Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps \((u_n,v_n)\) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form \(g_N -\beta dt^2\) for some Riemannian metric \(g_N\) and some positive function \(\beta \) on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.     

Curvature properties of φ-null Osserman Lorentzian S-manifolds

We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian S-manifold and the Jacobi operators with respect to particular spacelike unit vectors. We study the number of the eigenvalues of such operators on Lorentzian S-manifolds satisfying the φ-null Osserman condition, under suitable assumptions on the dimension of the manifold. Then, we provide in full generality a new curvature characterization for Lorentzian S-manifolds and we use it to obtain an algebraic decomposition for the Riemannian curvature tensor of φ-null Osserman Lorentzian S-manifolds.     

On the full holonomy group of Lorentzian manifolds

The classification of restricted holonomy groups of \(n\) -dimensional Lorentzian manifolds was obtained about ten years ago. However, up to now, not much is known about the structure of the full holonomy group. In this paper we study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. Based on the classification of the restricted holonomy groups of such manifolds, we prove several structure results about the full holonomy. We establish a construction method for manifolds with disconnected holonomy starting from a Riemannian manifold and a properly discontinuous group of isometries. This leads to a variety of examples, most of them being quotients of pp-waves with disconnected holonomy, including a non-flat Lorentzian manifold with infinitely generated holonomy group. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor.     

About the classification of the holonomy algebras of lorentzian manifolds

The classification of the holonomy algebras of Lorentzian manifolds can be reduced to the classification of the irreducible subalgebras h ? so(n) that are spanned by the images of linear maps from ? ⁿ to h satisfying some identity similar to the Bianchi identity. Leistner found all these subalgebras and it turned out that the obtained list coincides with the list of irreducible holonomy algebras of Riemannian manifolds. The natural problem is to give a simple direct proof of this fact. We give such a proof for the case of semisimple not simple Lie algebras h.     

Lorentzian Geometry of the Heisenberg Group

In this work we prove the existence of totally geodesic two-dimensional foliation on the Lorentzian Heisenberg group H ₃. We determine the Killing vector fields and the Lorentzian geodesics on H ₃.     

Lorentzian Legendrian mean curvature flow

In this paper we generalize the Legendrian mean curvature flow to Lorentzian geometry. More precisely, we study the case, where the ambient manifold is a Lorentzian Sasaki $\eta $ -Einstein manifold. For Legendrian curves we establish convergence results in Theorems 1.1 and 1.2 and we derive estimates for the Legendrian angle for arbitrary dimensions in Theorem 1.3.     

CURVATURE TENSORS IN A LORENTZIAN PARA SASAKIAN MANIFOLD

Abstract

The author and Mishra [1] have introduced some curvature tensors to study their relativistic and geometric properties. Matsumoto and Mihai [2] have introduced the notion of Lorentzian para Sasakian (LP-Sasakian) and studied certain transformations. In this paper some properties of curvature tensors, in a LP-Sasakian manifold, are studied.     

Schrödinger soliton from Lorentzian manifolds

In this paper, we introduce a new notion named as Schrödinger soliton. The so-called Schrödinger solitons are a class of solitary wave solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, then the Schrödinger soliton reduces to a harmonic map with potential from M into N. Especially, when the domain manifold M is a Lorentzian manifold, the Schrödinger soliton is a wave map with potential into N. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schrödinger soliton solution to the hyperbolic Ishimori system.     

The twistor equation in Lorentzian spin geometry

In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to dimension 7.Mathematics Subject Classification (2000): 53C15, 53C50in final form: 1 October 2003     

Commutability of homogenization and linearization at identity in finite elasticity and applications

We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.     

Second variation for L-minimal Legendrian submanifolds in pseudo-Sasakian manifolds

In this paper we provide the second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian–Sasakian manifolds and relate the L-stability of L-minimal Legendrian submanifolds in a Sasakian manifold M to their L-stability in an associated Lorentzian–Sasakian structure on M.     

A complete generalization of Göllnitz’s “big” theorem

In the spirit of Göllnitz’s “big” partition theorem of 1967, we present a new mod-6 partition identity. Alladi et al. provided a four-parameter refinement of Göllnitz’s big theorem in 1995 via a key identity of generating functions and the method of weighted words. By means of this technique, two similar mod-6 identities of this type were discovered—one by Alladi in 1999 and one by Alladi and Andrews in 2015. We finish the picture by presenting and proving the fourth and final possible mod-6 identity in this spirit. Furthermore, we provide a complete generalization of mod-n identities of this type. Finally, we apply a similar argument to generalize an identity of Alladi et al. from 2003.

     

Algebraic zero mean curvature hypersurfaces in de Sitter and anti de Sitter spaces

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter N-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature hypersurfaces in the de Sitter space.     

Clifford Product and Lorentzian Plane Displacements In 3-Dimensional Lorentzian Space

In this paper, by defining Clifford algebra product in 3-dimensional Lorentz space, L ³, it is shown that even Clifford algebra of L ³ corresponds to split quaternion algebra. Then, by using Lorentzian matrix multiplication, pole point of planar displacement in Lorentz plane L ² is obtained. In addition, by defining degenerate Lorentz scalar product for L ³ and by using the components of pole points of Lorentz plane displacement in particular split hypercomplex numbers, it is shown that the Lorentzian planar displacements can be represented as a special split quaternion which we call it Lorentzian planar split quaternion.      

Some remarks on Kong-Liu’s conjecture on Lorentzian metrics

For a(1+3)-dimensional Lorentzian manifold(M,g),the general form of solutions of the Einstein field equations takes that of type I,II,or III.For type I,there is a known result in Gu(2007).In this paper,we try to find the necessary and sufficient conditions for the Lorentzian metric to take the form of types II and III,and we show how to construct the new coordinate system.     

Les groupes oscillateurs et leurs reseaux

In this paper we study the geometry of oscillator groups: they are the only non commutative simply connected solvable Lie groups which have a biinvariant Lorentzian metric. We first study curvature and geodesics, and then give a full analysis of lattices - i.e. discrete co-compact subgroups - getting examples of compact Lorentzian homogeneous varieties.     

Singularity analysis of Lorentzian hypersurfaces on pseudo n‐spheres

This paper considers the one parameter families of extrinsic differential geometries of Lorentzian hypersurfaces on pseudo n‐spheres. We give a continuous relationship among the three types Gauss indicatrices by one parameter map. Meanwhile, we also give the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo n‐spheres by the Legendrian singularity theory. Copyright © 2014 John Wiley & Sons, Ltd.     

A timelike extension of Fermat's principle in general relativity and applications

We define a variational problem based on the arrival time functional for timelike curves on a Lorentzian manifold parameterized by a fixed constant multiple of their proper time. Under a causality assumption for the manifold , we prove that the stationary points of our problem are geodesics, obtaining an extension of the Fermat's Principle for light rays proven in [14] (see also [2]). Moreover, we study the compactness pr operties of the arrival time functional by global variational techniques. Under intrinsic assumptions on the metric of we get results of existence and multiplicity for geodesics with a given energy between an event and an observer of .