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Given an extendible spacetime one may ask how much, if any, uniqueness can in general be expected of the extension. Locally, this question was considered and comprehensively answered in a recent paper of Sbierski [22], where he obtains local uniqueness results for anchored spacetime extensions of similar character to earlier work for conformal boundaries by Chruściel [2]. Globally, it is known that non-uniqueness can arise from timelike geodesics behaving pathologically in the sense that there exist points along two distinct timelike geodesics which become arbitrarily close to each other interspersed with points which do not approach each other. We show that this is in some sense the only obstruction to uniqueness of maximal future boundaries: Working with extensions that are manifolds with boundary we prove that, under suitable assumptions on the regularity of the considered extensions and excluding the existence of such "intertwined timelike geodesics", extendible spacetimes admit a unique maximal future boundary extension. This is analogous to results of Chruściel for the conformal boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometric classifications of λ -self-expanders in three-dimensional Euclidean space. It is proved that for a 2-dimensional λ -self-expander X : M → R 3 in the 3-dimensional Euclidean space R 3 , if the Gauss curvature K is constant, then locally M is a plane R 2 , a cylinder S 1 ( λ ± λ 2 - 4 2) × R , a round sphere S 2 ( λ ± λ 2 - 8 4) and a generalized cylinder Γ × R , where Γ is an λ -self-expander curve. Furthermore, we give a classification of λ -self-expander M with constant squared norm of the second fundamental form, and show that M is locally a plane R 2 , a cylinder S 1 ( λ ± λ 2 - 4 2) × R and a round sphere S 2 ( λ ± λ 2 - 8 4) . [ABSTRACT FROM AUTHOR]

We find the local form of all non-closed Lorentzian Weyl manifolds (M , c , ∇) with recurrent curvature tensor. The recurrent curvature tensor turns out to be weighted parallel, i.e., the obtained spaces provide certain generalization of locally symmetric affine spaces for the Weyl geometry. If the dimension of the manifold is greater than 3, then the conformal structure is flat, and the recurrent Weyl structure is locally determined by a single function of one variable. Two local structures are equivalent if and only if the corresponding functions are related by a transformation from Aff 1 0 (R) × PSL 2 (R) × Z 2 . We find generators for the field of rational scalar differential invariants of this Lie group action. The global structure of the manifold M may be described in terms of a foliation with a transversal projective structure. It is shown that all locally homogeneous structures are locally equivalent, and there is only one simply connected homogeneous non-closed recurrent Lorentzian Weyl manifold. Moreover, there are 5 classes of cohomogeneity-one spaces, and all other spaces are of cohomogeneity-two. If dim M = 3 , the non-closed recurrent Lorentzian Weyl structures are locally determined by one function of two variables or two functions of one variable, depending on whether its holonomy algebra is 1- or 2-dimensional. In this case, two structures with the same holonomy algebra are locally equivalent if and only if they are related, respectively, by a transformation from an infinite-dimensional Lie pseudogroup or a 4-dimensional subgroup of Aff (R 3) . Again we provide generators for the field of rational differential invariants. We find a local expression for the locally homogeneous non-closed recurrent Lorentzian Weyl manifolds of dimension 3, and also of those of cohomogeneity one and two. In the end we give a local description of the non-closed recurrent Lorentzian Weyl manifolds that are also Einstein–Weyl. All of them are 3-dimensional and have a 2-dimensional holonomy algebra. [ABSTRACT FROM AUTHOR]

CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, Bartnik (Commun Math Phys 117(4):615–624, 1988) constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, Chruściel et al. (Commun Math Phys 257(1):29–42, 2005) constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology is also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik’s construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact Euclidean or hyperbolic three-manifold with any another compact Euclidean or hyperbolic three-manifold. Analogous examples in higher spacetime dimensions are also possible. We work with the Tolman–Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models. [ABSTRACT FROM AUTHOR]

In this paper, we define the evolute and focal surface of a spacelike framed curve with lightlike components in Minkowski 3-space. It is a generalization of the previous results of regular spacelike curves, since singularities are allowed in the original spacelike curves studied by spacelike framed curves with lightlike components. Meanwhile, we show a new geometric invariant to characterise singularities of the focal surface. Then, the classification theorem and recognition theorem for the singularities of the focal surface in generic are also given. [ABSTRACT FROM AUTHOR]

