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Nikos Georgiou

Publications and source records attributed to Nikos Georgiou.

At least 19 recordsLinked to original sources
arXiv2026

Isotropic submanifolds of $T\mathbb{S}^n$ and their focal sets

Families of oriented lines in $\mathbb{R}^{n+1}$ are studied via their identification with submanifolds of $T\mathbb{S}^n$. In particular, families of oriented lines which are orthogonal to submanifolds in $\mathbb{R}^{n+1}$ are shown to characterise those which are isotropic with respect to the canonical sympleptic structure on $T\mathbb{S}^n$. Families of lines that are tangent to a $k$-dimensional submanifold of $\mathbb{R}^{n+1}$ are then studied. For such families, isotropy is shown to be equivalent to the generating vector field being geodesic and hypersurface-orthogonal on the submanifold. The focal set in $\mathbb{R}^{n+1}$ of a family of lines is introduced, extending the classical definition for families normal to hypersurfaces, to general families of lines of arbitrary codimension. A formula is derived that expresses certain sectional curvatures of the focal set in terms of the signed distances between corresponding focal points. We then solve an inverse problem for the focal sets of hypersurfaces and show certain sectional and Ricci curvatures of the focal set are determined by the differences between the hypersurface's radii of curvature. This generalises a Theorem of Bianchi from 1874 - namely that surfaces in $\mathbb{R}^3$ of constant astigmatism have pseudo-spherical focal sets.

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arXiv2024

Minimal surfaces in the Riemannian product of surfaces

Minimal surfaces in the Riemannian product of surfaces of constant curvature have been considered recently, particularly as these products arise as spaces of oriented geodesics of 3-dimensional space-forms. This papers considers more general Riemannian products of surfaces and explores geometric and topological restrictions that arise for minimal surfaces. We show that generically, a totally geodesic surface in a Riemannian product is locally either a slice or a product of geodesics. If the Gauss curvatures of the factors are negative, it is proven that there are no minimal 2-spheres, while minimal 2-tori are Lagrangian with respect to both product symplectic structures. If the surfaces have non-zero bounded curvatures, we establish a sharp lower bound on the area of minimal 2-spheres and explore the properties of the Gauss and normal curvatures of general compact minimal surfaces.

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arXiv2022

Null hypersurfaces in 4-manifolds endowed with a product structure

In a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel, defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal non-trivial principal curvatures and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

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arXiv2021

Almost Paracomplex Structures on 4-Manifolds

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics of any real 3-dimensional space form admits both isometric and anti-isometric paracomplex structures. This paper considers the existence or otherwise of isometric and anti-isometric almost paracomplex structures $j$ on a pseudo-Riemannian 4-manifold $(M,g)$, such that $j$ is parallel with respect to the Levi-Civita connection of $g$. It is shown that if an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, then the scalar curvature of the metric must be zero. In addition, it is found that $j$ is parallel iff the eigenplanes are tangent to a pair of mutually orthogonal foliations by totally geodesic surfaces. The composition of a Riemannian metric with an isometric almost paracomplex structure $j$ yields a neutral metric $g'$. It is proven that if $j$ is parallel, then $g$ is Einstein iff $g'$ is conformally flat and scalar flat. The vanishing of the Hirzebruch signature is found to be a necessary topological condition for a closed 4-manifold to admit an Einstein metric with a parallel isometric paracomplex structure. Thus, while the K3 manifold admits an Einstein metric with an isometric paracomplex structure, it cannot be parallel. The same holds true for certain connected sums of complex projective 2-space and its conjugate.

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arXiv2019

A para-Kaehler structure in the space of oriented geodesics in a real space form

In this article, we construct a new para-Kähler structure $({\mathcal G},{\mathcal J},Ω)$ in the space of oriented geodesics ${\mathbb L}(M)$ in a non-flat, real space form $M$. We first show that the para-Kähler metric ${\mathcal G}$ is scalar flat and when $M$ is a 3-dimensional real space form, ${\mathcal G}$ is locally conformally flat. Furthermore, we prove that the space of oriented geodesics in hyperbolic $n$-space, equipped with the constructed metric ${\mathcal G}$, is minimally isometric embedded in the tangent bundle of the hyperbolic $n$-space. We then study the submanifold theory, and we show that ${\mathcal G}$-geodesics correspond to minimal ruled surfaces in the real space form. Lagrangian submanifolds (with respect to the canonical symplectic structure $Ω$) play an important role in the geometry of the space of oriented geodesics as they are the Gauss map of hypersurfaces in the corresponding space form. We demonstrate that the Gauss map of a non-flat hypersurface of constant Gauss curvature is a minimal Lagrangian submanifold. Finally, we show that a Hamiltonian minimal submanifold is locally the Gauss map of a hypersurface $Σ$ that is a critical point of the functional $\mathcal{F}(Σ)=\int_Σ\sqrt{|K|}\,dV$, where $K$ denotes the Gaussian curvature of $Σ$.

