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In this work, we focus on the evolution of the vortex filament flow ∂ γ ∂ t = ∂ γ ∂ s ∧ D ds ∂ γ ∂ s for spacelike and timelike curves in a 3-dimensional pseudo-Riemannian manifold. We study the relations between a partial differential equation and the vortex filament flow for spacelike and timelike curves. As a result, we prove that the vortex filament flow of the spacelike curve in a 3-dimensional pseudo-Riemannian manifold with constant sectional curvature is equivalent to the heat equation, and the flow of the timelike curve is equivalent to the non-linear Schrödinger equation. Also, we give some examples to illustrate the vortex filament flow. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the concept of an analytic prolongation of a local Riemannian metric. We propose a generalization of the notion of completeness realized as an analytic prolongation of an arbitrary Riemannian metric. Various Riemannian metrics are studied, primarily those related to the structure of the Lie algebra 픤 of all Killing vector fields for a local metric. We introduce the notion of a quasi-complete manifold, which possesses the property of prolongability of all local isometries to isometries of the whole manifold. A classification of pseudo-complete manifolds of small dimensions is obtained. We present conditions for the Lie algebra of all Killing vector fields 픤 and its stationary subalgebra 픥 of a locally homogeneous pseudo-Riemannian manifold under which a locally homogeneous manifold can be analytically prolonged to a homogeneous manifold. [ABSTRACT FROM AUTHOR]

For any pseudo-Riemannian hyperbolic space X over R , C , H or O , we show that the resolvent R (z) = (□ − z Id) − 1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators C c ∞ (X) → D ′ (X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D ′ (X) forms a representation of the isometry group of X , which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R (z) extends, and the residue representations can be infinite-dimensional. [ABSTRACT FROM AUTHOR]

In this paper, slant submanifolds of a pseudo-Riemannian manifold equipped with an almost paracontact structure are defined and studied. Some characterization theorems for slant submanifolds are obtained. We also present some examples of slant submanifolds when the ambient space is an almost paracontact metric manifold. [ABSTRACT FROM AUTHOR]

Given a homogeneous pseudo-Riemannian space (G / H , ⟨ , ⟩) , a geodesic γ : I → G / H is said to be two-step homogeneous if it admits a parametrization t = ϕ (s) (s affine parameter) and vectors X, Y in the Lie algebra g , such that γ (t) = exp (t X) exp (t Y) · o , for all t ∈ ϕ (I) . As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics ⟨ , ⟩ on the unimodular Lie group S L (2 , R) such that (S L (2 , R) , ⟨ , ⟩) is a two-step g.o. space. [ABSTRACT FROM AUTHOR]

We summarize and announce some recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces\G=H, beyond the classical setting where H is compact (e.g. theory of automorphic forms for arithmetic) or is trivial (e.g. Plancherel-type formula for semisimple symmetric spaces). [ABSTRACT FROM AUTHOR]

In this paper we introduce and consider the hyperbolic sets for the ows on pseudo-Riemannian manifolds. If is a hyperbolic set for a ow, then we show that at each point of we have a unique decomposition for its tangent space up to a distribution on the ambient pseudo-Riemannian manifold. We prove that we have such decomposition for many points arbitrarily close to a given member of. [ABSTRACT FROM AUTHOR]

Fino and Kath determined all possible holonomy groups of seven-dimensional pseudo-Riemannian manifolds contained in the exceptional, non-compact, simple Lie group G 2 ∗ via the corresponding Lie algebras. They are distinguished by the dimension of their maximal semi-simple subrepresentation on the tangent space, the socle. An algebra is called of Type I, II or III if the socle has dimension 1, 2 or 3, respectively. This article proves that each possible holonomy group of Type III can indeed be realized by a metric of signature (4, 3). For this purpose, metrics are explicitly constructed, using Cartan's methods of exterior differential systems, such that the holonomy of the manifold has the desired properties. [ABSTRACT FROM AUTHOR]