Semi-Riemannian manifolds that satisfy (homogeneous) linear differential conditions of arbitrary order r on the curvature are analyzed. They include, in particular, the spaces with (rth-order) recurrent curvature, (rth-order) symmetric spaces, as well as entire new families of semi-Riemannian manifolds rarely, or never, considered before in the literature—such as the spaces whose derivative of the Riemann tensor field is recurrent, among many others. Definite proof that all types of such spaces do exist is provided by exhibiting explicit examples of all possibilities in all signatures, except in the Riemannian case with a positive definite metric. Several techniques of independent interest are collected and presented. Of special relevance is the case of Lorentzian manifolds, due to its connection to the physics of the gravitational field. This connection is discussed with particular emphasis on Gauss–Bonnet gravity and in relation with Penrose limits. Many new lines of research open up and a handful of conjectures, based on the results found hitherto, is put forward. [ABSTRACT FROM AUTHOR]

We derive a variant of the Weierstrass–Enneper representation for generalized maximal immersions generated by a single harmonic function. We use this variant to construct examples of generalized maximal immersion from a single family. Subsequently, we proved a ‘superposition principle’ for generalized maximal immersions. It says that if we have given finitely many generalized maximal immersions, one can add their parametrizations to get another generalized maximal immersion under certain assumptions. We also study how the singularity of maxfaces behaves under superposition. [ABSTRACT FROM AUTHOR]

The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds M = G / H of signature (n - 2 , 2) such that a nilpotent analytic subgroup of G is transitive on M. For that we suppose that there is a reductive decomposition g = h ⊕ n (vector space direct sum) with [ h , n ] ⊂ n and n nilpotent. When the metric is nondegenerate on [ n , n ] we show that n is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on [ n , n ] we show that n is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature. [ABSTRACT FROM AUTHOR]

In this paper we assume that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and innovative approach, we investigate some curvature properties of perfect fluid spacetimes. Firstly, we provide a necessary condition for a Lorentzian manifold to be a perfect fluid spacetime. We prove that a conformally flat perfect fluid space-time is a spacetime of quasi constant sectional curvature. Also, we acquire that a conharmonically flat perfect fluid spacetime represents the radiation era. Next, we show that if a stiff matter fluid obeys Yang's equations, then the vorticity of the fluid vanishes. Moreover, we find that if the Ricci tensor is Killing, then the perfect fluid spacetime is (i) expansion-free and shear-free, and its flow is geodesic, however, not necessarily vorticity-free, and (ii) σ and p are constant. Besides, we establish that Ricci semi-symmetric or Ricci symmetric perfect fluid spacetimes represent either dark matter era, or phantom era. Finally, we acquire that if a perfect fluid is Ricci symmetric, then either the spacetime represents a dark matter era, or a static spacetime. [ABSTRACT FROM AUTHOR]

We describe all metrics geodesically compatible with a gl -regular Nijenhuis operator L. The set of such metrics is large enough so that a generic local curve γ is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of γ to be a g-geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g-geodesics is naturally equivalent to the Poisson action of R n on the cotangent bundle generated by the integrals coming from geodesic compatibility. [ABSTRACT FROM AUTHOR]

Similar to Nomizu–Pinkall's geometric characterization of the Cayley surface and Hu–Li–Zhang's characterization of the Cayley hypersurface, how can one characterize the generalized Cayley hypersurfaces? In this paper, by the affine α-connection of statistical manifolds, we study affine hypersurfaces with parallel cubic form relative to the affine α-connection. As the main results, we complete the classification of such hypersurfaces if its affine metric is either definite, or Lorentzian with α ≠ −1. Moreover, we give a new characterization of the generalized Cayley hypersurfaces to answer the question. [ABSTRACT FROM AUTHOR]

Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (G, X)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (G, X)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (G, X)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (G, X)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities. [ABSTRACT FROM AUTHOR]

Let L 4 be a 4-dimensional Lorentzian space with the sign (−,+,+,+). The aim of this study is to investigate the other missing algebraic forms of the constraint manifolds of 2C and 3C spatial open chains in L 4 . For this purpose, firstly, we obtain the structure equations of a spatial open chain using the equations of open chains of the Lorentz plane and Lorentz sphere. After then, using these structure equations, we search the algebraic forms of the constraint manifolds of 2C and 3C spatial open chains in Lorentzian 3-space with respect to the causal characters of the first link and the axis of rotation of the joint. [ABSTRACT FROM AUTHOR]

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and Stäckel matrices. [ABSTRACT FROM AUTHOR]

We give a detailed description of the geometry of isotropic space, in parallel to those of Euclidean space within the realm of Laguerre geometry. After developing basic surface theory in isotropic space, we define spin transformations, directly leading to the spinor representation of conformal surfaces in isotropic space. As an application, we obtain the Weierstrass-type representation for zero mean curvature surfaces, and the Kenmotsu-type representation for constant mean curvature surfaces, allowing us to construct many explicit examples. [ABSTRACT FROM AUTHOR]