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arXiv2018

A new geometric structure on tangent bundles

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown that $G$ is locally symmetric if and only if $g$ is locally symmetric. We then study submanifolds in $TN$ and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph to be minimal or Hamiltonian minimal in the tangent bundle $T{\mathbb R}^n$ of the Euclidean real space ${\mathbb R}^n$ are studied. Finally, using the cross product in ${\mathbb R}^3$ we show that the space of oriented lines in ${\mathbb R}^3$ can be minimally isometrically embedded in $T{\mathbb R}^3$.

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arXiv2016

The causal topology of neutral 4-manifolds with null boundary

This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null boundaries that arise from canonical 3- and 4-dimensional settings. A null hypersurface is foliated by its normal and, in the neutral case, inherits a pair of totally null planes at each point. This paper focuses on these plane bundles in a number of classical settings The first construction is the conformal compactification of flat neutral 4-space into the 4-ball. The null foliation on the boundary in this case is the Hopf fibration on the 3-sphere and the totally null planes in the boundary are integrable. The metric on the 4-ball is a conformally flat, scalar-flat, positive Ricci curvature neutral metric. The second constructions are subsets of the 4-dimensional space of oriented geodesics in a 3-dimensional space-form, equipped with its canonical neutral metric. We consider all oriented geodesics tangent to a given embedded strictly convex 2-sphere. The third is a neutral geometric model for the intersection of two surfaces in a 4-manifold. The surfaces are the sets of oriented normal lines to two round spheres in Euclidean 3-space, which form Lagrangian surfaces in the 4-dimensional space of all oriented lines. The intersection of the boundaries of their normal neighbourhoods form tori that we prove are totally real and Lorentz if the spheres do not intersect. We conclude with possible topological applications of the three constructions, including neutral Kirby calculus, neutral knot invariants and neutral Casson handles, respectively.

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arXiv2016

Hopf hypersurfaces in spaces of oriented geodesics

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the oriented geodesic space admits a canonical Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in the underlying space form. We prove that a tangent hypersurface is Hopf in the space of oriented geodesics with respect to this canonical (para-)Kaehler structure iff the underlying convex hypersurface is totally umbilic and non-flat. In the case of 3 dimensional space forms, however, there exists a second canonical complex structure which can also be used to define Hopf hypersurfaces. We prove that in this dimension, the tangent hypersurface of a convex hypersurface in the space form is always Hopf with respect to this second complex structure.

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arXiv2016

Minimal surfaces in the product of two dimensional real space forms endowed with a neutral metric

We investigate minimal surfaces in products of two-spheres ${\mathbb S}^2_p\times {\mathbb S}^2_p$, with the neutral metric given by $(g,-g)$. Here ${\mathbb S}^2_p\subset {\mathbb R}^{p,3-p}$ , and $g$ is the induced metric on the sphere. We compute all totally geodesic surfaces and we give a relation between minimal surfaces and the solutions of the Gordon equations. Finally, in some cases we give a topological classification of compact minimal surfaces.

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arXiv2014

On Hamiltonian minimal submanifolds in the space of oriented geodesics in real space forms

We prove that a deformation of a hypersurface in a $(n+1)$-dimensional real space form ${\mathbb S}^{n+1}_{p,1}$ induce a Hamiltonian variation of the normal congruence in the space ${\mathbb L}({\mathbb S}^{n+1}_{p,1})$ of oriented geodesics. As an application, we show that every Hamiltonian minimal sumbanifold in ${\mathbb L}({\mathbb S}^{n+1})$ (resp. ${\mathbb L}({\mathbb H}^{n+1})$) with respect to the (para-) Kaehler Einstein structure is locally the normal congruence of a hypersurface $Σ$ in ${\mathbb S}^{n+1}$ (resp. ${\mathbb H}^{n+1}$) that is a critical point of the functional ${\cal W}(Σ)=\int_Σ\left(Π_{i=1}^n|ε+k_i^2|\right)^{1/2}$, where $k_i$ denote the principal curvatures of $Σ$ and $ε\in\{-1,1\}$. In addition, for $n=2$, we prove that every Hamiltonian minimal surface in ${\mathbb L}({\mathbb S}^{3})$ (resp. ${\mathbb L}({\mathbb H}^{3})$) with respect to the (para-) Kaehler conformally flat structure is locally the normal congruence of a surface in ${\mathbb S}^{3}$ (resp. ${\mathbb H}^{3}$) that is a critical point of the functional ${\cal W}'(Σ)=\int_Σ\sqrt{H^2-K+1}$ (resp. ${\cal W}'(Σ)=\int_Σ\sqrt{H^2-K-1}\; $), where $H$ and $K$ denote, respectively, the mean and Gaussian curvature of $Σ$.

