Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations of these ruled surfaces. Some special ruled surfaces such as normal surface, binormal surface, tangent developable surface, rectifying developable surface and Darboux developable surface are worked. From those applications, we make use of such a work to interpret the Gaussian, mean curvatures of these surfaces and geodesic, normal curvature and geodesic torsion of the base curves with respect to these surfaces depend on three dimensional Lie groups.
We point out how some recent developments in the theory of constant scalar curvature Kähler metrics can be used to clarify the existence issue for such metrics in the special case of geometrically ruled complex surfaces.
For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is "doubly ruled" (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define "doubly ruled parameterized surfaces" in any Riemannian manifold, independent of Riemannian su…
We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising sequence.
We establish that, over certain ground fields, the set of osculating tangents of Cayley's ruled cubic surface gives rise to a (maximal partial) spread which is also a dual (maximal partial) spread. It is precisely the Betten-Walker spreads that allow for this construction. Every infinite Betten-Walker spread is not an algebraic set of lines, but it turns into such a set by adding just one pencil of lines.
In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the relationships between invariants of Mannheim timelike surface offsets. Furthermore, we give the conditions for these surface offsets to be developable.
In this study, we define some new types of ruled surfaces called slant ruled surfaces. We give some characterizations for a regular ruled surface to be a slant ruled surface in Euclidean 3- space. We show that if the slant ruled surface is developable then the striction curve is a general helix or a slant helix according to the kind of surface. Moreover, we give the relationships between slant ruled surfaces and some offset surfaces such as Bertrand offsets and Mannheim offsets.
In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim offsets of the spacelike ruled surfaces by means of dual Darboux frame. We obtain relationships between the invariants of Mannheim spacelike offset surfaces and offset angle, offset distance. Furthermore, we give conditions for these surface offsets to be developable.
In this study, we give the relationships between the conical curvatures of ruled surfaces drawn by the unit vectors of the ruling, central normal and central tangent of a regular ruled surface in the Euclidean -space. We obtain the differential equations characterizing slant ruled surfaces and if the reference ruled surface is a slant ruled surface, we give some conditions for the surfaces drawn by the central normal and the central tangent vectors to be slant ruled surfaces.
In this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space E_1^3. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in E_1^3. Moreover, we obtain some corollaries which give the relationships between a non-null slant ruled surface and its striction line in E_1^3.
In this study, we introduce Darboux slant ruled surfaces in the Euclidean 3-space which is defined by the property that the Darboux vector of orthonormel frame of ruled surface makes a constant angle with a fixed, non-zero direction. We obtain the characterizations of Darboux slant ruled surfaces regarding the conical curvature kappa and give the relations between Darboux slant ruled surfaces and some other slant ruled surfaces.
In this study, we define a family of ruled surfaces in the Euclidean 3-space E^3 and called similar ruled surfaces. We obtain some properties of these special surfaces and we show that developable ruled surfaces form a family of similar ruled surfaces if and only if the striction curves of the surfaces are similar curves with variable transformation.
We classify ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is also proved that all translation minimal surfaces are ruled.
In this study, we consider the notion of similar ruled surface for timelike and spacelike ruled surfaces in Minkowski 3-space. We obtain some properties of these special surfaces in E_1^3 and we show that developable ruled surfaces in E_1^3 form a family of similar ruled surfaces if and only if the striction curves of the surfaces are similar curves with variable transformation. Moreover, we obtain that cylindrical surfaces and conoids form two families of similar ruled surfaces in E_1^3.
In this paper, we study the special curves and ruled surfaces on helix hypersurface whose tangent planes make a constant angle with a fixed direction in Euclidean n-space Besides, we observe some special ruled surfaces in and give requirement of being developable of the ruled surface. Also, we investigate the helix surface generated by a plane curve in Euclidean 3-space .
We study ruled surfaces in R3 which are obtained from dual spher- ical indicatrix curves of dual Frenet vector fields. We find the Gaussian and mean curvatures of the ruled surfaces and give some results of being Wein- garten surface.
We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.
In this study we give definitions and characterizations of transversal surfaces of timelike ruled surfaces. We study some special cases such as the striction curve is a geodesic, an asymptotic line or a line of curvature. Moreover, we obtain developable conditions for transversal surfaces of a timelike ruled surface.