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature (p, q), with min (p , q) ≥ 2 . Building on a recent article [17], provided that such a structure is locally equivalent to its model X , the main question treated here is the completeness of the associated (G , X) -structure. Because of the similarities between the model spaces of projective geometry and non-Lorentzian conformal geometry, a number of arguments apply in both contexts. We therefore present the proofs in parallel. The conclusion is that in both cases, when the real rank is maximal, the manifold is globally equivalent to either the model space X or its double cover. [ABSTRACT FROM AUTHOR]

We introduce an analogue to the amalgamation of metric spaces into the setting of Lorentzian pre-length spaces. This provides a very general process of constructing new spaces out of old ones. The main application in this work is an analogue of the gluing theorem of Reshetnyak for CAT(k) spaces, which roughly states that gluing is compatible with upper curvature bounds. Due to the absence of a notion of spacelike distance in Lorentzian pre-length spaces we can only formulate the theorem in terms of (strongly causal) spacetimes viewed as Lorentzian length spaces. [ABSTRACT FROM AUTHOR]

In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz–Minkowski space that are homothetic self-similar solutions for the inverse mean curvature flow. This classification shows the existence of two classes of non-cylindrical homothetic solitons: one with lightlike rulings and another one with non-lightlike rulings. [ABSTRACT FROM AUTHOR]

Recently, Lorentzian length spaces have been introduced inspired by length spaces. One of the main objects of study in these spaces is a time separation function τ , which is closely linked to their causal structure. In analogy to the metric d in length spaces, τ can express basic notions and many results in the setting of Lorentzian length spaces. In this paper, the concept of conformal Lorentzian length spaces is introduced and a novel version of limit curve theorem is proven. Finally, the global hyperbolic and causally simple Lorentzian length spaces are characterized. [ABSTRACT FROM AUTHOR]

The paper aims to investigate the general properties of pseudo Z-symmetric Lorentzian manifolds with semi-symmetric metric ρ -connection ∇ ¯ and examine compatibility conditions. Moreover, such a manifold is applied to general relativity and its physical consequences are given. [ABSTRACT FROM AUTHOR]

The Dini helicoid is a surface obtained by a screw motion of the tractrix. In this paper, we consider various analogs of the Dini helicoid in the three-dimensional Minkowski space. As profiles, we take nontrivial pseudo-Euclidean analogs of the tractrix different from pseudo-Euclidean circles. We prove that on analogs of the Dini helicoid in a the pseudo-Euclidean space, one of the following metrics is induced: the metric of the Lobachevsky plane, the metric of the de Sitter plane, or a degenerate metric. [ABSTRACT FROM AUTHOR]

In this study, we investigate the geometric and algebraic aspects of defocusing Complex Coupled Dispersionless (CCD) equations in Minkowski space. We give the conditions for obtaining CCD equations from moving spacelike or timelike space curves using the Frenet and Darboux frames. Besides, we present the Lax pairs that indicate that the corresponding CCD equations are integrable. Finally, we evaluate the conserved quantities for CCD equations from the geometric point of view. [ABSTRACT FROM AUTHOR]

In this paper, we locally classify the surfaces immersed into the non-flat (Riemannian or Lorentzian) 3-space forms satisfying the condition □ H → = λ H → for a real number λ , where H → is the mean curvature vector field and □ denotes the Cheng–Yau operator of the surface. We obtain the classification result by proving, at a first step, that the mean curvature function must be constant and, in a second step, we complete the classification. [ABSTRACT FROM AUTHOR]

We study mixed generalized quasi-Einstein warped products with respect to affine connection for arbitrary dimension n ≥ 3 . Then we prove a mixed generalized quasi-Einstein manifold is a generalized quasi-Einstein manifold with respect to affine connection under some certain conditions. In the last section we also give some examples of warped product on mixed generalized quasi-Einstein manifold. [ABSTRACT FROM AUTHOR]

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condition on the measure for this problem to have a solution. In this paper, we address Alexandrov's problem for convex bodies in the hyperbolic space H m + 1 . After defining the Gauss curvature measure of an arbitrary hyperbolic convex body, we completely solve Alexandrov's problem in this setting. Contrary to the Euclidean case, we also prove the uniqueness of such a convex body. The methods for proving existence and uniqueness of the solution to this problem are both new. [ABSTRACT FROM AUTHOR]

