arXiv subjects
Publications and source records attributed to Pascual Lucas.
The main goal of this paper is to show that helix surfaces and the Enneper surface are the only surfaces in the 3-dimensional Euclidean space $R^3$ whose isogonal lines are generalized helices and pseudo-geodesic lines.
In this paper we characterize concircular helices in $R^3$ by means of a differential equation involving their curvature and torsion. We find a full description of concircular surfaces in $R^3$ as a special family of ruled surfaces, and we show that $M$ in $R^3$ is a proper concircular surface if and only if either $M$ is parallel to a conical surface or $M$ is the normal surface to a spherical curve. Finally, we characterize the concircular helices as geodesics of concircular surfaces.
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.
Let $M$ be an orientable surface immersed in the De Sitter space $S_1^3$ in $R^4_1$ or anti de Sitter space $H_1^3$ in $R^4_2$. In the case that $M$ is of $L_1$-2-type we prove that the following conditions are equivalent to each other: $M$ has a constant principal curvature; $M$ has constant mean curvature; $M$ has constant second mean curvature. As a consequence, we also show that an $L_1$-2-type surface is either an open portion of a standard pseudo-Riemannian product, or a $B$-scroll over a null curve, or else its mean curvature, its Gaussian curvature and its principal curvatures are all non-constant.
In this paper we generalize the notion of helix in the three-dimensional Euclidean space, which we define as that curve $C$ for which there is an $F$-constant vector field $W$ along $C$ that forms a constant angle with a fixed direction $V$ (called an axis of the helix). We find the natural equation and the geometric integration of helices $C$ where the $F$-constant vector field $W$ is orthogonal to its axis.
An algebraic background in order to study the integrability properties of pseudo-null curve motions in a $3$-dimensional Lorentzian space form is developed. As an application we delve into the relationship between the Burgers' equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.
We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinite many local symmetries for the null localized induction equation is provided.
A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation is finally discussed from this point of view.
We study hypersurfaces either in the De Sitter space $§_1^{n+1}\subset\R_1^{n+2}$ or in the anti De Sitter space $\H_1^{n+1}\subset\R_2^{n+2}$ whose position vector $ψ$ satisfies the condition $L_kψ=Aψ+b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface, for a fixed $k=0,...,n-1$, $A$ is an $(n+2)\times(n+2)$ constant matrix and $b$ is a constant vector in the corresponding pseudo-Euclidean space. For every $k$, we prove that when $A$ is self-adjoint and $b=0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, open pieces of standard pseudo-Riemannian products in $§_1^{n+1}$ ($§_1^m(r)\times§^{n-m}(\sqrt{1-r^2})$, $\H^m(-r)\times§^{n-m}(\sqrt{1+r^2})$, $§_1^m(\sqrt{1-r^2})\times§^{n-m}(r)$, $\H^m(-\sqrt{r^2-1})\times§^{n-m}(r)$), open pieces of standard pseudo-Riemannian products in $\H_1^{n+1}$ ($\H_1^m(-r)\times§^{n-m}(\sqrt{r^2-1})$, $\H^m(-\sqrt{1+r^2})\times§_1^{n-m}(r)$, $§_1^m(\sqrt{r^2-1})\times\H^{n-m}(-r)$, $\H^m(-\sqrt{1-r^2})\times\H^{n-m}(-r)$) and open pieces of a quadratic hypersurface $\{x\in\mathbb{M}_{c}^{n+1}\;|\;Rx,x=d\}$, where $R$ is a self-adjoint constant matrix whose minimal polynomial is $t^2+at+b$, $a^2-4b\leq 0$, and $\mathbb{M}_{c}^{n+1}$ stands for $§_1^{n+1}\subset\R_1^{n+2}$ or $\H_1^{n+1}\subset\R_2^{n+2}$. When $H_k$ is constant and $b$ is a non-zero constant vector, we show that the hypersurface is totally umbilical, and then we also obtain a classification result (see Theorem 2).
The simplest (2+1)-dimensional mechanical systems associated with light-like curves, already studied by Nersessian and Ramos, are reconsidered. The action is linear in the curvature of the particle path and the moduli spaces of solutions are completely exhibited in 3-dimensional Minkowski background, even when the action is not proportional to the pseudo-arc length of the trajectory.