Special Slant Surfaces and a Basic Inequality

A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $?detilde M^2(4?silon)$ with constant holomorphic sectional curvature 4?, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H²(p) ≥ 2K(p) ? 2(1 + 3 cos²θ)?. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented.     

Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit and approximate solutions of second‐order evolution differential equations in Hilbert space

The explicit closed‐form solutions for a second‐order differential equation with a constant self‐adjoint positive definite operator coefficient A (the hyperbolic case) and for the abstract Euler–Poisson–Darboux equation in a Hilbert space are presented. On the basis of these representations, we propose approximate solutions and give error estimates. The accuracy of the approximation automatically depends on the smoothness of the initial data. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 111–131, 1999     

Minimal helix surfaces in <Emphasis Type="Italic">N</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℝ

An immersed surface M in N ⁿ ×ℝ is a helix if its tangent planes make constant angle with ∂ _t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N ⁿ ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.     

Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori

The multiplier spectral curve of a conformal torus f : T ² → S ⁴ in the 4-sphere is essentially (Bohle et al., Conformal maps from a 2-torus to the 4-sphere. arXiv:0712.2311) given by all Darboux transforms of f. In the particular case when the conformal immersion is a Hamiltonian stationary torus ${f: T^2 \to\mathbb{R}^4}$ in Euclidean 4-space, the left normal N : M → S ² of f is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of N, as defined in Hitchin (J Differ Geom 31(3):627–710, 1990). We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian.     

Slant helix of order n and sequence of Darboux developables of principal-directional curves

In this paper, we consider the sequence of the principal-directional curves of a curve γ and define the slant helix of order n (n-SLH) of the curve in Euclidean 3-space. The notion is an extension of the notion of slant helix. We present an important formula that determines if the nth principal-directional curve of γ can be the slant helix of order n (n ≥ 1). As an application of singularity theory, we study the singularities classifications of the Darboux developable of nth principal-directional curve of γ . It is demonstrated that the formula plays a key role in characterizing the singularities of the Darboux developables of the nth principal-directional curve of a curve γ .     

Position vectors of slant helices in Euclidean 3-space

In this paper, position vector of a slant helix with respect to standard frame in Euclidean space E³ is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine a position vector of an arbitrary slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric representation of a Salkowski curve, anti-Salkowski curve and a curve of constant precession, as examples of a slant helices, by means of intrinsic equations.     

Pseudo-spherical Darboux images and lightcone images of principal-directional curves of nonlightlike curves in Minkowski 3-space

Choosing an alternative frame, which is the Frenet frame of the principal-directional curve along a nonlightlike Frenet curve γ , we define de Sitter Darboux images, hyperbolic Darboux images, and lightcone images generated by the principal directional curves of nonlightlike Frenet curves and investigate geometric properties of these associated curves under considerations of singularity theory, contact, and Legendrian duality. It is shown that pseudo-spherical Darboux images and lightcone images can occur singularities (ordinary cusp) characterized by some important invariants. More interestingly, the cusp is closely related to the contact between nonlightlike Frenet curve γ and a slant helix, the principal-directional curve ψ of γ and a helix or the principal-directional curve ψ and a slant helix. In addition, some relations of Legendrian dualities between C-curves and pseudo-spherical Darboux images or lightcone images are shown. Some concrete examples are provided to illustrate our results.     

On the Quaternionic B 2-Slant Helices in the Euclidean Space E 4

In this paper we give a new definition of harmonic curvature functions in terms of B ₂ and we define a new kind of slant helix which we call quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ by using the new harmonic curvature functions. Also we define a vector field D which we call Darboux quaternion of the real quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ and we give a new characterization such as: "a: I ì \mathbb R ? E⁴{``\alpha : I \subset {\mathbb R} \rightarrow E^4} is a quaternionic B ₂–slant helix ${\Leftrightarrow H^\prime_2 -KH_{1} = 0"}${\Leftrightarrow H^\prime_2 -KH_{1} = 0"} where H ₂, H ₁ are harmonic curvature functions and K is the principal curvature function of the curve α.     

Position vector of a time-like slant helix in Minkowski 3-space

In this paper, position vector of a time-like slant helix with respect to standard frame of Minkowski space E³₁ is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary time-like slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the representation of a time-like Salkowski and time-like anti-Salkowski curves as examples of a slant helices, by means of intrinsic equations. Moreover, we present some new characterizations of slant helices and illustrate some examples of our main results.     

Contributions to differential geometry of isotropic curves in the complex space

This work deals with classical differential geometry of isotropic curves in the complex space C⁴. First, we study spherical isotropic curves and pseudo helices. Besides, in this section we introduce some special isotropic helices (type-1, type-2 and type-3 isotropic slant helices) and express some characterizations of them in terms of É. Cartan equations. Thereafter, we prove that position vector of an isotropic curve satisfies a vector differential equation of fourth order. Finally, we investigate position vector of an arbitrary curve with respect to É. Cartan frame by a system of complex differential equations whose solution gives components of the position vector. Solutions of the mentioned system and vector differential equation have not yet been found. Therefore, in terms of special cases, we present some special characterizations.     

Zerlegung der Darboux-Drehung in Zwei Ebene Drehungen

The Darboux rotation for space curves in Euclidean space E³ is decomposed into two simultaneous rotations. The axes of these simultaneous rotations are joined by a simple mechanism. One of these axes is a parallel of the principal normal of the curve, the direction of the other is the direction of the Darboux vector of the curve. This decomposition of the Darboux rotation yields a necessary condition for the curve to be closed.

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Herrn Prof. Dr.Oswald Giering zum 60. Geburtstag gewidmet     

Ruled surfaces generated by some special curves in Euclidean 3-Space

In this paper, a family of ruled surfaces generated by some special curves using a Frenet frame of that curves in Euclidean 3-space is investigated. Some important results are obtained in the case of general helices as well as slant helices. Moreover, as an application, circular general helices, spherical general helices, Salkowski curves and circular slant helices, which illustrate the results, are provided and graphed.     

Über direkt integrierbare Kurven und strenge Böschungskurven im ℝn sowie geschlossene verallgemeinerte Hyperkreise

A curve in Euclidean space ?ⁿ is called “directly integrable”, if it can be explicitly calculated from the curvatures in a specified way. A necessary and sufficient condition for a curve to be directly integrable is that all its curvatures are real multiples of a single real function. Directly integrable curves in an odd-dimensional space ?ⁿ (n=2q+1) can be interpreted as generalized helices. In the case of even-dimensional space ?ⁿ (n=2p), we give a simple necessary and sufficient condition for a directly integrable curve to be closed.     

A short proof of the generalized Bonnet theorem

In three‐dimensional Euclidean space E³, the Bonnet theorem says that a curve on a ruled surface in three‐dimensional Euclidean space, having two of the following properties, has also a third one, namely, it can be a geodesic, that it can be the striction line, and that it cuts the generators under constant angle. In this work, in n dimensional Euclidean space Eⁿ, a short proof of the theorem generalized for (k + 1) dimensional ruled surfaces by Hagen in 4 is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Biharmonic curves in 3-dimensional Sasakian space forms

We show that every proper biharmonic curve in a 3-dimensional Sasakian space form of constant holomorphic sectional curvature H is a helix (both of whose geodesic curvature and geodesic torsion are constants). In particular, if H ≠  1, then it is a slant helix, that is, a helix which makes constant angle α with the Reeb vector field with the property . Moreover, we construct parametric equations of proper biharmonic herices in Bianchi–Cartan–Vranceanu model spaces of a Sasakian space form.      

On the pullback equation

We discuss the existence of a diffeomorphism such that

φ^*(g)=f