In the present paper, a new type of ruled surfaces called osculating-type (OT)-ruled surface is introduced and studied. First, a new orthonormal frame is defined for OT-ruled surfaces. The Gaussian and the mean curvatures of these surfaces are obtained and the conditions for an OT-surface to be flat or minimal are given. Moreover, the Weingarten map of an OT-ruled surface is obtained and the normal curvature, the geodesic curvature and the geodesic torsion of any curve lying on surface are obtained. Finally, some examples related to helices and slant helices are introduced.
In this paper, we consider non developable ruled surface with spacelike ruling, timelike ruling, respectively. We give the relations between the structure functions with the curvature and torsion of the striction line of the timelike and spacelike non developable ruled surfaces. Also, we have calculated the gaussian and mean curvatures of timelike and spacelike non developable ruled surfaces using the structure functions.
This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a counter-example on the problem of Bernstein type.
It is shown that if a minimal ruled surface admits a Kähler Yamabe minimizer, then this metric must be generalized Kähler-Einstein and the underlying holomorphic vector bundle of the ruled surface must be quasi-stable.
In this paper, Legendre curves on unit tangent bundle are given using rotation minimizing (RM) vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classifed.
In this paper, we define a new type of ruled surface called ruled surface by using the alternative frame of a base curve. Then, we study its differential geometric properties such as striction line, distribution parameter, fundamental forms, Gaussian and mean curvatures. Moreover, we find geodesic curvatures, normal curvatures, and geodesic torsions of the base curve and the striction line of the new ruled surface. Finally, we give two related examples.
In this paper, we investigate the ruled surfaces generated by a straight line according to rotation minimizing frame (RMF). Using this frame of a straight line, we obtained the necessary and sufficient conditions when the ruled surface is developable. Also, we give some new results and theorems related to be the asymptotic curve, the geodesic curve and the line of curvature of the base curve on the ruled surface.
In this study, we give the dual characterizations of Mannheim offsets of the ruled surface in terms of their integral invariants and the new characterization of the Mannheim offsets of developable surface. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants.
In this paper, using the classifications of timelike and spacelike ruled surfaces, we define and study the Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space. We give the conditions for spacelike offset surfaces to be developable.
In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed curve in dual Lorentzian space.
In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space $E^3$. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.
In this paper, we obtain the characterizations of Mannheim offsets of the timelike ruled surface with spacelike rulings in dual Lorentzian space. We give the relations between terms of their integral invariants and also we give the new characterization of the Mannheim offsets of developable timelike ruled surface. Moreover, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of timelike ruled surfaces and their integral invariants.
This paper deals with skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$ which are equipped with polar normalizations, that is, relative normalizations such that the relative normal at each point of the ruled surface lies on the corresponding polar plane. We determine the invariants of a such normalized ruled surface and we study some properties of the Tchebychev vector field and the support vector field of a polar normalization. Furthermore, we study a special polar normalization, the relative image of which degenerates into a curve.
We consider relative normalizations of ruled surfaces with non-vanishing Gaussian curvature $K$ in the Euclidean space $\mathbb{R} ^{3}$, which are characterized by the support functions $^{\left( α\right) }q=\left \vert K\right \vert ^α$ for $α\in \mathbb{R}$ (Manhart's relative normalizations). All ruled surfaces for which the relative normals, the Pick invariant or the Tchebychev vector field have some specific properties are determined. We conclude the paper by the study of the affine normal image of a non-conoidal ruled surface.
New examples of extremal Kähler metrics on blow-ups of parabolic ruled surfaces are constructed. The method is based on the gluing construction of Arezzo, Pacard and Singer. This enables to endow ruled surfaces of the form $\mathbb{P}(\mathcal{O}\oplus L)$ with special parabolic structures such that the associated iterated blow-up admits an extremal metric of non-constant scalar curvature.
The aim of this paper is to present a new perspective on the generation of developable trajectory ruled surfaces in Minkowski 3-space. Involute trajectory ruled surfaces generated by the Frenet trihedron, moving along spacelike involutes of a given timelike space curve, is stated according to Lorentzian timelike angle between tangent vector and unit vector of direction Darboux vector of this timelike space curve. Also some new results and theorems related to the developability of the involute trajectory timelike ruled surfaces are obtained. Finally we illustrate these surfaces by presenting on…
In this study, we investigate the existence theorems for timelike ruled surfaces in Minkowski 3-space. We obtain a general system and give the existence theorems for a timelike ruled surface according to Gaussian curvature, distribution parameter and strictional distance. Moreover, we give some special cases such as the directirx of the surface is a geodesic, an asymptotic line, a line of curvature or a general helix.
