Rotational linear Weingarten surfaces of hyperbolic type

A linear Weingarten surface in Euclidean space ℝ³ is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c ∈ ℝ. Such a surface is said to be hyperbolic when a ² + 4bc < 0. In this paper we study rotational linear Weingarten surfaces of hyperbolic type giving a classification under suitable hypothesis. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in ℝ³ that consists of surfaces with self-intersections whose generating curves are periodic. Partially supported by MEC-FEDER grant no. MTM2007-61775.     

Homogeneous surfaces in Lie sphere geometry

In this paper, we study surfaces of S ³ in the context of Lie sphere geometry. We construct invariants with respect to Lie sphere transformations on the surfaces, which determine the surfaces up to a Lie sphere transformation. Finally we classify completely the homogeneous surfaces in S ³ with respect to the Lie sphere transformation group of S ³.     

On the geometry of linear Weingarten hypersurfaces in the hyperbolic space

In this paper, we apply some forms of generalized maximum principles in order to study the geometry of complete linear Weingarten hypersurfaces with nonnegative sectional curvature immersed in the hyperbolic space. In this setting, under the assumption that the mean curvature attains its maximum, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder.     

On the geometry of linear Weingarten spacelike hypersurfaces in the de Sitter space

Our purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in the de Sitter space $\mathbb{S}_1^{n + 1} $ . In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of the hyperbolic cylinders of $\mathbb{S}_1^{n + 1} $ . In the compact case, we obtain a rigidity result concerning to a such hypersurface according to the length of its second fundamental form.     

Weingarten surfaces and sine-Gordon equation

A relation between Weingarten surfaces with conditionK - 2mH +m ² +l ² = 0 and solutions of sine-Gordon equations is established.     

Special Weingarten surfaces foliated by circles

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles that satisfy a Weingarten condition of type aH + bK = c, where a,b and c are constant, and H and K denote the mean curvature and the Gauss curvature respectively. We prove that such a surface must be a surface of revolution, one of the Riemann minimal examples, or a generalized cone. Authors’ address: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain     

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.     

Hyperbolic linear Weingarten surfaces in ℝ<Superscript>3</Superscript>

A hyperbolic linear Weingarten surface in ℝ³ is a surface M whose mean and Gaussian curvatures satisfy the relationship 2aH +bK = c for real numbers a, b, c such that a²+bc < 0. In this work we obtain a representation for such a surface in terms of its Gauss map when, more generally, a, b, c are functions on M. We also study the completeness of such surfaces and describe a procedure to construct complete examples from solutions of the sine-Gordon equation. The first author is partially supported by Junta de Comunidades de Castilla-La Mancha, Grant no. PAI-05-034. The first and second authors are partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746.     

Ruled Weingarten surfaces in Minkowski 3-space

We characterize all ruled surfaces in Minkowski 3-space with a relation between the Gauss and mean curvature (Weingarten surfaces). It turns out that, except if the rulings are in a null direction, these are given by Lorentzian screw motions of straight lines. However, if the rulings are always in a null direction, then every ruled surface is Weingarten. Received: 9 February 1998 / Revised version: 20 December 1998     

Laguerre minimal surfaces,isotropic geometry and linear elasticity

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy \(\int (H^2-K)/K d\!A\). They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of \(\int (H^2-K)d\!A\), which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.     

Ruled Weingarten surfaces in the Galilean space

In this paper we study ruled Weingarten surfaces in the Galilean space. Weingarten surfaces are surfaces having a nontrivial funcional relation between their Gaussian and mean curvature. We describe some further examples of Weingarten surfaces.      

Introducing Weingarten cyclic surfaces in R3

In this paper, Cyclic surfaces are introduced using the foliation of circles of curvature of a space curve. The conditions on a space curve such that these cyclic surfaces are of type Weingarten surfaces or HK-quadric surfaces are obtained. Finally, some examples are given and plotted.     

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ?³ orthogonal to the geodesics of Σ.We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ?³ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ?³.     

Ruled surfaces of Weingarten type in Minkowski 3-space

In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K_II, H, H_II}, where K is the Gaussian curvature, K_II is the second Gaussian curvature, H is the mean curvature, and H_II is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK_II + bH + cH_II + dK is constant along each ruling for some constants a, b, c, d with a² + b² + c² ≠ 0.     

Riemannian geometry of Lie algebroids

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.     

Weingarten rotation surfaces in 3-dimensional de Sitter space

In the 3-dimensional de Sitter Space , a surface is said to be a spherical (resp. hyperbolic or parabolic) rotation surface, if it is a orbit of a regular curve under the action of the orthogonal transformations of the 4-dimensional Minkowski space which leave a timelike (resp. spacelike or degenerate) plane pointwise fixed. In this paper, we give all spacelike and timelike Weingarten rotation surfaces in .     

Lie symmetries and CR geometry

This paper is divided into three parts. Part I develops a general, new theory (inspired by modern CR geometry) of Lie symmetries of completely integrable pde systems, viewed from their associated submanifolds of solutions. Part II constructs general combinatorial formulas for the prolongations of vector fields to jet spaces. Part III explicitly characterizes the flatness of some systems of the second order. The results presented here are original and were not published elsewhere; most formulas of Parts II and III were verified by means of Maple Release 7. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 47, Complex Analysis, 2007.