This paper addresses the equivalence problem of conic submanifolds in the tangent bundle of a smooth 2-dimensional manifold. Those are given by a quadratic relation between the velocities and are treated as nonholonomic constraints whose admissible curves are trajectories of the corresponding control systems, called quadratic systems. We deal with the problem of characterising and classifying conic submanifolds under the prism of feedback equivalence of control systems, both control-affine and fully nonlinear. The first main result of this work is a complete description of non-degenerate conic submanifolds via a characterisation under feedback transformations of the novel class of quadratic control-affine systems. This characterisation can explicitly be tested on structure functions defined for any control-affine system and gives a normal form of quadratisable systems and of conic submanifolds. Then, we consider the classification problem of regular conic submanifolds (ellipses, hyperbolas, and parabolas), which is treated via feedback classification of quadratic control-nonlinear systems. Our classification includes several normal forms of quadratic systems (in particular, normal forms not containing functional parameters as well as those containing neither functional nor real parameters) and, as a consequence, gives a classification of regular conic submanifolds. [ABSTRACT FROM AUTHOR]

We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive a complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show the vanishing of the Q-curvature. [ABSTRACT FROM AUTHOR]

Infinitely many new examples of compact Lorentzian surfaces without conjugate points are given. Further, we study the existence and the stability of this property among Lorentzian metrics with a Killing field. We obtain a new obstruction and prove that the Clifton–Pohl torus and some of our examples are as stable as possible. This shows that in constrast with the Riemannian Hopf theorem, the absence of conjugate points in the Lorentzian setting is neither "special" nor rigid. [ABSTRACT FROM AUTHOR]

The Hadamard expansion describes the singularity structure of Green’s operators associated with a normally hyperbolic operator P in terms of Riesz distributions (fundamental solutions on Minkowski space, transported to the manifold via the exponential map) and Hadamard coefficients (smooth sections in two variables, corresponding to the heat Kernel coefficients in the Riemannian case). In this paper, we derive an asymptotic expansion analogous to the Hadamard expansion for powers of advanced/retarded Green’s operators associated with P, as well as expansions for advanced/retarded Green’s operators associated with P - z for z ∈ C . These expansions involve the same Hadamard coefficients as the original Hadamard expansion, as well as the same or analogous (with built-in z-dependence) Riesz distributions. [ABSTRACT FROM AUTHOR]

The main result in this paper is the generalisation of the fundamental equations of a Riemannian submersion presented in the 1966 article by O'Neill (Michigan Math J 13:459–469, 1966) to the context of pseudo-Finsler submersions. In the meantime, we also explore some basic properties of the O'Neill fundamental tensors of the submersion and study Finsler submersions with totally geodesic fibers. [ABSTRACT FROM AUTHOR]

Let M s 2 be an orientable surface immersed in the De Sitter space S 1 3 ⊂ R 1 4 or anti de Sitter space H 1 3 ⊂ R 2 4 . In the case that M s 2 is of L 1 -2-type we prove that the following conditions are equivalent to each other: M s 2 has a constant principal curvature; M s 2 has constant mean curvature; M s 2 has constant second mean curvature. As a consequence, we also show that an L 1 -2-type surface is either an open portion of a standard pseudo-Riemannian product, or a B-scroll over a null curve, or else its mean curvature, its Gaussian curvature and its principal curvatures are all non-constant. [ABSTRACT FROM AUTHOR]

Two homogeneous pseudo-Riemannian manifolds (G / H , d s 2) and (G ′ / H ′ , d s ′ 2) belong to the same real form family if their complexifications (G C / H C , d s C 2) and (G C ′ / H C ′ , d s C ′ 2) are isometric. The point is that in many cases, a particular space (G / H , d s 2) has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if (G / H , d s 2) is a geodesic orbit space with a reductive decomposition g = h ⊕ m , then the same holds all the members of its real form family. In particular,our understanding of compact geodesic orbit Riemannian manifolds gives information on geodesic orbit pseudo-Riemannian manifolds. We also prove similar results for naturally reductive spaces, for commutative spaces, and in most cases for weakly symmetric spaces. We end with a discussion of inclusions of these real form families, a discussion of D’Atri spaces, and a number of open problems. [ABSTRACT FROM AUTHOR]

On the occasion of Sir Roger Penrose's 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a pedagogical introduction to the classical theorems with an emphasis on the analytical side of the arguments. We especially concentrate on focusing results for causal geodesics under appropriate geometric and initial conditions, in the smooth and in the low regularity case. The latter comprise the main technical advance that leads to the proofs of C 1 -singularity theorems via a regularisation approach that allows to deal with the distributional curvature. We close with an overview on related lines of research and a future outlook. [ABSTRACT FROM AUTHOR]

