Planar Geodesics Research Articles

Surfaces in Sol3 Space Foliated by Circles

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol3. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in totally geodesic planes and we classify all such surfaces under the assumption of minimality or flatness.

Minimal translation surfaces in hyperbolic space

In the half-space model of hyperbolic space, that is, $${\mathbb{R}^3_{+}=\{(x,y,z)\in\mathbb{R}^3;z > 0\}}$$ with the hyperbolic metric, a translation surface is a surface that writes as z = f(x) + g(y) or y = f(x) + g(z), where f and g are smooth functions. We prove that the only minimal translation surfaces (zero mean curvature in all points) are totally geodesic planes.

Affine planar geodesic immersions with maximal codimension

This work gives a classification theorem for affine immersions with planar geodesics in the case where the codimension is maximal. Vrancken classified parallel affine immersions in this case and obtained, among others, generalized Veronese submanifolds. In this work it is shown that the immersions with planar geodesics are the same as the parallel ones in the considered case. A geometric interpretation of parallel immersions is also given: The affine immersions with pointwise planar normal sections (with respect to the equiaffine transversal bundle) are parallel. This result is verified for surfaces in ℝ4 and for immersions with the maximal codimension.

Generalized Kerr spacetime with an arbitrary mass quadrupole moment: geometric properties versus particle motion

An exact solution of Einstein's field equations in empty space first found in 1985 by Quevedo and Mashhoon is analyzed in detail. This solution generalizes Kerr spacetime to include the case of matter with an arbitrary mass quadrupole moment and is specified by three parameters, the mass M, the angular momentum per unit mass a and the quadrupole parameter q. It reduces to the Kerr spacetime in the limiting case q = 0 and to the Erez–Rosen spacetime when the specific angular momentum a vanishes. The geometrical properties of such a solution are investigated. Causality violations, directional singularities and repulsive effects occur in the region close to the source. Geodesic motion and accelerated motion are studied on the equatorial plane which, due to the reflection symmetry property of the solution, also turns out to be a geodesic plane.

Erratum to “Affine surfaces in R4 with planar geodesics with respect to the affine metric”

In this work a mistake in the paper is corrected. There is also a new proof of the main theorem which classifies the non-degenerate affine surfaces in R 4 having planar geodesics with respect to the affine metric.

The embedded singly periodic Scherk-Costa surfaces

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Open image in new window with finite total curvature.

Integral geometry of equidistants in hyperbolic space

We generalize the classical formulas of integral geometry, by getting integral geometric formulas for the intersection of a fixed compact hypersurface of hyperbolic space and a moving totally umbilical hypersurface. In particular we compute the mean value of the volume, the total mean curvatures and the Euler characteristic of these intersections when the totally umbilical hypersurface moves over all the intersecting positions. Analogous formulas are given for totally umbilical hypersurfaces contained in totally geodesic planes of ℍn.

Existence and non-existence for a mean curvature equation in hyperbolic space

There exists a well-known criterion for the solvability of theDirichlet Problem for the constant mean curvature equation in bounded smooth domains in Euclidean space. This classicalresult was established by Serrin in 1969. Focusing the DirichletProblem for radial vertical graphs P.-A. Nitsche has establishedan existence and non-existence results on account of a criterionbased on the notion of a hyperbolic cylinder. In this work wecarry out a similar but distinct result in hyperbolic spaceconsidering a different Dirichlet Problem based on another systemof coordinates. We consider a non standard cylindergenerated by horocycles cutting orthogonally a geodesic plane$\mathcal P$ along the boundary of a domain $\Omega\subset \mathcal P.$ We provethat a non strict inequality between the mean curvature $\mathcal H'_{\mathcal C}(y)$of this cylinder along $\partial \Omega$ and the prescribed meancurvature $\mathcal H(y),$ i.e $\mathcal H'_{\mathcal C}(y)\geq |\mathcal H(y)|, \forally\in\partial\Omega$ yields existence of our Dirichlet Problem. Thuswe obtain existence of surfaces whose graphs have prescribed meancurvature $\mathcal H(x)$ in hyperbolic space taking a smooth prescribedboundary data $\varphi.$ This result is sharp because if ourcondition fails at a point $y$ a non-existence result can beinferred.

The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H 3 , and solve both of them. We construct the unique mean curvature one surface in H 3 that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.

Affine surfaces in R4with planar geodesics with respect to the affine metric

It is proved that non-degenerate surfaces in R 4 with planar geodesics are only the complex paraboloid and the product of two parabolas.

Meromorphic extension of the Selberg zeta function for Kleinian groups via thermodynamic formalism

AbstractWe prove the existence of a piecewise analytic expanding map associated to certain Kleinian groups without parabolics acting in the 3-dimensional hyperbolic space. These groups have a fundamental domain ℛ with the property that the geodesic planes containing each face are part of the tesselation. We use this map together with the methods of thermodynamic formalism to give another proof that the Selberg zeta function for such groups has a meromorphic extension to ℂ.

Surfaces of a Euclidean space with Helical or planar geodesics through a point

Surfaces of a Euclidean space with Helical or planar geodesics through a point

A surface which contains planar geodesics

It is proved that a complete surface in E3 is a sphere or a plane if it contains at least four geodesics through each point which are plane curves.

Submanifolds with proper $d$-planar geodesics immersed in complex projective spaces

Submanifolds with proper $d$-planar geodesics immersed in complex projective spaces

Some examples of Riemannian almost-product manifolds

Foliations with bundle-like metrics, conformal, minimal and totally geodesic foliations and minimal and geodesic plane fields have been subject to recent study. A. M. Naveira has fitted all these classes into a general scheme and has gotten thirty six classes of riemannian almostproduct manifolds. In this paper we give strict examples of these classes, showing that none of them is vacuous, and that the inclusion relations among them are strict. The basic riemannian manifolds for the construction of these examples are submanifolds of C and H (C = complex field, H = quaternion field), and we use the canonical complex structures on these vector spaces. Perhaps the most interesting examples are those of minimal foliations which are not totally geodesic foliations.

Manifolds with planar geodesics

Manifolds with planar geodesics