Invariant surfaces with coordinate finite-type Gauss map in simply isotropic space

We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only. To understand the contrast between distinct choices of an isotropic Gauss map, here we study surfaces with a Gauss map whose coordinates are eigenfunctions of the surface Laplace-Beltrami operator. We take into account two choices, the so-called minimal and parabolic normals, and show that when applied to simply isotropic invariant surfaces the condition that the coordinates of the corresponding Gauss map are eigenfunctions leads to planes, certain cylinders, or surfaces with constant isotropic mean curvature. Finally, we also investigate (non-necessarily invariant) surfaces with harmonic Gauss map and show this characterizes constant mean curvature surfaces.Comment: 24 pages (23 in the published version), 3 figures; Keywords: Simply isotropic space, Gauss map, helicoidal surface, parabolic revolution surface, invariant surface, Cayley-Klein geometr

Translation surfaces in the Galilean space

In this paper we describe, up to a congruence, translation surfaces in the Galilean space having constant Gaussian and mean curvatures as well as translation Weingarten surfaces. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature K that are not cylindrical surfaces, and translation surfaces with constant mean curvature H ≠ 0 that are not ruled

Bour’s theorem and helicoidal surfaces with constant mean curvature in the Bianchi–Cartan–Vranceanu spaces

In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space R3 to the case of helicoidal surfaces in the Bianchi– Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones

Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces

We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then generalizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Gaussian and mean curvatures, we solve the corresponding problem of prescribed curvature for invariant surfaces whose generating curves lie on a plane containing the degenerated direction

The Bour's Theorem for invariant surfaces in three-manifolds

In this paper, we apply techniques of the equivariant geometry to give a positive answer to the conjecture that a generalized Bour's Theorem holds for surfaces that are invariant under the action of a one-parameter group of isometries of a three-dimensional Riemannian manifold.Comment: 17 page

Invariant surfaces with generalized elastic profile curves

150 p.En esta memoria estamos principalmente interesados en estudiar la conexión entre las curvas críticas para energías de tipo elásticas adecuadas (lo que posteriormente llamaremos curvas elásticas generalizadas) y las supercies invariantes que posean ciertas propiedades geométricas interesantes (como por ejemplo, supercies de evolución binormal, superficies de curvatura media constante, superficies de Weingarten lineales,...). Además, se establecerá una conexión similar entre las superficies invariantes críticas para energías de tipo-Willmore (véase la definición de superficies tipo-Willmore en la ágina XIII) y curvas elásticas generalizadas

Invariant surfaces with generalized elastic profile curves

150 p.En esta memoria estamos principalmente interesados en estudiar la conexión entre las curvas críticas para energías de tipo elásticas adecuadas (lo que posteriormente llamaremos curvas elásticas generalizadas) y las supercies invariantes que posean ciertas propiedades geométricas interesantes (como por ejemplo, supercies de evolución binormal, superficies de curvatura media constante, superficies de Weingarten lineales,...). Además, se establecerá una conexión similar entre las superficies invariantes críticas para energías de tipo-Willmore (véase la definición de superficies tipo-Willmore en la ágina XIII) y curvas elásticas generalizadas

Afina geometrija minimalnih ploha Minkowskog u {R_1}^3

On the one hand we give results concerning affine invariants of the focal surfaces of Minkowski minimal surfaces. Secondly we investigate the behavior of affine invariants in case of association of Minkowski minimal surfaces.S jedne strane mi dajemo rezultate vezane uz afine invarijante žarišnih ploha minimalnih ploha Minkowskog, dok s druge istražujemo karakteristike afinih invarijanata u slučaju pridruživanja kod minimalnih ploha Minkowskog