An Example of Bianchi Transformation in E4 (original) (raw)
On certain surfaces in the isotropic 4-space
2016
The isotropic space is a special ambient space obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we establish a method to calculate the second fundamental form of the surfaces in the isotropic 4-space. Further, we classify some surfaces (the spherical product surfaces and Aminov surfaces) in the isotropic 4-space with vanishing curvatures.
Some geometrical aspects of Bianchi type-I space time
Theoretical and Applied Mechanics, 2002
A problem of spatially homogeneous Bianchi type-I space-time is investigated in Einstein theory without source of gravitation. Some geometrical natures of the space-time are discussed.
Chen Rotational Surfaces of Hyperbolic or Elliptic Type in the Four-dimensional Minkowski Space
2010
We study the class of spacelike surfaces in the four-dimensional Minkowski space whose mean curvature vector at any point is a non-zero spacelike vector or timelike vector. These surfaces are determined up to a motion by eight invariant functions satisfying some natural conditions. The subclass of Chen surfaces is characterized by the condition one of these invariants to be zero. In the present paper we describe all Chen spacelike rotational surfaces of hyperbolic or elliptic type.
On translation surfaces in 4-dimensional Euclidean space
Acta et Commentationes Universitatis Tartuensis de Mathematica, 2016
We consider translation surfaces in Euclidean spaces. Firstly, we give some results of translation surfaces in the 3-dimensional Euclidean space E 3. Further, we consider translation surfaces in the 4-dimensional Euclidean space E 4. We prove that a translation surface is flat in E 4 if and only if it is either a hyperplane or a hypercylinder. Finally we give necessary and sufficient condition for a quadratic triangular Bézier surface in E 4 to become a translation surface.
Proceedings of The American Mathematical Society, 1985
We provide answers (Theorem C) to some questions concerning surfaces in R4 and maps into the quadric Q2 raised by D. Hoffman and R. Osserman. Let S be an oriented surface immersed in R4. The Gauss map of S is the map G of 5 into G(2,4), the Grassmannian of oriented two-planes in R4, given by G(p) = TpS. G(2,4) can be identified with Q2, the complex quadric in CP3, and in turn Q2 is biholomorhic to CP1 X CP1. If we give CP3 the Fubini-Study metric of constant holomorphic sectional curvature 2, then the induced metric on Q2 is given by 2|dw1|2/(l +K|2) + 2|dw2|2/(l +k2|2) ,
On spacelike surfaces in 4-dimensional
2012
On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann type result. Two remarkable families of examples of spacelike surfaces in a lightcone are explicitly constructed. Finally, several results which involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a lightcone are obtained.
Mathematical Physics, Analysis and Geometry, 2014
In this paper, we consider a class of spacelike rotational surfaces in Minkowski space E 4 1 with 2-dimensional axis. There are three types of rotational surfaces with 2-dimensional axis, called rotational surfaces of elliptic, hyperbolic or parabolic type. We obtain all flat spacelike rotational surfaces of elliptic and hyperbolic types with pointwise 1-type Gauss map. We also determine flat spacelike rotational surface of parabolic type with pointwise 1-type Gauss map of the first kind. Then, we conclude that there exists no flat spacelike rotational surface of parabolic type in E 4 1 with pointwise 1-type Gauss map of the second kind.
Pseudo-spherical and pseudo-hyperbolic submanifolds via the quadric representation, I
Archiv der Mathematik, 1997
We consider the quadric representation of a submanifold into non flat pseudo-Riemannian space forms. Then we classify submanifolds such that the mean curvature vector field of its quadric representation is proper for the Laplacian. We have got a complete characterization of hypersurfaces whose quadric representation satisfies DH lH mf ÿ f 0 . As for surfaces into De Sitter and anti De Sitter worlds we have also found nice characterizations for minimal B-scrolls and complex circles.
Proceedings Book of International Workshop on Theory of Submanifolds, 2017
In this work, we study two families of rotational surfaces in the pseudo-Euclidean space E 4 2 with profile curves lying in 2-dimensional planes. First, we obtain a classification of pseudo-umbilical spacelike surfaces and timelike surfaces in these families. Then, we show that in this classification there exists no a pseudo-umbilical rotational surface in E 4 2 with pointwise 1type Gauss map of second kind. Finally, we determine such pseudo-umbilical rotational surfaces in E 4 2 having pointwise 1-type Gauss map of first kind.
A note on proper curvature collineations in Bianchi type VIII and IX space-times
Gravitation and Cosmology, 2010
Curvature collineations of Bianchi type IV space-times are investigated using the rank of the 6 6 × Riemann matrix and direct integration technique. From the above study it follows that the Bianchi type IV space-times possesses only one case when it admits proper curvature collineations. It is shown that proper curvature collineations form an infinite dimensional vector space.
On Generalized Spherical Surfaces in Euclidean Spaces
arXiv: Differential Geometry, 2016
In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean (n+1)−(n+1)-(n+1)−space mathbbEn+1\mathbb{E}^{n+1}mathbbEn+1. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces mathbbE3\mathbb{E}^{3}mathbbE3 and % \mathbb{E}^{4} respectively. We have shown that the generalized spherical surfaces of first kind in mathbbE4\mathbb{E}^{4}mathbbE4 are known as rotational surfaces, and the second kind generalized spherical surfaces are known as meridian surfaces in mathbbE4\mathbb{E}^{4}mathbbE4. We have also calculated the Gaussian, normal and mean curvatures of these kind of surfaces. Finally, we give some examples.
