Related papers: Legendre curves on 3-dimensional $C_{12}$-Manifold…

The main purpose of this paper is to present the spherical characterization of Legendre curves in $3$-dimensional quasi-Sasakian pseudo-metric manifolds. Furthermore, null Legendre curves are also characterized in this class of manifold.

In $(2n+1)$-dimensional non-Sasakian contact metric manifolds, we consider Legendre curves whose mean curvature vector fields are $\mathcal{C}$-parallel or $\mathcal{C}$-proper in the tangent or normal bundles. We obtain the curvature…

Using convex integration we give a constructive proof of the well-known fact that every continuous curve in a contact $3$-manifold can be approximated by a Legendrian curve.

We show that if a smooth projective curve $C\subset\mathbb P^3$ (over an algebraically closed field of characteristic zero) is Legendrian with respect to a contact structure (it is well known that a contact structure on $\mathbb P^3$ is…

Object of study in the present paper are slant and Legendre null curves in 3-dimensional Sasaki-like almost contact B-metric manifolds. For the examined curves we express the general Frenet frame and the Frenet frame for which the original…

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…

The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this…

We give a procedure to ``average'' canonically $C^1$-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant…

In this paper, we consider Legendre trajectories of trans-$S$-manifolds. We obtain curvature characterizations of these curves and give a classification theorem. We also investigate Legendre curves whose Frenet frame fields are linearly…

In the present paper, we define and study $C$-parallel and $C$-proper slant curves of $S$-manifolds. We prove that a curve $\gamma $ in an $S$-manifold of order $r\geq 3,$ under certain conditions, is $C$-parallel or $C$-parallel in the…

We investigate not only the associated curves of regular plane curves, but also those of Legendre curves. As associated curves, we consider Bertrand regular plane curves and Bertrand Legendre curves. These curves contain parallel, evolute…

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…

The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e.…

Almost contact manifolds with B-metric are considered. Of special interest are the so-called vertical classes of the almost contact B-metric manifolds. Curvature properties of these manifolds are studied. An example of 5-dimensional…

It has been proposed that equilibrium thermodynamics is described on Legendre submanifolds in contact geometry. It is shown in this paper that Legendre submanifolds embedded in a contact manifold can be expressed as attractors in phase…

We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in…

The object of investigations are almost contact B-metric structures on 3-dimensional Lie groups considered as smooth manifolds. There are established the existence and some geometric characteristics of these manifolds in all basic classes.…

Almost contact manifolds with B-metric are considered. A special linear connection is introduced, which preserves the almost contact B-metric structure on these manifolds. This connection is investigated on some classes of the considered…

Three-dimensional almost contact B-metric manifolds are constructed by a three-parametric family of Lie groups. It is established the class of the investigated manifolds which has an important geometrical interpretation. It is determined…

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of…