arXiv research

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.

The study shows that certain curve graphs are hierarchically hyperbolic but not Gromov hyperbolic.

Introduces hyperbolic generalized framed surfaces and their properties.

The generic singularities and bifurcations are classified for one-parameter families of curves with frames in a space form, the Euclidean space, the elliptic space or the hyperbolic space via projective geometry. Two kinds of frames are considered, adapted frames and osculating frames, in terms of certain differential …

Extends Kanai's result to higher dimensions for negatively curved manifolds.

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a…

Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.

The study characterizes canal hypersurfaces formed by non-null curves with parallel frame in Minkowski space-time.

Rotation minimizing vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even the curvature vanishes. Nevertheless, many other geometric properties have not been studied. In the present paper, RM vector fields along a curve immersed …

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 r…

Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.

This thesis is devoted to the Differential Geometry of curves and surfaces along with applications in Quantum Mechanics. In the 1st part we introduce the well known Frenet frame. Later, we show that the curvature function is a lower bound for the scalar angular velocity of any other orthonormal moving frame, from which…

Study of generalized Bishop frames on curves in 4D space.

We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss "bicycle curves" using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment AB that can move so that it re…

This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ ha…

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.

The main drawback of the Frenet frame is that it is undefined at those points where the curvature is zero. Further- more, in the case of planar curves, the Frenet frame does not agree with the standard framing of curves in the plane. The main drawback of the Bishop frame is that the principle normal vector N is not in …

It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math …

Study of generalized Bishop frames on time-like curves in 4D Lorentz space.

New method studies moving points on curves using rotating frames.

Defines non-parabolic curves in spatial hybrid space with applications.

The paper defines evolutes and involutes for framed curves and their properties.

Generalized Frenet frames for singular space curves

The paper explores Bertrand and framed curves in 3D space.

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

Bertrand framed surfaces defined in Euclidean 3-space with applications.

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…

Discussing moving frames for curve and surface invariants.

For a regular curve on a spacelike surface in Lorentz-Minkowski $3$-space, we have a moving frame along the curve which is called a Lorentzian Darboux frame. We introduce five special vector fields along the curve associated to the Lorentzian Darboux frame and investigate their singularities.

In this paper, we study spinor Bishop equations of curves in E^3. We research the spinor formulations of curves according to Bishop frames in E^3. Also, the relation between spinor formulations of Bishop frames and Frenet frame are expressed.

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-secti…

The Frenet frame is generally known an orthonormal vector frame for curves. But, it does not always meet the needs of curve characterizations. In this study, with the help of associated curves of any spatial curve we obtained a new orthonormal frame which has the property that the second vector makes a constant angle w…

A 3D space of hyperbolic manifolds is connected but not path-connected.

Study curves of constant breadth in a specific 3D manifold.

Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.

In this work, we give parallel transport frame of a curve and we introduce the relations between the frame and Frenet frame of the curve in 4-dimensional Euclidean space. The relation which is well known in Euclidean 3-space is generalized for the first time in 4-dimensional Euclidean space. Then we obtain the conditio…

In this study, we determine some special Smarandache curves according to Darboux frame in E3. We give some characterizations and consequences of Smarandache curves.

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-…

In this paper, we investigate Mannheim pairs, Frenet-Mannheim curves and Weakened Mannheim curves with respect to the modified orthogonal frame in Euclidean 3-space(E 3 ). We obtain some characterizations of these curves.

Quaternionic curves with specific torsion properties don't exist.

Framework for isometric immersions of planar regions from framed curves.

We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this fr…

In this paper, we have first given easily the characterization of special curves with the help of the Rotation minimizing frame (RMF). Also, rectifying-type curves are generalized n-dimensional space $R_{n}$.

In this paper, we study helices and the Bertrand curves. We obtain some of the classification results of these curves with respect to the modified orthogonal frame in Euclidean 3-spaces.

In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…

In this study, we consider AW(k)-type curves according to the Bishop Frame in Euclidean space E^3. We give the relations between the Bishop curvatures k_1, k_2 of a curve in E^3.