such solutions could play a pivotal role in the study of quantum gravity, where lower-dimensional systems often serve as simplified models.

In addition, the geometric differences in curvature classification between these spacetimes provide insights into the broader structure of gravitational theories. For instance, the LBH, with its singular nature, might belong to distinct mathematical classifications (e.g., different levels of Einstein-type spacetimes), while the BBH, as a regular solution, offers a contrasting framework. Understanding these differences allows researchers to test the robustness of curvature-based spacetime classifications across dimensions.

Another critical application of LBH lies in black hole thermodynamics and holography. The asymptotically AdS nature of the LBH ties directly to the AdS/CFT correspondence, where the curvature properties of the bulk spacetime influence the dual field theory. Comparing this with the curvature regularization of the BBH in asymptotically flat spacetime opens pathways to explore how singularities impact thermodynamic properties, entropy, and quantum field behavior near horizons.

Moreover, the curvature properties determine observable phenomena, such as gravitational lensing and shadow formation. While the singular nature of LBH may lead to distinct observational signatures, the regularity of BBH modifies lensing patterns and shadow characteristics. These differences have practical implications for astrophysical observations, offering potential tests for distinguishing between singular and regular black holes.

In summary, the comparison of the curvature properties of the LBH and the BBH bridges the gap between lower-dimensional theoretical studies and higher-dimensional astrophysical models. It offers a deeper understanding of the role of dimensionality, regularization, and source terms in shaping spacetime geometry, with implications for quantum gravity, holography, black hole thermodynamics, and observational black hole physics.

There are several kinds of symmetry in various black holes, such as axial symmetry, translation symmetry, spherical symmetry, conformal symmetry, Schrödinger symmetry, gauge symmetry, supersymmetry, and also some other hidden symmetries. Axial symmetry refers to invariance under rotations about a specific axis, and such a kind of symmetry is observed in rotating solutions like the Kerr black hole and the Kerr–Newman black hole, which are axially symmetric around their spin axes [72]. Translational symmetry occurs in black holes with planar horizons, such as AdS black branes, where the spacetime is invariant under translations along horizon directions. Spherical symmetry, on the other hand, refers to invariance under rotations in all directions. This symmetry characterizes black holes that are static and have isotropic horizons, such as the Schwarzschild black hole and the Reissner–Nordström black hole [2]. Conformal symmetry involves transformations that preserve angles but not necessarily distances. It appears in the near-horizon geometry of extremal black holes and in the AdS/CFT correspondence, which links gravitational theories in anti-de Sitter (AdS) space to CFTs. Notable examples include the near-horizon geometry of extremal Reissner–Nordström–AdS black holes [73]. Schrödinger symmetry, a non-relativistic symmetry, applies in certain limits of black hole physics, particularly in non-relativistic AdS/CFT correspondence. It relates to scale invariance and time-reversal transformations in spacetime [74]. Gauge symmetry involves invariance under transformations of fields, such as the electromagnetic potential in charged black holes. This type of symmetry is crucial in coupling gravity with other forces, as seen in the Reissner–Nordström solution. Supersymmetry, which links bosons and fermions, leads to unique black hole solutions known as Bogomol’nyi–Prasad–Sommerfield (BPS) black holes. These solutions preserve some fraction of supersymmetry, often exhibit extremality, and possess special properties. For example, the Myers–Perry black hole [75] in  generalizes the Kerr solution with extra rotational parameters.

Again, hidden symmetries [76] in black holes refer to deeper, underlying properties of spacetime that are not immediately apparent from the metric but play a crucial role in simplifying the equations of motion. For instance, in black holes such as the Kerr solution, hidden symmetries are associated with Killing tensors, mathematical objects that uncover conserved quantities beyond those linked to explicit symmetries like stationarity or axial symmetry. These hidden symmetries offer profound insights into the structure and dynamics of spacetime.

It is important to note that the conformal and Riemann curvature operators do not commute in LBH spacetime, and hence it is useful in CFT, and this commutator, basically, produced a Tachibana tensor due to the Weyl conformal curvature tensor. The LBH spacetime exhibits the Ricci symmetry, whereas it does not possess local symmetry and hence its geodesics symmetry at any point is not isometry, but matter distribution obeys a certain kind of symmetry due to its Ricci symmetry. And the matter distribution is incoherent due to its rotational symmetry (cyclic parallel) of the Ricci tensor. Hence the classification of curvature properties will be certainly enriched in the understanding of different kinds of hidden symmetries of the LBH spacetime.

The curvature classification , , , , and  implies various types of hidden symmetries in LBH, which can be related to the Killing tensors. Another interesting fact in LBH spacetime is that it is Ein, and hence its 2nd level Ricci tensor is a linear combination of the Ricci tensor and its metric tensor (see Theorem 3.1), and consequently, its gravitational field is extremely strong. But it is neither quasi-Einstein nor Roter type. This is a very strange geometric property of this black hole. These classification results of the curvature properties of LBH spacetime may be applicable to determine the nature of its pseudosymmetry.

The article is arranged as follows: In Section 2, we delve into key definitions of geometric structures that are essential for our investigation into the properties of the LBH spacetime. Section 3 is dedicated to a detailed examination of the LBH spacetime, where we derive and discuss several intriguing results. In Section 4, we analyze the geometric properties associated with the energy–momentum tensor of the LBH spacetime. Section 5 is devoted to the study of the nature of Ricci solitons and Ricci–Yamabe solitons present in the LBH spacetime. In Section 6, we compare the LBH spacetime to the BBH spacetime, focusing on their respective geometric structures. Finally, in Section 7, applications of LBH and BBH spacetime have been analyzed together with the application of the comparison and the curvature classification of LBH spacetime and also specified the application of the methodology utilized in the paper.