3-dimensional Space Forms Research Articles

The Sasakian statistical structures of constant ϕ-sectional curvature on Sasakian space forms

In this paper, we investigate the Sasakian statistical structures of constant ϕ-sectional curvature based on Sasakian space forms. We obtain the classification of this kind of Sasakian statistical structures. Our classification results show that the Sasakian statistical structures of constant ϕ-sectional curvature on a Sasakian space form with dimension higher than 3 must be almost-trivial; on a 3-dimensional Sasakian space form, in addition to the almost-trivial Sasakian statistical structure, there exist other Sasakian statistical structures which satisfy the constant ϕ-sectional curvature condition. We also point out that a rigidity result for cosymplectic statistical structures of constant ϕ-sectional curvature on 3-dimensional cosymplectic space forms in [11] can be improved to the corresponding classification result.

Homogeneous Riemannian Structures in Thurston Geometries and Contact Riemannian Geometries

We give explicit parametrizations for all the homogeneous Riemannian structures on model spaces of Thurston geometry. As an application, we give all the homogeneous contact metric structures on $3$-dimensional Sasakian space forms.

Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures

The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to non-null framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by $\mathbb{M}_q^n(c)$ the $n$-dimensional space form of index $q=0,1$ and constant curvature $c\neq 0$. We introduce two types of framed Bertrand curves and framed Mannheim curves in $\mathbb{M}_q^3(c)$ by using two different moving frames: the general moving frame and the Frenet-type frame. We investigate geometric properties of these framed Bertrand and framed Mannheim curves in $\mathbb{M}_q^3(c)$ that may have singularities. We then give characterizations for a non-null framed curve to be a framed Bertrand curve or to be a framed Mannheim curve. We show that in special cases these characterizations reduce to the well-known classical formulas: $\lambda \kappa+\mu \tau=1$ for Bertrand curves and $\lambda(\kappa^2+\tau^2)=\kappa$ for Mannheim curves. We provide several examples to support our results, and we visualize these examples by using the Hopf map, the hyperbolic Hopf map, and the spherical projection.

Local rigidity of constant mean curvature hypersurfaces in space forms

In this paper, we study the local rigidity of constant mean curvature (CMC) hypersurfaces. Let x:Mn→Mn+1(c),n≥4, be a piece of immersed constant mean curvature hypersurface in the (n+1)-dimensional space form Mn+1(c). We prove that if the scalar curvature R is constant and the number g of the distinct principal curvatures satisfies g≤3, then Mn is an isoparametric hypersurface. Further, if Mn is a minimal hypersurface, then Mn is a totally geodesic hypersurface for c≤0, and Mn is either a Cartan minimal hypersurface, a Clifford minimal hypersurface, or a totally geodesic hypersurface for c>0, which solves the high dimensional version of Bryant Conjecture.

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi’s rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

Deformations of cuspidal edges in a 3-dimensional space form

Deformations of cuspidal edges in a 3-dimensional space form

The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

<abstract><p>In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.</p></abstract>

The cosymplectic statistical structures of constant ϕ-sectional curvature on cosymplectic space forms

In this paper, we study the cosymplectic statistical structures of constant ϕ-sectional curvature based on cosymplectic space forms. For higher than 3-dimensional cosymplectic space forms, we prove that the cosymplectic statistical structure of constant ϕ-sectional curvature on which must be unique and we determine it completely. For 3-dimensional cosymplectic space forms, we give an example to show that the cosymplectic statistical structures of constant ϕ-curvature are not unique, and we also obtain a rigidity theorem in this case.

Concircular Hypersurfaces and Concircular Helices in Space Forms

In this paper, we find a full description of concircular hypersurfaces in space forms as a special family of ruled hypersurfaces. We also characterize concircular helices in 3-dimensional space forms by means of a differential equation involving the concircular factor and their curvature and torsion, and we show that the concircular helices are precisely the geodesics of the concircular surfaces.

Minimal Legendrian submanifolds in Sasakian space forms with C-parallel second fundamental form

It is known that Cheng-He-Hu [3] gave a complete classification of the n-dimensional C-totally real (or equivalently, Legendrian) minimal submanifolds in the (2n+1)-dimensional Sasakian space form N2n+1(c) with constant sectional curvature. According to this classification and the proof therein, such minimal Legendrian submanifolds are all of C-parallel second fundamental form. The purpose of this paper is to extend the above result partly, and as the main results, we classify all the minimal Legendrian submanifolds in N2n+1(c) with C-parallel second fundamental form in cases n=2,3,4.

