3-dimensional Space Forms Research Articles

A Symmetry Result on Submanifolds of Space Forms and Applications

In this paper we prove a symmetry result on submanifolds of codimension one in a (n + 1)-dimensional space form, related to the geodesic distance and to the normal curvature of some fixed vector field. As applications we will prove sphere characterization type theorems for Kahler manifolds endowed with a toric group action.

Surfaces with parallel mean curvature in S3 x R and H3 x R

We prove a Simons type equation for non-minimal surfaces with par- allel mean curvature vector (pmc surfaces) in M 3 (c) × R, where M 3 (c) is a 3- dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.

Bertrand curves in 3-dimensional space forms

We define Bertrand curves in a 3-dimensional Riemannian manifold and prove that a Frenet curve α with the curvature κ and the torsion τ in a 3-dimensional simply connected space form is a Bertrand curve if and only if it satisfies τ=0 or κ+aτ=b for constants a and b≠0.

Spectral theory on 3-dimensional hyperbolic space and Hermitian modular forms

Abstract We study some arithmetics of Hermitian modular forms of degree two by applying the spectral theory on 3-dimensional hyperbolic space. This paper presents three main results: (1) a 3-dimensional analogue of Katok–Sarnak's correspondence, (2) an analytic proof of a Hermitian analogue of the Saito–Kurokawa lift by means of a converse theorem, (3) an explicit formula for the Fourier coefficients of a certain Hermitian Eisenstein series.

On the extrema of Dirichlet’s first eigenvalue of a family of punctured regular polygons in two dimensional space forms

Let \( \wp_{1}, \wp_0 \) be two regular polygons of n sides in a space form M 2(κ) of constant curvature κ = 0,1 or − 1 such that \(\wp_0 \subset \wp_1\) and having the same center of mass. Suppose \( \wp_{0} \) is circumscribed by a circle C contained in \( \wp_{1} .\) We fix \(\wp_1\) and vary \(\wp_0\) by rotating it in C about its center of mass. Put \(\Omega=(\wp_{1} \setminus \wp_{0})^{0},\) the interior of \( \wp_{1} \setminus \wp_{0} \) in M 2(κ). It is shown that the first Dirichlet’s eigenvalue λ 1( Ω) attains extremum when the axes of symmetry of \( \wp_{0} \) coincide with those of \( \wp_{1}.\)

Constant angle surfaces in product spaces

We classify all the surfaces in M2(c1)×M2(c2) for which the tangent space TpM2 makes constant angles with Tp(M2(c1)×{p2}) (or equivalently with Tp({p1}×M2(c2)) for every point p=(p1,p2) of M2. Here M2(c1) and M2(c2) are 2-dimensional space forms, not both flat. As a corollary we give a classification of all the totally geodesic surfaces in M2(c1)×M2(c2).

Kähler immersions of homogeneous Kähler manifolds into complex space forms

In this paper we study the homogeneous Kaehler manifolds (h.K.m.) which can be Kaehler immersed into finite or infinite dimensional complex space forms. On the one hand we completely classify the h.K.m. which can be Kaehler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. On the other hand, we extend known results about Kaehler immersions into the finite dimensional complex projective space to the infinite dimensional setting

Lorentzian stationary surfaces in 4-dimensional space forms of index 2

We discuss the necessary and sufficient conditions for the existence of Lorentzian stationary surfaces in 4-dimensional space forms of index 2, and isometric stationary deformations preserving normal curvature.

Critical curves for the total normal curvature in surfaces of 3-dimensional space forms

A variational problem closely related to the bending energy of curves contained in surfaces of real 3-dimensional space forms is considered. We seek curves in a surface which are critical for the total normal curvature energy (and its generalizations). We start by deriving the first variation formula and the corresponding Euler–Lagrange equations of these energies and apply them to study critical special curves (geodesics, asymptotic lines, lines of curvature) on surfaces. Then, we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear Weingarten surface. Finally, we give some classification and existence results for this family of rotation surfaces.