We study n-dimensional pseudo-Riemannian spaces Vn(gij) with an arbitrary signature that admit projective motions, i.e., groups of continuous transformations preserving geodesics. In particular, we find the metric of a pseudo-Riemannian space of special type and establish important projective-group properties of this space. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. $${{\text {SO}}}_0(p+r, q)/({{\text {SO}}}_0(p,q) \times {{\text {SO}}}(r))\,(p,q,r \ge 1, \ q{:}\,\text {odd})$$ and $${{\text {SL}}}(p+q, {\mathbb {C}})/{{\text {SU}}}(p,q)\,(p,q \ge 1)$$ , of a homogeneous space that does not admit a compact Clifford-Klein form. To construct these examples, we apply Cartan's theorem on relative Lie algebra cohomology of reductive pairs and the theory of $$\varepsilon $$ -families of semisimple symmetric pairs. [ABSTRACT FROM AUTHOR]

A homogeneous Riemannian manifold (M = G/K, g) is called a space with homogeneous geodesics or a G-g.o. space if every geodesic γ(t) of M is an orbit of a one-parameter subgroup of G. that is γ(t) = exp(tX) · o, for some non zero vector X in the Lie algebra of G. We give an exposition on the subject, by presenting techniques that have been used so far and a wide selection of previous and recent results. We discuss generalization to two-step homogeneous geodesics. We also present some open problems. [ABSTRACT FROM AUTHOR]

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group. [ABSTRACT FROM AUTHOR]

Some basic inequalities, involving the scalar curvature and the mean curvature, for a pseudo-Riemannian submanifold of a pseudo-Riemannian manifold are obtained. We also find inequalities for spacelike submanifolds. Equality cases are also discussed. [ABSTRACT FROM AUTHOR]

We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Γ \ G / H , beyond the classical cases where H is compact (automorphic forms) or Γ is trivial (analysis on symmetric spaces). For any non-Riemannian reductive symmetric space X = G / H on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Γ whose action on X is “sufficiently proper”, we construct L 2 -eigenfunctions of the Laplacian on X Γ : = Γ \ X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to X Γ of sums, over the Γ-orbits, of eigenfunctions of the Laplacian on X . We prove that the Poincaré series we construct still converge, and define nonzero L 2 -functions, after any small deformation of Γ inside G , for a large class of groups Γ. Thus the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface. We actually construct joint L 2 -eigenfunctions for the whole commutative algebra of invariant differential operators on X Γ . [ABSTRACT FROM AUTHOR]

Conformal Killing forms are a natural generalization of conformal Killing vector fields. These forms have applications in physics related to hidden symmetries, conserved quantities, symmetry operators, or separation of variables. In this paper, we prove two vanishing theorems of conformal Killing forms on a space-like totally umbilical submanifold of a Lorentzian manifold. Finally, we show an application of these results to electrodynamics in the General Relativity Theory. [ABSTRACT FROM AUTHOR]

Abstract: We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a -homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds. [Copyright &y& Elsevier]

Space-like surfaces with special second fundamental forms have been an important tool in the study of pseudo-Riemannian manifolds. This paper focuses on adapting some theorems of Riemannian geometry in the large to the study of geometry of space-like surfaces of pseudo-Riemannian manifolds, specifically, various comparison theorems that use sectional and Ricci curvatures. The purpose of this paper is to establish a global geometry of complete spacelike totally umbilical and maximal surfaces in pseudo-Riemannian manifolds. [ABSTRACT FROM AUTHOR]

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold ( M,g) is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of ( M,g). We characterize the following simply connected pseudo-Riemannian manifolds that admit these subbundles in terms of their holonomy algebras: Riemannian manifolds, Lorentzian manifolds, pseudo-Riemannian manifolds with irreducible holonomy algebras, and pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we investigate the regularity for certain elliptic systems without an -antisymmetric structure. As applications, we prove some regularity results for weakly harmonic maps from the unit ball () into certain pseudo-Riemannian manifolds. [Copyright &y& Elsevier]

It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold with the same recurrence form. [ABSTRACT FROM AUTHOR]

Abstract: The family of all the submanifolds of a given Riemannian or pseudo-Riemannian manifold is large enough to classify them into some interesting subfamilies such as minimal (maximal), totally geodesic, Einstein, etc. Most of these have been extensively studied by many authors, but as far as we know, no paper has hitherto been published on the class of isotropic submanifolds. The purpose of this paper is therefore to gain a better understanding of this interesting class of submanifolds that arise naturally in mathematics and physics by studying their relationships with other closely distinguished families. [Copyright &y& Elsevier]