General rotational surfaces in the 4-dimensional Minkowski space
TURKISH JOURNAL OF MATHEMATICS, 2014
General rotational surfaces as a source of examples of surfaces in the fourdimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with special invariants. We describe analytically the flat general rotational surfaces and the general rotational surfaces with flat normal connection. We classify completely the minimal general rotational surfaces and the general rotational surfaces consisting of parabolic points.
The clairaut's theorem on rotational surfaces in pseudo Euclidean 4-space with index 2
2021
In this study, the Clairaut’s theorem on the hyperbolic and elliptic rotational surfaces generated by using a curve and matrices are expressed in E 2 . Therefore, the time-like geodesic curves are used to generate these surfaces of rotation in 4-dimensional semi-Euclidean space. Also, in these surfaces of rotation, the geodesics are completely characterized according to results of the Clairaut’s theorem.
Generalized Rotation Surfaces in E 4
Results in Mathematics, 2012
In the present study we consider generalized rotation surfaces imbedded in an Euclidean space of four dimensions. We also give some special examples of these surfaces in E 4. Further, the curvature properties of these surfaces are investigated. We give necessary and sufficient conditions for generalized rotation surfaces to become pseudo-umbilical. We also show that every general rotation surface is Chen surface in E 4. Finally we give some examples of generalized rotation surfaces in E 4 .
Meridian Surfaces of Elliptic or Hyperbolic Type in the Four-dimensional Minkowski Space
2014
We consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. We call these surfaces meridian surfaces of elliptic or hyperbolic type, respectively. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants and give the complete classification of the meridian surfaces with constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces and the meridian surfaces with parallel normal bundle.
Pseudo-umbilical surfaces in euclidean spaces
Kodai Mathematical Journal, 1971
Recently, the author introduced the notion of αth curvatures of first and second kinds for surfaces in higher dimensional euclidean space [2, 3]. The main purpose of this paper is to study these curvatures more detail. In §1, we derive some integral formulas for the αth curvatures of first and second kinds. In §2, we get some applications of these formulas to pseudo-umbilical surfaces. §1. Integral formulas for αth curvatures. Let M 2 be an oriented closed Riemannian surface with an isometric immersion x: M 2-+E 2+N. Let F(M 2) and F(E 2+N) be the bundles of orthonormal frames of M 2 and E 2+N respectively. Let B be the set of elements b=(p, e l9 e 2 , •••, e 2+N) such that (p, e ly e 2)eF(M 2) and (x(p\ e ly •••, e 2+N)ζF(E 2+N) whose orientation is coherent with that of E 2+N > identifying e t with dx(eι\ i=l, 2. Then B-+M 2 may be considered as a principal bundle with fibre O(2)xSO(N) and x: B-*F(E 2+N) is naturally defined by x(b)=(x(p), e ίt •••, e 2+N). The structure equations of E 2+N are given by (1) where ω' A , ω' AB are differential 1-forms on F(E 2+N). Let ω A , ω AB be the induced 1forms on B from ω' Al ω AB by the mapping x. Then we have
Special Structures on General Rotational Surfaces in Pseudo-Euclidean 4-Space With Neutral Metric
Research Square (Research Square), 2023
On general rotational surfaces in the pseudo-Euclidean 4-dimensional space of neutral signature, we describe the behavior of geometric objects, such as Killing vector fields (and in particular homothetic vector fields), divergence-free vector fields, co-closed and harmonic one-forms, and also harmonic functions. We classify geodesic and parallel vector fields, geodesic curves, concircular vector fields and concircular functions, and also concurrent vector fields and functions whose gradient is concurrent. Our results are new, as they have not been obtained in the Euclidean and Minkowski framework. The tools here are taken from both differential geometry and partial and ordinary differential equations. This topic could be of interest to many fields of mathematics, physics, engineering, architecture.
Bianchi type A hyper-symplectic and hyper-Kähler metrics in 4D
Classical and Quantum Gravity, 2011
We present a simple explicit construction of hyper-Kähler and hyper-symplectic (also known as neutral hyper-Kähler or hyper-parakähler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kähler and hyper-symplectic structures in dimension four. Contents 1. Introduction 1 2. Hyper-Kähler metrics in dimension four 2 2.1. The group SU (2), Bianchi type IX 5 2.2. The group SU (1, 1), Bianchi type V III 6 2.3. The Heisenberg group H 3 , Bianchi type II, Gibbons-Hawking class 7 2.4. Rigid motions of euclidean 2-plane-Bianchi V II 0 7 2.5. Rigid motions of Lorentzian 2-plane-Bianchi type V I 0 8 2.6. Contractions 10 3. Hyper symplectic metrics in dimension 4 10 3.1. Bianchi type IX hyper-symplectic metrics and hyper-Kähler Bianchi type V III hyper-Kähler metrics 12 3.2. Bianchi type V III hyper-symplectic metrics and hyper-Kähler Bianchi type IX hyper-Kähler metrics 12 3.3. Bianchi type II hyper-symplectic metrics and hyper-Kähler Bianchi type II hyper-Kähler metrics 12 3.4. Bianchi type V II 0 hyper-symplectic metrics and hyper-Kähler Bianchi type V I 0 metrics 13 3.5. Bianchi type V I 0 hyper-symplectic metrics and hyper-Kähler Bianchi type V II 0 metrics 13 4. Conclusions 13 References 14