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We study symmetric minimal surfaces in the three-dimensional Heisenberg group Nil3 using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will present a general scheme for how to construct minimal surfaces in Nil3 with non-trivial geometry. Special emphasis will be put on equivariant minimal surfaces. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group Iso◦(Nil3) of Nil3.

In this paper, we study the semi-symmetry and pseudo-symmetry of almost cosymplectic [Formula: see text]-manifolds. First, we prove that an [Formula: see text]-almost cosymplectic [Formula: see text]-manifold [Formula: see text] is semi-symmetric if and only if it is cosymplectic. Here by an [Formula: see text]-almost cosymplectic [Formula: see text]-manifold, we mean an almost cosymplectic [Formula: see text]-manifold whose characteristic vector field [Formula: see text] is a harmonic unit vector field. If an almost cosymplectic [Formula: see text]-manifold [Formula: see text] whose fundamental endomorphism field [Formula: see text] is parallel in the direction of the characteristic vector field [Formula: see text] ([Formula: see text]), then it is [Formula: see text]-almost cosymplectic. In particular, an almost cosymplectic [Formula: see text]-manifold [Formula: see text] satisfying [Formula: see text] is semi-symmetric if and only if it is cosymplectic. Next, we prove that pseudo-symmetric [Formula: see text]-almost cosymplectic [Formula: see text]-manifolds are certain generalized almost cosymplectic [Formula: see text]-spaces.

The $$\kappa $$ κ -curve is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or $$\epsilon \kappa $$ ϵ κ -curves.$$\kappa $$ κ -curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bézier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using $$\epsilon \kappa $$ ϵ κ -curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.

In this paper, we classify almost cosymplectic 3-manifolds with pseudo-parallel characteristic Jacobi operator. The only simply connected and complete non-cosymplectic almost cosymplectic 3-manifold with pseudo parallel characteristic Jacobi operator is the Minkowski motion group.

We show that the statistical manifold of normal distributions is homogeneous. In particular, it admits a $2$-dimensional solvable Lie group structure. In addition, we give a geometric characterization of the Amari-Chentsov $\alpha$-connections on the Lie group., Comment: 9 pages, no figure

Demoulin surfaces in real projective $3$-space are investigated. Our result enable us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian $6$-symmetric space., 18 pages, v2 several typos fixed

Magnetic curves represent trajectories of charged particles moving on a Riemannian manifold under the action of a magnetic field. The study of magnetic curves in arbitrary Riemannian manifolds was developed in early 1990's, even though related works can be found earlier. Recently there are interesting results on magnetic curves in Euclidean space, Sasakian manifolds, cosymplectic manifolds, Sol space and quasi Sasakian manifolds. In this talk we consider magnetic curves with respect to the almost cosymplectic structure of the Sol space.

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups., 25 pages

The Ricci tensor, phi-Ricci tensor and the characteristic Jacobi operator on cosymplectic 3-manifolds are investigated.

We study magnetic trajectories corresponding to contact magnetic fields in 3-dimensional quasi-Sasakian manifolds. We show that they are slant curves, that is their contact angles are constant. We prove that such magnetic curves are geodesics for a certain linear connection for which all four structure tensor fields are parallel.

In this paper we investigate contact magnetic curves on the unit tangent bundle UM of a Riemannian manifold M and we write the magnetic equations. In the case when M is a space form M(c), it follows that every contact normal magnetic curve is slant when $$c=1$$ , while for $$c\ne 1$$ a contact normal magnetic curve is slant if and only if it satisfies a conservation law. We perform a detailed study of contact magnetic curves in $$U{{\mathbb {S}}}^2$$ .

The class of log-aesthetic curves includes the logarithmic spiral, clothoid, and involute of a circle. Although most of these curves are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them, thereby presenting many applications in industrial and graphic design. The use of the log-aesthetic curves in practical design, however, is still limited. Therefore, we should extend its formula to obtain curves that solve various practical design problems such as $$G^n$$ Hermite interpolation, deformation, smoothing, data-point fitting, and blending plural curves. In this paper, we present a systematic approach to representing log-aesthetic curves via similarity geometry. In turn, this research provides a unified framework for various studies on log-aesthetic curves, particularly of log-aesthetic curve formulation.