We provide the tangent bundle TM of pseudo-Riemannian manifold (M, g) with the Sasaki metric gs and the neutral metric gn. First we show that the holonomy group Hs of (TM, gs) contains the one of (M, g). What allows us to show that if (TM, gs) is indecomposable reducible, then the basis manifold (M, g) is also indecomposable-reducible. We determine completely the holonomy group of (TM, gs) according to the one of (M, g). Secondly we found conditions on the base manifold under which (TM, gs) (respectively (TM, gn) ) is Kählerian, locally symmetric or Einstein manifolds. (TM, gn) is always reducible. We show that it is indecomposable if (M, g) is irreducible. [ABSTRACT FROM AUTHOR]

We consider a class of almost geodesic mappings, namely, almost geodesic mappings of class π, and obtain conditions under which almost Hermitian manifolds admit almost geodesic mappings of class π. We prove that an almost Hermitian manifold admits a π-mapping with respect to a Riemannian connection if and only if it is an NK-manifold. We obtain a condition on the defining form ψ of any nontrivial π( e)-mapping under which a proper NK-structure is taken to a proper NK-structure. [ABSTRACT FROM AUTHOR]

Abstract: We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form , to higher dimension and codimension. We consider a complete spacelike graphic submanifold with parallel mean curvature, defined by a map between two Riemannian manifolds and of sectional curvatures and , respectively. We take on the pseudo-Riemannian product metric . Under the curvature conditions, and , we prove that, if the second fundamental form of satisfies an integrability condition, then is totally geodesic, and it is a slice if at some point. For bounded , and hyperbolic angle , we conclude that must be maximal. If is a maximal surface and , we show is totally geodesic with no need for further assumptions. Furthermore, is a slice if at some point , , and if is flat and at some point , then the image of lies on a geodesic of . [Copyright &y& Elsevier]

Abstract: In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space. [Copyright &y& Elsevier]

Abstract: Several notions of isotropy of a (pseudo-)Riemannian manifold have been introduced in the literature, in particular, the concept of pseudo-isotropic immersion. The aim of this paper is to look more closely at this notion of pseudo-isotropy and then to study the rigidity of this class of immersion into the pseudo-Euclidean space. It is worth pointing out that we first obtain a characterization of the pseudo-isotropy condition by using tangent vectors of any causal character. Then, rigidity theorems for pseudo-isotropic immersions are proved, and in particular, some well known results for the Riemannian case arise. Later, we bring together the notions of pseudo-isotropy, intrinsically and extrinsically isotropic manifolds, and prove interesting relations among them. Finally, we pay special attention to the case of codimension two Lorentz surfaces. [Copyright &y& Elsevier]

We consider the six-dimensional pseudo-Riemannian space V 6(gij) with the signature [++−−−−], which admits projective motions, i.e., continuous transformation groups preserving geodesics. We find necessary and sufficient conditions for the six-dimensional rigid h-spaces to have constant curvature. [ABSTRACT FROM AUTHOR]

The geometry of the symplectic structures and Fubini-Study metric is discussed. Discussion in the paper addresses geometry of Quantum Mechanics in the classical phase space. Also, geometry of Quantum Mechanics in the projective Hilbert space has been discussed for the chosen Quantum states. Since the theory of classical gravity is basically geometric in nature and Quantum Mechanics is in no way devoid of geometry, the explorations pertaining to more and more geometry in Quantum Mechanics could prove to be valuable for larger objectives such as understanding of gravity. [ABSTRACT FROM AUTHOR]

