One-harmonic invariant vector fields on three-dimensional Lie groups

We determine all left-invariant vector fields on three-dimensional Lie groups which define harmonic sections of the corresponding tangent bundles, equipped with the complete lift metric.     

Characterizing killing vector fields of standard static space-times

We provide a global characterization of the Killing vector fields of a standard static space-time by a system of partial differential equations. By studying this system, we determine all the Killing vector fields in the same framework when the Riemannian part is compact. Furthermore, we deal with the characterization of Killing vector fields with zero curl on a standard static space-time.     

Harmonic morphisms and bicomplex manifolds

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of complex-harmonic morphism between complex-Riemannian manifolds and showing how these are given by bicomplex-holomorphic functions when the codomain is one-bicomplex dimensional. By taking real slices, we recover well-known compactifications for the three possible real cases. On the way, we discuss some interesting conformal compactifications of complex-Riemannian manifolds by interpreting them as bicomplex manifolds.     

Two-symmetric Lorentzian manifolds

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen–Wallach spaces, they have commutative holonomy.     

Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces

We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollary 2.3, Corollary 2.4 and Corollary 2.6), biharmonic maps between spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal multiplications and eigenmaps. We also study biharmonic graphs of maps, derive the equation for a function whose graph is a biharmonic hypersurface in a Euclidean space, and give an equivalent formulation of Chen’s conjecture on biharmonic hypersurfaces by using the biharmonic graph equation (Theorem 4.1) which paves a way for the analytic study of the conjecture.     

Biminimal properly immersed submanifolds in the Euclidean spaces

We consider a complete nonnegative biminimal   submanifold M$M$ (that is, a complete biminimal submanifold with λ≥0$\lambda \ge 0$) in a Euclidean space E^N${\mathbb{E}}^N$. Assume that the immersion is proper  , that is, the preimage of every compact set in E^N${\mathbb{E}}^N$ is also compact in M$M$. Then, we prove that M$M$ is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0$\lambda <0$, we construct examples of biminimal submanifolds and curves.     

Holomorphic vector fields of compact pseudo-Kähler manifolds

It is well known that a pseudo-Kähler structure is a natural generalization of the Kähler structure. In this paper, we consider holomorphic vector fields of a compact pseudo-Kähler manifold from the viewpoint of Kähler manifolds.     

On the geometrized Skyrme and Faddeev models

The higher power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to _σ2${\sigma }_2$-energy, introduced by Eells and Sampson (1964) [1]. This paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical points are outlined.     

Biminimal hypersurfaces in a sphere

In this paper, we mainly concentrate on the biminimal hypersurfaces in a sphere. First, we obtain some rigidity theorems for biminimal hypersurfaces. Then we give a classification of non-minimal biminimal isoparametric hypersurfaces in a sphere.     

Bernstein theorems for space-like graphs with parallel mean curvature and controlled growth

In this paper, we obtain an Ecker–Huisken-type result for entire space-like graphs with parallel mean curvature.     

On surfaces whose twistor lifts are harmonic sections

We study surfaces whose twistor lifts are harmonic sections, and characterize these surfaces in terms of their second fundamental forms. As a corollary, under certain assumptions for the curvature tensor, we prove that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. For surfaces in four-dimensional Euclidean space, a lower bound for the vertical energy of the twistor lifts is given. Moreover, under a certain assumption involving the mean curvature vector field, we characterize a surface in four-dimensional Euclidean space in such a way that the twistor lift is a harmonic section, and its vertical energy density is constant.     

Construction of a central extension of a Lie group from its class of symplectic cohomology

Bargmann’s group is a central extension of Galilei group motivated by quantum-theoretical considerations. Bargmann’s work suggests that one of the reasons of the failure of naïve attemps to construct actions on quantum wave functions has a cohomologic origin. It is this point, we develop in the context of Lie groups with symplectic actions. Studying the co-adjoint representation of a central extension of a group G$G$, we highlight the link between the extension cocycles and the symplectic cocycles of G$G$. Also, each extension coboundary corresponds to a symplectic coboundary. Finally, we emphasize the condition to be satisfied by the extension cocycle for the class of symplectic cohomology of the extension being null. The method is illustrated by application to Physics.     

A classification of Einstein lightlike hypersurfaces of a Lorentzian space form

We study Einstein lightlike hypersurfaces of a semi-Riemannian manifold of constant curvature c$c$, whose shape operator is conformal to the shape operator of its screen distribution. Our main result is a classification theorem for Einstein lightlike hypersurfaces of Lorentzian space forms.     

A criterion for global hyperbolicity of left-invariant Lorentz metrics on Lie groups

We show that every left-invariant Lorentz metric on a non-abelian simply connected Lie group is globally hyperbolic whenever its restriction to the commutator ideal of the Lie algebra is positive definite. We also show that a left-invariant Lorentz metric on the three-dimensional Heisenberg group is globally hyperbolic if and only if its restriction to the center of the Lie algebra is positive definite or degenerate.     

The spectral geometry of the canonical Riemannian submersion of a compact Lie group

Let G$G$ be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy$m(x,y)=xy$ be the multiplication operator. We show the associated fibration m:G×G→G$m:G\times G\to G$ is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull-backs of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions for the pull-back of a form on the base to be harmonic on the total space.     

Homogeneous Lorentzian Spaces Whose Null-geodesics are Canonically Homogeneous

It is shown that a homogeneous Lorentzian space for which every null-geodesic is canonically homogeneous, admits a non-vanishing homogeneous Lorentzian structure belonging to the class .