On Deformable Minimal Hypersurfaces in Space Forms

The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss–Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with vanishing Gauss–Kronecker curvature and scalar curvature bounded from below.

A measure of symmetry for the moduli of spherical minimal immersions

A Grunbaum type of measure of symmetry is calculated and estimated for the DoCarmo-Wallach moduli spaces for eigenmaps and spherical minimal immersions. The DeTurck-Ziller classification of minimal imbeddings of 3-dimensional space forms is used to obtain exact determination of the measure for the SU(2)-equivariant moduli.

Parabolic Geodesics in Sasakian 3-Manifolds

AbstractWe give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms.

Second Eigenvalue of Paneitz Operators and Mean Curvature

For $n\geq 7$, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact $n$-dimensional submanifolds in an $(n+p)$-dimensional space form.

Spinorial Characterizations of Surfaces into 3-dimensional Pseudo-Riemannian Space Forms

We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in $\mathbb{R}^{2,1}$ to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well as for spacelike and timelike immersions of surfaces of signature (0,2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.

Real hypersurfaces which are contact in a nonflat complex space form

In an $n$ $(\geqq2)$-dimensional nonflat complex space form $\widetilde{M}_n(c)(=\mathbb{C}P^n(c)$ or $\mathbb{C}H^n(c)$), we classify real hypersurfaces $M^{2n-1}$ which are contact with respect to the almost contact metric structure $(\phi,\xi,\eta,g)$ induced from the K\"ahler structure $J$ and the standard metric $g$ of the ambient space $\widetilde{M}_n(c)$. Our theorems show that this contact manifold $M^{2n-1}$ is congruent to a homogeneous real hypersurface of $\widetilde{M}_n(c)$.

Spectral deformation and Bäcklund transformation of constrained Willmore surfaces

The class of constrained Willmore surfaces in space-forms forms a Möbius invariant class of surfaces with strong links to the theory of integrable systems. This paper is dedicated to an overview on the topic. We define a spectral deformation, by the action of a loop of flat metric connections, and Bäcklund transformations, by applying a dressing action. We establish a permutability between spectral deformation and Bäcklund transformation and verify that all these transformations corresponding to the zero multiplier preserve the class of Willmore surfaces. We show that, for special choices of parameters, both spectral deformation and Bäcklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, and, in particular, the class of constant mean curvature surfaces in 3-dimensional space-forms.

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

For constant mean curvature surfaces of class C 2 immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the unique ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.

Biharmonic Riemannian submersions from 3-manifolds

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form of non-positive curvature into a surface is biharmonic if and only if it is harmonic.

On complete spacelike submanifolds with second fundamental form locally timelike in a semi-Riemannian space form

In this work we obtain a Simon’s type inequality for a suitable tensor and apply it in order to obtain some results on a complete n -dimensional spacelike submanifold M n with parallel mean curvature vector and second fundamental form locally timelike in an ( n + p ) -dimensional semi-Riemannian space form, N q n + p ( c ) of index q ( 1 ≤ q ≤ p ).

The Gauss-Bonnet-Grotemeyer Theorem in space forms

In 1963, K.P.~Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let $M$ be an oriented closed surface in theEuclidean space $\mathbb R^3$ with Euler characteristic $\chi(M)$, Gauss curvature $G$ and unit normal vector field $\vec n$. Grotemeyer'sidentity replaces the Gauss-Bonnet integrand $G$ by the normal moment $ ( \vec a \cdot \vec n )^2G$, where $a$ is a fixed unit vector: $\int_M(\vec a\cdot \vec n)^2 Gdv=\frac{2 \pi}{3}\chi(M) $. We generalize Grotemeyer's result to oriented closed even-dimensionalhypersurfaces of dimension $n$ in an $(n+1)$-dimensional space form $N^{n+1}(k)$.