Three-dimensional Space Forms Research Articles

Kirchhoff elastic rods in three-dimensional space forms

The Kirchhoff elastic rod is one of the mathematical models of thin elastic rods, and is characterized as a critical point of the energy functional obtained by adding the effect of twisting to the bending energy. In this paper, we investigate Kirchhoff elastic rods in three-dimensional space forms. In particular, we give explicit formulas of Kirchhoff elastic rods in the three-sphere and in the three-dimensional hyperbolic space in terms of Jacobi sn function and the elliptic integrals.

Generalized Weierstrass representations of surfaces with the constant Gauss curvature in pseudo-Riemannian three-dimensional space forms

We know that there exists the correspondence between one of the nonumbilical Riemannian surface with constant Gauss curvature K in M¯3(K¯)(K≠K¯), the nonumbilical spacelike or timelike surfaces with constant Gauss curvature K in M¯13(K¯)(K≠K¯), and a solution of one of the following differential equations: the sine-Gordon, sinh-Gordon, sine-laplace, sinh-laplace, and cosh-Gordon equations. In this paper, we consider the initial value problems of these equations and give the generalized Weierstrass representations of these surfaces that depend only on the initial values of these equations.

Ribaucour Transformations for Hypersurfaces in Space Forms

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.

Particles with curvature and torsion in three-dimensional pseudo-Riemannian space forms

We consider the motion of relativistic particles described by an action which is a function of the curvature and torsion of the particle path. The Euler–Lagrange equations and the dynamical constants of the motion are expressed in a simple way in terms of a suitable coordinate system. The moduli spaces of solutions in a three-dimensional pseudo-Riemannian space form are completely exhibited.

A new variational characterization of 𝑛-dimensional space forms

A Riemannian manifold ( M n , g ) (M^n,g) is associated with a Schouten ( 0 , 2 ) (0,2) -tensor C g C_g which is a naturally defined Codazzi tensor in case ( M n , g ) (M^n,g) is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional F k [ g ] = ∫ M σ k ( C g ) d v o l g \mathcal {F}_k[g]=\int _M\sigma _k(C_g)dvol_g defined on M 1 = { g ∈ M | V o l ( g ) = 1 } \mathcal {M}_1=\{g\in \mathcal {M}|Vol(g)=1\} , where M \mathcal {M} is the space of smooth Riemannian metrics on a compact smooth manifold M M and { σ k ( C g ) , 1 ≤ k ≤ n } \{\sigma _k(C_g),\ 1\leq k\leq n\} is the elementary symmetric functions of the eigenvalues of C g C_g with respect to g g . We prove that if n ≥ 5 n\geq 5 and a conformally flat metric g g is a critical point of F 2 | M 1 \mathcal {F}_2|_{\mathcal {M}_1} with F 2 [ g ] ≥ 0 \mathcal {F}_2[g]\geq 0 , then g g must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky’s very recent theorem that the critical point of F 2 | M 1 \mathcal {F}_2|_{\mathcal {M}_1} with F 2 [ g ] ≥ 0 \mathcal {F}_2[g]\geq 0 characterized the three-dimensional space forms.

Determining the shape of the universe using discrete sources

Suppose we have identified three clusters of galaxies as being topological copies of the same object. How does this information constrain the possible models for the shape of our universe? It is shown here that, if our universe has flat spatial sections, these multiple images can be accommodated within any of the six classes of compact orientable three-dimensional flat space forms. Moreover, the discovery of two more triples of multiple images in the neighbourhood of the first one would allow the determination of the topology of the universe, and in most cases the determination of its size.

$B$-Scrolls with Non-Diagonalizable Shape Operators

We study some Lorentzian surfaces in the three-dimensional Lorentzian space forms whose shape operators are not diagonalizable at least at one point. It is related to the so-called notion of 2-type surfaces. A local classification theorem in this respect is obtained.

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H 3 . We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1.

A new variational characterization of three-dimensional space forms

A new variational characterization of three-dimensional space forms

General Helices in the Three-Dimensional Lorentzian Space Forms

We present some Lancret-type theorems for general helices in the three-dimensional Lorentzian space forms which show remarkable differences with regard to the same question in Riemannian space forms. The key point will be the problem of solving natural equations. We give a geometric approach to that problem and show that general helices in the three-dimensional Lorentz-Minkowskian space are geodesics either of right general cylinders or of flat B-scrolls. In this sense the anti De Sitter and De Sitter worlds behave as the spherical and hyperbolic space forms, respectively.

Solutions of the Betchov-Da Rios soliton equation: a Lorentzian approach

The purpose of this paper is to find out explicit solutions of the Betchov-Da Rios soliton equation in three-dimensional Lorentzian space forms. We start with non-null curves and obtain solutions living in certain flat ruled surfaces in l 3 and h 1 3, as well as in r 3 and s 3. Next we take a null curve and have got solutions lying in the associated B-scrolls in l 3, s 1 3 and h 1 3. It should be pointed out that we extend previous results already obtained, and as far as we know, this is the first time that solutions in the De Sitter 3-space appear in the literature. Soliton solutions are characterized as null geodesics in B-scrolls.

Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.

Static fields on three-dimensional space forms in terms of contour integrals of twistor functions

Contour integral formulae for massless fields on R 3 are described. These are used to give local explicit expressions for harmonic morphisms defined on domains of R 3. The construction is generalized to the three-sphere and hyperbolic three-space.

Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms

AbstractA complete classification is given of harmonic morphisms to a surface and conformal foliations by geodesics, with or without isolated singularities, of a simply-connected space form. The method is to associate to any such a holomorphic map from a Riemann surface into the space of geodesics of the space form. Properties such as nonintersecting fibres (or leaves) are translated into conditions on the holomorphic mapping which show it must have a simple form corresponding to a standard example.