We consider multicomponent local Poisson structures of the form P 3 + P 1 , under the assumption that the third-order term P 3 is Darboux–Poisson and nondegenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular,we show that each Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with edges and vertices labelled by numerical marks. They are also naturally related to certain pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the corresponding Nijenhuis pencils. [ABSTRACT FROM AUTHOR]

In this article, we present a new tensor that generalizes the conharmonic curvature tensor, termed the ψ-conharmonic curvature tensor. In the first part, we derive some fundamental geometrical properties of a pseudo ψ-conharmonically symmetric manifold (briefly,) and give some fascinating outcomes. Among others, we establish that an Einstein manifold is of zero scalar curvature. Besides, we investigate ψ-conharmonically flat perfect fluid and spacetimes, respectively. As a result, we acquire some significant theorems. Finally, we construct several non-trivial examples to establish the existence of. [ABSTRACT FROM AUTHOR]

The paper is devoted to explore the trajectories of charged particles varying under the influence of Lorentz force of Killing magnetic fields (i.e., closed 2-form corresponding to Killing vector fields) in Egorov 3-spaces. Predominantly, we characterize Killing vector fields and hence determine magnetic trajectories in some Killing magnetic fields on Egorov 3-spaces. We also obtain magnetic trajectories in some Killing magnetic backgrounds on flat Egorov 3-spaces. [ABSTRACT FROM AUTHOR]

We study the geometric properties of certain Codazzi tensors for their own sake, and for their appearance in the recent theory of Cotton gravity. We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdziński. We also give necessary and sufficient conditions for a space-time to host a current-flow Codazzi tensor. In particular, we study the static and spherically symmetric cases, which include the Nariai and Bertotti-Robinson metrics. The latter are a special case of Yang Pure space-times, together with spatially flat FRW space-times with constant curvature scalar. We apply these results to the recent Cotton gravity by Harada. We show that the equation of Cotton gravity is Einstein's equation modified by the presence of a Codazzi tensor, which can be chosen freely and constrains the space-time where the theory is staged. In doing so, the tensor (chosen in forms appropriate for physics) implies the form of the Ricci tensor. The two tensors specify the energy-momentum tensor, which is the source in the equation of Cotton gravity for the metric implied by the Codazzi tensor. For example, we show that the Stephani, Nariai and Bertotti-Robinson space-times are characterized by a "current flow" Codazzi tensor. Because of it, they solve Cotton gravity with physically sensible energy-momentum tensors. Finally, we discuss Cotton gravity in constant curvature space-times. [ABSTRACT FROM AUTHOR]

In this work, we prove a synthetic splitting theorem for globally hyperbolic Lorentzian length spaces with global non-negative timelike curvature containing a complete timelike line. Inspired by the proof for smooth spacetimes (Beem et al. in Global differential geometry and global analysis 1984, Springer, pp. 1–13, 1985), we construct complete, timelike asymptotes which, via triangle comparison, can be shown to fit together to give timelike lines. To get a control on their behaviour, we introduce the notion of parallelity of timelike lines in the spirit of the splitting theorem for Alexandrov spaces as proven in Burago et al. (A course in metric geometry, vol 33, American Mathematical Society, Providence, 2001) and show that asymptotic lines are all parallel. This helps to establish a splitting of a neighbourhood of the given line. We then show that this neighbourhood has the timelike completeness property and is hence inextendible by a result in Grant et al. (Ann Glob Anal Geom 55(1):133–147, 2019), which globalises the local result. [ABSTRACT FROM AUTHOR]

We classify the Lorentzian manifolds of dimension n ≥ 3 admitting some diagonalizable operators which satisfy the Codazzi equation. This classification is applied to characterize three-dimensional weakly-Einstein Lorentzian manifolds which fall in the conformal class of the flat metrics. [ABSTRACT FROM AUTHOR]

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form N t m + 1 (c) of dimension m + 1 and index t ( m ≥ 2 and 1 ≤ t ≤ m ). We shall study hypersurfaces M t ′ m which are polyharmonic of order r (briefly, r-harmonic), where r ≥ 3 and either t ′ = t or t ′ = t - 1 . Let A denote the shape operator of M t ′ m . Under the assumptions that M t ′ m is CMC and Tr A 2 is a constant, we shall obtain the general condition which determines that M t ′ m is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces ( r ≥ 3 ). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form. [ABSTRACT FROM AUTHOR]