On minimal immersions of a singular non-CSC extremal Kähler metric into 3-dimensional space forms

On minimal immersions of a singular non-CSC extremal Kähler metric into 3-dimensional space forms

Killing submersions and magnetic curves

We investigate magnetic curves in Killing submersions and we show that the bundle curvature is constant along magnetic curves with respect to the Killing vector field if and only if all vertical tubes derived from these magnetic curves are of constant mean curvature. Then we examine magnetic Jacobi fields along horizontal Killing magnetic curves in the total space of a Killing submersion. We generalize some important results obtained by O'Neill for horizontal geodesics to horizontal magnetic curves. Finally, we study magnetic Jacobi fields along horizontal Killing magnetic trajectories in 3-dimensional Sasakian space forms.

On the uniqueness of complete biconservative surfaces in 3-dimensional space forms

Biconservative surfaces are surfaces with divergence-free stress-bienergy tensor. Simply connected, complete, non-$CMC$ biconservative surfaces in $3$-dimensional space forms were constructed working in extrinsic and intrinsic ways. Then, one raises the question of the uniqueness of such surfaces. In this paper we give a positive answer to this question.

Quartic differentials and harmonic maps in conformal surface geometry

We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightlike mean curvature in 3-dimensional lightcones. We then investigate Bryant's quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconformal or orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply the previous result to characterise surfaces with such a property.

On η-biharmonic hypersurfaces with constant scalar curvature in higher dimensional pseudo-Riemannian space forms

In this paper, η-biharmonic hypersurfaces Mrn with constant scalar curvature in a pseudo-Riemannian space form are studied. Under the assumption that Mrn has diagonalizable shape operator with at most six distinct principal curvatures, we prove that Mrn has constant mean curvature. As an application, we obtain a partial classification result of these hypersurfaces and show that (nc)-biharmonic hypersurfaces must be minimal.

Special Slant Surfaces with Non-constant Mean Curvature in 2-Dimensional Complex Space Forms

In the late 1990s, B. Y. Chen introduced the notion of special slant surfaces in K\"{a}hler surfaces and classified non-minimal proper special slant surfaces with constant mean curvature in $2$-dimensional complex space forms. In this paper, we completely classify proper special slant surfaces with non-constant mean curvature in $2$-dimensional complex space forms.

Rotation hypersurfaces in semi-Riemannian space forms of arbitrary index

Let n, ν be integers such that n≥2 and 0≤ν≤n+1. Let Mc,νn+1 be a (n+1)-dimensional semi-Riemannian space form of constant sectional curvature c and index ν. We give the parametrizations of rotation hypersurfaces in Mc,νn+1. Adding the condition for the r-th mean curvature to be constant, we show the study of these hypersurfaces is reduced to the study of a function on two variables. As an illustrative example, we study the particular case of rotation hypersurfaces with constant scalar curvature in Mc,νn+1, determining the values of n,c,ν for which these hypersurfaces are isoparametric. In the final part we study the variation of the second fundamental form of rotation hypersurfaces in order to obtain some rigidity results.

Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form

Let \(\mathscr{G}\) be the manifold of all (unparametrized) oriented lines of \(\mathbb{R}^{3}\). We study the controllability of the control system in \(\mathscr{G}\) given by the condition that a curve in \(\mathscr{G}\) describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed \(\alpha \). Actually, we pose the analogous more general problem by means of a control system on the manifold \(\mathscr{G}_{\kappa }\) of all the oriented complete geodesics of the three dimensional space form of curvature \(\kappa \): \(\mathbb{R}^{3}\) for \(\kappa =0\), \(S^{3}\) for \(\kappa =1\) and hyperbolic 3-space for \(\kappa =-1\). We obtain that the system is controllable if and only if \(\alpha ^{2}\neq \kappa \). In the spherical case with \(\alpha =\pm 1\), an admissible curve remains in the set of fibers of a fixed Hopf fibration of \(S^{3}\).We also address and solve a sort of Kendall’s (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.

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.

Four-dimensional steady gradient Ricci solitons with 3-cylindrical tangent flows at infinity

In this paper we consider 4-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed 4-manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of R with a 3-dimensional spherical space form. We also classify the tangent flows at infinity of 4-dimensional steady soliton singularity models in general.