Legendre Curves Research Articles

Dualities and envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐spaces

AbstractWe consider envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one‐parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one‐parameter families of frontals in hyperbolic and de Sitter 2‐space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes.

Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.

On interpolating sesqui-harmonic Legendre curves in Sasakian space forms

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain a proper example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

Surfaces of Revolution of Frontals in the Euclidean Space

For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves. Moreover, we give properties of surfaces of revolution with singularities and cones.

A Note on f-biharmonic Legendre Curves in S-Space Forms

In this paper, we study f-biharmonic Legendre curves in S-space forms. Our aim is to find curvature conditions for these curves and determine their types, i.e., a geodesic, a circle, a helix or a Frenet curve of osculating order r with specific curvature equations. We also give a proper example of f-biharmonic Legendre curves in the S-space form R^(2m+s)(−3s), with m = 2 and s = 2.

Envelopes of one-parameter families of framed curves in the Euclidean space

As a one-parameter family of singular space curves, we consider a one-parameter family of framed curves in the Euclidean space. Then we define an envelope for a one-parameter family of framed curves and investigate properties of envelopes. Especially, we concentrate on one-parameter families of framed curves in the Euclidean 3-space. As applications, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves and one-parameter families of spherical Legendre curves, respectively.

Variance of arithmetic sums and L-functions in 𝔽q[t

We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in $\mathbb{F}_q(t)$, in the limit as $q\to\infty$. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when $q\to\infty$, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual $k$-divisor function, when the L-function in question has degree one. They illustrate the role played by the degree of the L-functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over $\mathbb{F}_q(t)$, and we illustrate them by examining in some detail the generalised $k$-divisor functions associated with the Legendre curve.

C-parallel and C-proper slant curves of S-manifolds

In the present paper, we define and study C-parallel and C-proper slant curves of S-manifolds. We prove that a slant curve in an S-manifold of order r ? 3, under certain conditions, is C-parallel or C-parallel in the normal bundle if and only if it is a non-Legendre slant helix or Legendre helix, respectively. Moreover, under certain conditions, we show that is C-proper or C-proper in the normal bundle if and only if it is a non-Legendre slant curve or Legendre curve, respectively. We also give two examples of such curves in R2m+s(-3s).

Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves

A Huff curve over a field K is an elliptic curve defined by the equation $$ax(y^2-1)=by(x^2-1)$$ where $$a,b\in K$$ are such that $$a^2\ne b^2$$ . In a similar fashion, a general Huff curve over K is described by the equation $$x(ay^2-1)=y(bx^2-1)$$ where $$a,b\in K$$ are such that $$ab(a-b)\ne 0$$ . In this note we express the number of rational points on these curves over a finite field $${\mathbb {F}_q}$$ of odd characteristic in terms of Gaussian hypergeometric series $$\displaystyle {_2F_1}(\lambda ):={_2F_1}\left( \begin{matrix} \phi &{}\phi &{} \epsilon \end{matrix}\Big | \lambda \right) $$ where $$\phi $$ and $$\epsilon $$ are the quadratic and trivial characters over $${\mathbb {F}_q}$$ , respectively. Consequently, we exhibit the number of rational points on the elliptic curves $$y^2=x(x+a)(x+b)$$ over $${\mathbb {F}_q}$$ in terms of $${_2F_1}(\lambda )$$ . This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of $${_2F_1}$$ . Finally, we present the exact value of $$_2F_1(\lambda )$$ for different $$\lambda $$ ’s over a prime field $${\mathbb {F}_p}$$ extending previous results of Greene and Ono.

Dynamics on [formula omitted] and the Hopf fibration

In the present paper we investigate the dynamics of the S3−component of a J−trajectory in R×S3. We obtain several geometric properties for the projection curve on S2(12) via the Hopf map and we generate some examples with Mathematica. We also prove that the curves in S3 are geodesics on the Hopf tubes over the projection curve on the 2−sphere if and only if they are Legendre curves.