We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space $\mathbb{R}^{3}$. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve. Moreover, we study the Laplace normal image of a non-conoidal ruled surface whose relative normals lie on the asymptotic plane.
This paper deals with skew ruled surfaces $\varPhi$ in the Euclidean space $\mathbb{E}^{3}$ which are right normalized, that is they are equipped with relative normalizations, whose support function is of the form $q(u,v) = \frac{f(u) + g(u)\, v}{w(u,v)}$, where $w^2(u,v)$ is the discriminant of the first fundamental form of $\varPhi$. This class of relatively normalized ruled surfaces contains surfaces such that their relative image $\varPhi^{*}$ is either a curve or it is as well as $\varPhi$ a ruled surface whose generators are, additionally, parallel to those of $\varPhi$. Moreover we inve…
In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed spacelike curve with timelike binormal in dual Lorentzian space.
In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=α\langle N,\mathbf{x}\rangle+λ$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $α$ and $λ$ are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.
In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed spacelike curve with a spacelike binormal in dual Lorentzian space.
In this paper, we study ruled surfaces and quadrics in the 3-dimensional Euclidean space which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite $III$-type, respectively.
We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally contactomorphic to the complex plane.
This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature and the curvature of the relative metric of a relatively normalized ruled surface $\varPhi$ and in section 3 we introduce some special normalizations of it. All ruled surfaces and their corresponding normalizations that make $\varPhi$ an improper or a proper relative sphere are determined in section 4. In the last section we study ruled surfaces, which are \e…
Let $X$ be a ruled surface over a curve of genus $g$. We prove that $X$ has a scalar-flat Hermitian metric if and only if $g\geq 2$ and $m(X)>2-2g$ where $m(X)$ is an intrinsic number depends on the complex structure of $X$.
We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-space. We also describe ruled Lagrangian surfaces. Finally we characterize those cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces…
In this study, we have obtained the distribution parameter of a ruled surface generated by a straight line in Frenet trihedron moving along a timelike curve and also along another curve with the same parameter. At this time, the Frenet frames of these timelike curves are not the same. We have moved the director vector of the first curve along the second curve. It is shown that the ruled surface is developable if and only if the base timelike curve is helix. In addition, some smilarities and differences are presented with theorems and results.
It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three dimensional Heisenberg group whose mean curvature is zero with respect to both of the standard Riemannian metric and the standard Lorentzian metric.
In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space. Firstly, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the Mannheim offsets of a timelike ruled surface may be timelike or spacelike. Furthermore, we characterize the developable of Mannheim offset of a timelike ruled surface by the derivative of the conical curvature of the directing cone.
We consider a skew ruled surface $Φ$ in the Euclidean space $E^{3}$ and relative normalizations of it, so that the relative normals at each point lie in the corresponding asymptotic plane of $Φ$. We call such relative normalizations and the resulting relative images of $Φ$ \emph{asymptotic}. We determine all ruled surfaces and the asymptotic normalizations of them, for which $Φ$ is a relative sphere (proper or inproper) or the asymptotic image degenerates into a curve. Moreover we study the sequence of the ruled surfaces ${Ψ_{i}}_{i\in \mathbb{N}}$, where $Ψ_{1}$ is an asymptotic image of $Φ$ …
We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e., their position vector $\boldsymbol{x}$ satisfies the relation $Δ^{III}\boldsymbol{x}=\varLambda \boldsymbol{x}$ where $\varLambda $ is a square matrix of order 3. We show that helicoids and spheres are the only surfaces in $E^3$ satisfying the preceding relation.
We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the Gaussian and mean curvatures. We also provide model-independent proof for the known fact that extrinsically flat surfaces in space forms are ruled. This proof allows us to identify the necessary and sufficient condition the curvature tensor must satisfy for an extrinsically flat surface in a generic 3d manifold to be ruled. Further, we show that if a 3d manifol…
