Curves in three dimensional Riemannian space forms

We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian) and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space.

Bertrand curves in 3-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 \( \wp_{1}.\)

BackgroundCommon existing phylogenetic tree visualisation tools are not able to display readable trees with more than a few thousand nodes. These existing methodologies are based in two dimensional space.ResultsWe introduce the idea of visualising phylogenetic trees in three dimensional hyperbolic space with the Walrus graph visualisation tool and have developed a conversion tool that enables the conversion of standard phylogenetic tree formats to Walrus' format. With Walrus, it becomes possible to visualise and navigate phylogenetic trees with more than 100,000 nodes.ConclusionWalrus enables desktop visualisation of very large phylogenetic trees in 3 dimensional hyperbolic space. This application is potentially useful for visualisation of the tree of life and for functional genomics derivatives, like The Adaptive Evolution Database (TAED).

We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from Bäcklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem.

Geometry of linear and angular momenta of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>N</mml:mi></mml:math>-particles in the Riemannian space forms and de Sitter space

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in m m -dimensional Euclidean space, or a set of n n -by- m m systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in m m -dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of n n -by- m m systems of linear forms. Consequently, a class of m m -dimensional results in Diophantine approximation can be bootstrapped to corresponding n n -by- m m -dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

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.

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 $\Sigma$ in ${\mathbb S}^{n+1}$ (resp. ${\mathbb H}^{n+1}$) that is a critical point of the functional ${\cal W}(\Sigma)=\int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}$, where $k_i$ denote the principal curvatures of $\Sigma$ and $\epsilon\in\{-1,1\}$. 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 ${\cal W}'(\Sigma)=\int_\Sigma\sqrt{H^2-K+1}$ (resp. ${\cal W}'(\Sigma)=\int_\Sigma\sqrt{H^2-K-1}\; $), where $H$ and $K$ denote, respectively, the mean and Gaussian curvature of $\Sigma$.

Rotation hypersurfaces in semi-Riemannian space forms of arbitrary index

an estimate (Theorem 3) which is sharper than those mentioned above with the additional hypothesis of an upper curvature bound but without global hypothesis on the injectivity radius. Moreover, this estimate is sharp with e quality holding only in the case of spheres of constant curvature and gives bounds for higher eigenvalues if the dimension is at least three. If the manifold is homeomorphic to certain -dimensional spherical space forms and the bounds on the injectivity radius and sectional curvature hold globally, then we can give an upper bound for the th eigenvalue (Theorem 4). A sharp lower bound for the sum of the reciprocals of the first t hree eigenvalues of 2 was made by Hersch [10] in terms of area alone. P. Yang and S.-T. Yau [17] generalized this estimate to compact surfaces in terms of genus and area. Both of these results give an upper bound for the first eigenvalue whi ch is sharp in the case of spheres. P. Li and S.-T. Yau [12] reproduced this estimate by employing a conformal invariant, the conformal area. They were also able to give a sharp estimate in the case of the real projective plane and for the conformal class of the square, flat torus also only in terms of area. In higher dimensions their estima tes require ( ) to be conformally equivalent to an immersed, minimal submanifold of the standard sphere of dimension ≥ . Examples of H. Urakawa [15] and J. Dodziuk [5] show that for dimension at least three, there does not exist an upper bound for the first eigenvalue in terms of volume alone. Other upper estimates for λ1 involve either a lower bound on curvature [3], [7], or hold only for surfaces [6], [9], [10], [14]. See [11] for a survey of eigenvalue bounds. We will use the following notation. The metric ball of radius at a point

In this paper we consider minimal Lagrangian submanifolds in $n$-dimensional complex space forms. More precisely, we study such submanifolds which, endowed with the induced metrics, write as a Riemannian product of two Riemannian manifolds, each having constant sectional curvature. As the main result, we give a complete classification of these submanifolds.

Curve motion inducing modified Korteweg-de Vries systems

Helices and constant procession curves are special examples of slant curves. However, there is no example of a $k$-slant curve for a positive integer $k\geq 2$ in three dimensional Euclidean spaces. Furthermore, the position vector of a $k$-slant curve for a positive integer $k\geq 2$ has not been known thus far. In this paper, we propose a method for constructing $k$-slant curves in three dimensional Euclidean spaces. We then show that spherical $k$-slant curves and $N_{k}$-constant procession curves can be derived from circles, for $k \in \mathbb{N}$, the set of all nonnegative integers. In addition, we provide a new proof of the spherical curve characterization and define a curve in the sphere called a spherical prime curve. Afterward, we apply $k$-slant curves to magnetic curves. Finally, we discuss the need for further research.

We prove that a sphere-foliated minimal or constant mean curvature hypersurface in hyperbolic space of dimension > 5 is one of the following: hypersurface of rotation around a geodesic, geodesic hyperplane, horosphere, equidistant hypersurface, or a geodesic sphere in the upper half-space model. And we show that a sphere-foliated minimal or constant mean curvature hypersurface in sphere of dimension > 5 is either a hypersurface of rotation or a hypersphere. We also show that a hypersurface of nonzero constant mean curvature in Lorentz-Minkowski space foliated by spheres in space-like hyperplanes is either a hypersurface of rotation or a pseudo-hyperbolic space and that maximal space-like hypersurfaces foliated by spheres in hyperplanes are rotational if the ambient space has dimension > 4.