On Legendre curves in normed planes

Legendre curves are smooth plane curves which may have singular points, but still have a well defined smooth normal (and corresponding tangent) vector field. Because of the existence of singular points, the usual curvature concept for regular curves cannot be straightforwardly extended to these curves. However, Fukunaga, and Takahashi defined and studied functions that play the role of curvature functions of a Legendre curve, and whose ratio extend the curvature notion in the usual sense. Going into the same direction, our paper is devoted to the extension of the concept of circular curvature from regular to Legendre curves, but additionally referring not only to the Euclidean plane. For the first time we will extend the concept of Legendre curves to normed planes. Generalizing in such a way the results of the mentioned authors, we define new functions that play the role of circular curvature of Legendre curves, and tackle questions concerning existence, uniqueness, and invariance under isometries for them. Using these functions, we study evolutes, involutes, and pedal curves of Legendre curves for normed planes, and the notion of contact between such curves is correspondingly extended, too. We also provide new ways to calculate the Maslov index of a front in terms of our new curvature functions. It becomes clear that an inner product is not necessary in developing the theory of Legendre curves. More precisely, only a fixed norm and the associated orthogonality (of Birkhoff type) are necessary.

Remarks on Legendrian self-linking

The Thurston-Bennequin invariant provides one notion of self-linking for any homologically trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in R3. Our definition is based upon reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.

Periodic J-trajectories on R×S3

In this paper we find the equations of motion for a J -trajectory in the product space R × S 3 . We study the periodicity of J -trajectories whose projection on the 3-sphere is a Legendre curve and we find a periodicity criterion. We obtain also the contact angle for the projection on S 3 of a J -trajectory.

Framed Surfaces in the Euclidean Space

A framed surface is a smooth surface in the Euclidean space with a moving frame. The framed surfaces may have singularities. We treat smooth surfaces with singular points, that is, singular surfaces more directly. By using the moving frame, the basic invariants and curvatures of the framed surface are introduced. Then we show that the existence and uniqueness for the basic invariants of the framed surfaces. We give properties of framed surfaces and typical examples. Moreover, we construct framed surfaces as one-parameter families of Legendre curves along framed curves. We give a criteria for singularities of framed surfaces by using the curvature of Legendre curves and framed curves.

Legendre curves on 3-dimensional Kenmotsu manifolds admitting semisymmetric metric connection

The object of the present paper is to study biharmonic Legendre curves, locally ?-symmetric Legendre curves and slant curves in 3-dimensional Kenmotsu manifolds admitting semisymmetric metric connection. Finally, we construct an example of a Legendre curve in a 3-dimensional Kenmotsu manifold.

Curvature and torsion of a legendre curve in (e;d) Trans-Sasakian manifolds

Curvature and torsion of a legendre curve in (e;d) Trans-Sasakian manifolds

Envelopes of legendre curves in the unit spherical bundle over the unit sphere

Envelopes of legendre curves in the unit spherical bundle over the unit sphere

Complete minimal surfaces densely lying in arbitrary domains of ℝn

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology. Moreover, we show that every domain in $\mathbb{R}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface. Our method of proof can be adapted to give analogous results for non-orientable minimal surfaces in $\mathbb{R}^n$ $(n\ge 3)$, complex curves in $\mathbb{C}^n$ $(n\ge 2)$, holomorphic null curves in $\mathbb{C}^n$ $(n\ge 3)$, and holomorphic Legendrian curves in $\mathbb{C}^{2n+1}$ $(n\in\mathbb{N})$.

Hypergeometric properties of genus 3 generalized Legendre curves

Inspired by a result of Manin, we study the relationship between certain period integrals and the trace of Frobenius of genus 3 generalized Legendre curves. We show that both of these properties can be computed in terms of “matching” classical and finite field hypergeometric functions, a phenomenon that has also been observed in elliptic curves and many higher dimensional varieties.

Darboux Charts Around Holomorphic Legendrian Curves and Applications

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,\xi)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds, and we find a holomorphic Darboux chart around any contractible isotropic Stein submanifolds in an arbitrary complex contact manifold.