Author
Gabriel Ruiz-Hernández
Other affiliations: Massachusetts Institute of Technology, Instituto Nacional de Matemática Pura e Aplicada
Bio: Gabriel Ruiz-Hernández is an academic researcher from National Autonomous University of Mexico. The author has contributed to research in topics: Mean curvature & Hypersurface. The author has an hindex of 9, co-authored 29 publications receiving 252 citations. Previous affiliations of Gabriel Ruiz-Hernández include Massachusetts Institute of Technology & Instituto Nacional de Matemática Pura e Aplicada.
Papers
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TL;DR: In this article, the authors give necessary and sufficient conditions for a manifold M to be immersed as an helix in some Euclidean space, and a characterization of helix submanifolds related to the solutions of the so called eikonal differential equation.
Abstract: A submanifold of $${\mathbb {R}^n}$$
whose tangent space makes constant angle with a fixed direction d is called a helix. In the first part of the paper we study helix hypersurfaces. We give a local description of how these hypersurfaces are constructed. As an application we construct (nonflat) minimal helices hypersurfaces in $${\mathbb {R}^n}$$
for n > 3. In the second part we give a characterization of helix submanifolds related to the solutions of the so called eikonal differential equation. As a corollary we give necessary and sufficient conditions for a manifold M to be immersed as an helix in some Euclidean space. In the third part of this paper we study r-helices submanifolds. That is to say submanifolds such that its tangent space makes a constant angle with r linearly independent directions.
57 citations
TL;DR: In this paper, the authors give a method to find every solution to the eikonal PDE on a Riemannian manifold locally, and give a local construction of arbitrary Euclidean helix submanifolds of any dimension and codimension.
Abstract: A submanifold of Rn whose tangent space makes constant angle with a fixed direction d is called a helix. Helix submanifolds are related with the eikonal PDE equation. We give a method to find every solution to the eikonal PDE on a Riemannian manifold locally. As a consequence we give a local construction of arbitrary Euclidean helix submanifolds of any dimension and codimension. Also we characterize the ruled helix submanifolds and in particular we describe those which are minimal.
32 citations
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TL;DR: In this article, it was shown that CMC constant angle hypersurfaces in Euclidean space are either totally geodesic or cylinders, and they were shown to be Delaunay surfaces.
Abstract: Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful characterizations. In particular, we relate them with transnormal functions and eikonal equations. With the further condition of having constant mean curvature (CMC) we obtain a characterization of the canonical principal direction surfaces in Euclidean space as Delaunay surfaces. We also prove that CMC constant angle hypersurfaces in a product $\mathbb{R}\times N$ are either totally geodesic or cylinders.
26 citations
TL;DR: In this article, it was shown that CMC constant angle hypersurfaces in a product R×N are either totally geodesic or cylinders, and with the further condition of having constant mean curvature (CMC) surfaces in Euclidean space as Delaunay surfaces.
Abstract: Given a vector field X in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to X if the projection of X onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful characterizations. In particular, we relate them with transnormal functions and eikonal equations. With the further condition of having constant mean curvature (CMC) we obtain a characterization of the canonical principal direction surfaces in Euclidean space as Delaunay surfaces. We also prove that CMC constant angle hypersurfaces in a product R×N are either totally geodesic or cylinders.
26 citations
TL;DR: In this article, the authors study translation graphs with zero mean curvature, that is, minimal translation graphs, by imposing natural conditions on ϕ and ψ, like harmonicity, minimality and eikonality (constant norm of the gradient).
Abstract: The graph of a function f defined in some open set of the Euclidean space of dimension (p + q) is said to be a translation graph if f may be expressed as the sum of two independent functions ϕ and ψ defined in open sets of the Euclidean spaces of dimension p and q, respectively. We obtain a useful expression for the mean curvature of the graph of f in terms of the Laplacian, the gradient of ϕ and ψ as well as of the mean curvatures of their graphs. We study translation graphs having zero mean curvature, that is, minimal translation graphs, by imposing natural conditions on ϕ and ψ, like harmonicity, minimality and eikonality (constant norm of the gradient), giving several examples as well as characterization results.
26 citations
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TL;DR: In this article, the authors give necessary and sufficient conditions for a manifold M to be immersed as an helix in some Euclidean space, and a characterization of helix submanifolds related to the solutions of the so called eikonal differential equation.
Abstract: A submanifold of $${\mathbb {R}^n}$$
whose tangent space makes constant angle with a fixed direction d is called a helix. In the first part of the paper we study helix hypersurfaces. We give a local description of how these hypersurfaces are constructed. As an application we construct (nonflat) minimal helices hypersurfaces in $${\mathbb {R}^n}$$
for n > 3. In the second part we give a characterization of helix submanifolds related to the solutions of the so called eikonal differential equation. As a corollary we give necessary and sufficient conditions for a manifold M to be immersed as an helix in some Euclidean space. In the third part of this paper we study r-helices submanifolds. That is to say submanifolds such that its tangent space makes a constant angle with r linearly independent directions.
57 citations
TL;DR: A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector as discussed by the authors.
Abstract: A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector. In this work we study and classify these surfaces. In particular, we show that they are flat. Next we prove that a tangent developable surface (resp. cylinder, cone) is a constant angle surface if and only if the generating curve is a helix (resp. a straight line, a circle).
50 citations
TL;DR: In this article, the authors find all constant slope surfaces in the Euclidean 3-space, namely, those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point.
Abstract: In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely, those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bidimensional analog of the generalized helices. Some pictures are drawn by using the parametric equations we found.
44 citations
TL;DR: In this paper, the authors consider hypersurfaces in the Euclidean space E 4 defined as the sum of a curve and a surface whose mean curvature vanishes and give a classification of these surfaces.
42 citations
TL;DR: In this article, the authors find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point.
Abstract: In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bi-dimensional analogue of the generalized helices. Some pictures are drawn by using the parametric equations we found.
37 citations