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arXiv2014

Lagrangian immersions in the product of Lorentzian two manifold

For Lorentzian 2-manifolds $(Σ_1,g_1)$ and $(Σ_2,g_2)$ we consider the two product para-Kähler structures $(G^ε,J,Ω^ε)$ defined on the product four manifold $Σ_1\timesΣ_2$, with $ε=\pm 1$. We show that the metric $G^ε$ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures $κ_1,κ_2$ of $g_1,g_2$, respectively, are both constants satisfying $κ_1=-εκ_2$ (resp. $κ_1=εκ_2$). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product $γ_1\timesγ_2\subsetΣ_1\timesΣ_2$, where $γ_1$ and $γ_2$ are curves of constant curvature. We study Lagrangian surfaces in the product $d{\mathbb S}^2\times d{\mathbb S}^2$ with non null parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and $H$-minimal surfaces.

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arXiv2013

Marginally trapped surfaces in spaces of oriented geodesics

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds is marginally trapped. In the Euclidean case we show that Lagrangian rotationally symmetric sections are marginally trapped and construct an explicit family of marginally trapped Lagrangian tori. In the hyperbolic case we explore the relationship between marginally trapped and Weingarten surfaces, and construct examples of marginally trapped surfaces with various properties.

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arXiv2013

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

Let $(Σ_1,g_1)$ and $(Σ_2,g_2)$ be connected, complete and orientable Riemannian two manifolds. Consider the two canonical Kähler structures $(G^ε,J,Ω^ε)$ on the product 4-manifold $Σ_1\timesΣ_2$ given by $ G^ε=g_1\oplus εg_2$, $ε=\pm 1$ and $J$ is the canonical product complex structure. Thus for $ε=1$ the Kähler metric $G^+$ is Riemannian while for $ε=-1$, $G^-$ is of neutral signature. We show that the metric $G^ε$ is locally conformally flat iff the Gauss curvatures $κ(g_1)$ and $κ(g_2)$ are both constants satisfying $κ(g_1)=-εκ(g_2)$. We also give conditions on the Gauss curvatures for which every $G^ε$-minimal Lagrangian surface is the product $γ_1\timesγ_2\subsetΣ_1\timesΣ_2$, where $γ_1$ and $γ_2$ are geodesics of $(Σ_1,g_1)$ and $(Σ_2,g_2)$, respectively. Finally, we explore the Hamiltonian stability of projected rank one Hamiltonian $G^ε$-minimal surfaces.

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arXiv2012

Hamiltonian stability of Hamiltonian minimal Lagrangian submanifolds in pseudo- and para-Kähler manifolds

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular we observe that a minimal Lagrangian submanifold L in a Ricci-flat pseudo- or para-Kähler manifold is H-stable, i.e. its second variation is definite and L therefore a local extremizer of the volume with respect to Hamiltonian variations. We also give a stability criterion for spacelike minimal Lagrangian submanifolds in para-Kähler manifolds, similar to Oh's stability criterion for minimal Lagrangian manifolds in Kähler-Einstein manifolds. Finally, we determine the H-stability of a series of examples of H-minimal Lagrangian submanifolds: the product of n circles of arbitrary radii in complex space C^n is H-unstable with respect to any indefinite flat Hermitian metric, while the product of n hyperbolas in para-complex vector space D^n is H-stable for n=1,2 and H-unstable for n > 2. Recently, minimal Lagrangian surfaces in the space of geodesics of three-dimensional space forms have been characterized; on the other hand, a class of H-minimal Lagrangian surfaces in the tangent bundle of a Riemannian, oriented surface has been identified. We discuss the H-stability of all these examples.

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arXiv2010

On maximal surfaces in the space of oriented geodesics of hyperbolic 3-space

We study area-stationary, or maximal, surfaces in the space ${\mathbb L}({\mathbb H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in ${\mathbb L}({\mathbb H}^3)$ is a maximal surface. We then classify Lagrangian maximal surfaces $Σ$ in ${\mathbb L}({\mathbb H}^3)$ and prove that the family of parallel surfaces in ${\mathbb H}^3$ orthogonal to the geodesics $γ\inΣ$ form a family of equidistant tubes around a geodesic.

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arXiv2009

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ${\mathbb{H}}^3$ orthogonal to the geodesics of $Σ$. We prove that the induced metric on a Lagrangian surface in ${\mathbb{L}}({\mathbb{H}}^3)$ has zero Gauss curvature iff the orthogonal surfaces in ${\mathbb{H}}^3$ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in ${\mathbb{L}}({\mathbb{H}}^3)$ and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ${\mathbb{H}}^3$.

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arXiv2008

Totally Null Surfaces in Neutral Kaehler 4-Manifolds

We study the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual ($α$-planes) or anti-self-dual ($β$-planes) and so we consider $α$-surfaces and $β$-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the $α$-planes are integrable and $α$-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The $β$-surfaces are less known and our interest is mainly in their description. In particular, we classify the $β$-surfaces of the neutral Kaehler metric on $TN$, the tangent bundle to a Riemannian 2-manifold $N$. These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which we show that the $β$-surfaces are affine tangent bundles to curves of constant geodesic curvature on $S^2$ and $H^2$, respectively. In addition, we construct the $β$-surfaces of the space of oriented geodesics of hyperbolic 3-space.

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