We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive perspective of the metric of quantum states with a focus on the geometry in the background independent quantum mechanics. We also explore the possibilities of geometrical formulations of quantum mechanics beyond the quantum state space and Kähler manifold. The metric of quantum states in the classical configuration space with the pseudo-Riemannian signature and its possible applications are explored. On contrary to the common perception that a metric for quantum state can yield a natural metric in the configuration space when the limit ℏ→0, we obtain the metric of quantum states in the configuration space without imposing the limiting condition ℏ→0. Here Planck’s constant ℏ is absorbed in the quantity like Bohr radii $\frac{1}{2mZ\alpha}\sim a_{0}$ . While exploring the metric structures associated with Hydrogen like atom, we witness another interesting finding that the invariant lengths appear in the multiple of Bohr’s radii as: ds 2= a ( ∇ Ψ)2. [ABSTRACT FROM AUTHOR]

Abstract: Here we give a natural extension of the duality principle for the curvature tensor of pointwise pseudo-Riemannian Osserman manifolds. We proved that this extended duality principle holds under certain additional assumptions. Also, it is proved that duality principle holds for every four-dimensional Osserman manifold. [Copyright &y& Elsevier]

Abstract: We modify the “Kaplan example” (a six-dimensional nilpotent Lie group which is a Riemannian g.o. space) and we obtain two pseudo-Riemannian homogeneous spaces with noncompact isotropy group. These examples have the property that all geodesics are homogeneous up to a set of measure zero. We also show that the (incomplete) geodesic graphs are strongly discontinuous at the boundary, i.e., the limits along certain curves are always infinite. [Copyright &y& Elsevier]

Abstract: Let be a real semisimple Lie algebra with Killing form B and a B-nondegenerate subalgebra of of maximal rank. We give a description of all -invariant decompositions such that , and are subalgebras. It is reduced to a description of parabolic subalgebras of with given reductive part . This is obtained in terms of crossed Satake diagrams. As an application, we get a classification of invariant bi-Lagrangian (or equivalently para-Kähler) structures on homogeneous manifolds of a semisimple group G. [Copyright &y& Elsevier]

We introduce the notion of contactly geodesic transformation of the metric of an almost-contact metric structure as a contact analog of holomorphically geodesic transformations of the metric of an almost-Hermitian structure. A series of invariants of such transformations is obtained. We prove that such transformations preserve the normality property of an almost-contact metric structure. We prove that cosymplectic and Sasakian manifolds, as well as Kenmotsu manifolds, do not admit nontrivial contactly geodesic transformations of the metric, which is a contact analog of the well-known result for Kählerian manifolds due to Westlake and Yano. [ABSTRACT FROM AUTHOR]

A generalized definition of the metric of quantum states is proposed by using the techniques of differential geometry. The metric of quantum state space derived earlier by Anandan is reproduced and verified here by this generalized definition. The metric of quantum states in the configuration space and its possible geometrical framework are explored. Also, invariance of the metric of quantum states under local gauge transformations, coordinate transformations, and the relativistic transformations is discussed. [ABSTRACT FROM AUTHOR]

Abstract: We study the existence of isotropic -immersions in a pseudo-Riemannian manifold following the Convex Integration techniques of Gromov. [Copyright &y& Elsevier]

We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov. [ABSTRACT FROM AUTHOR]

The relativity of motion and covariance of equation of motion in Newtonian-Riemannian space-time, some relationship between Newton's mechanics in N-R space-time and the general relativity, their difference and identity are discussed. [ABSTRACT FROM AUTHOR]

The relativity of motion and covariance of equation of motion in Newtonian-Riemannian space-time, some relationship between Newton's mechanics in N-R space-time and the general relativity, their difference and identity are discussed. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called “Geodesic Lemma”, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds. [Copyright &y& Elsevier]

In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass n complete null (light-like) geodesics. If, for the function ψ the metric ψ − 2 g is also Einstein, then ψ is a constant, and conformal mapping is homothetic. Note that Kiosak and Matveev previously assumed that all light-lines were complete. If the Einstein manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš and Kühnel). [ABSTRACT FROM AUTHOR]

In the present paper, we study geodesic mappings of special pseudo-Riemannian manifolds called V n (K) -spaces. We prove that the set of solutions of the system of equations of geodesic mappings on V n (K) -spaces forms a special Jordan algebra and the set of solutions generated by concircular fields is an ideal of this algebra. We show that pseudo-Riemannian manifolds admitting a concircular field of the basic type form the class of manifolds closed with respect to the geodesic mappings. [ABSTRACT FROM AUTHOR]