We study contact magnetic curves in the unit tangent sphere bundle over the Euclidean plane. In particular, we obtain all contact magnetic curves which are slant.

In a previous paper, we introduced the notion of magnetic vector fields. More precisely, we consider a vector field ? as a map from a Riemannian manifold into its tangent bundle endowed with the usual almost K?hlerian structure and we find necessary and sufficient conditions for ? to be a magnetic map with respect to ? itself and the K?hler 2-form. In this paper we give new examples of magnetic vector fields.

A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bezier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose -curve to introduce symmetry concept ...

We investigate curvatures of normal almost contact Riemannian 3-manifolds. In particular, we show that Kenmotsu 3-manifolds of constant scalar curvature are of constant curvature −1.

We study the Grassmann geometry of surfaces when the ambient space is a 3-dimensional unimodular Lie group with left invariant metric, that is, it is one of the 3-dimensional commutative Lie group, the 3-dimensional Heisenberg group, the groups of rigid motions on the Euclidean or the Minkowski planes, the special unitary group $SU(2)$, and the special real linear group $SL(2,\mathbb R)$.

The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schr\"odinger equation. In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schr\"odinger equation. We also present explicit formulas for both smooth and discrete curves in terms of $\tau$ functions of the two-component KP hierarchy., Comment: 38 pages

In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a 3-dimensional Sasakian space form, embedded as a totally geodesic submanifold of the Sasakian space form of dimension (2n+1).

In this paper we classify the magnetic trajectories with respect to contact magnetic fields in cosymplectic manifolds of arbitrary dimension. We classify Killing magnetic curves in product spaces M 2 × R , recalling also explicit description of magnetic curves in E 3 , S 2 × R and H 2 × R . Finally, we prove a reduction theorem for magnetic curves in the cosymplectic space form M ¯ 2 n ( k ) × R , in order to show that the (2n+1)-dimensional case reduces to the 3-dimensional one.

In this note we present a simple alternative proof for the Bernstein problem in the threedimensional Heisenberg group Nil3 by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch- Rosenberg diòerential.

Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing modified Korteweg-de Vries (mKdV) equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the [Formula: see text] functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class gives regular solutions to the defocusing mKdV equation. Some pictures illustrating the typical dynamics of the curves are presented.

Magnetic curves represent trajectories of charged particles moving on a Riemannian manifold under the action of a magnetic field. A vector field X is a Killing vector field if the Lie derivative with respect to X of the ambient space metric g vanishes. The trajectories corresponding to the Killing magnetic fields are called the Killing magnetic curves. In this talk we study the Killing magnetic curves in the 3-dim Sol space.

It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe it. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criterion for periodicity of magnetic curves on the unit sphere ${\mathbb{S}}^3$.

In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler–Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.

In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space $\mathbb{E}^3$ with H=0 and $H \neq 0$, respectively. These surface classes have been investigated intensively in the literature. For the case $0 \leq H < 1$ there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present a generalized Weierstra{\ss} type representation for surfaces of constant mean curvature in $\mathbb{H}^3$ with particular emphasis on the case of mean curvature $0\leq H < 1$. In particular, the generalized Weierstra{\ss} type representation presented in this paper enables us to construct simultaneously minimal surfaces (H=0) and non-minimal constant mean curvature surfaces ($0, Comment: 37 pages, 4 figures. v3: Various typos fixed. v4: Proposition D.1 has been fixed

We study slant curves in almost contact metric 3-manifolds equipped with canonical connection $$\widetilde{\nabla }$$ . Moreover, we study slant curves in almost contact metric 3-manifolds whose mean curvature is proper with respect to the canonical connection, that is, $$\widetilde{\Delta }\widetilde{H}=\lambda \widetilde{H}$$ or proper in the normal bundle, $$\widetilde{\Delta }^\perp \widetilde{H}=\lambda \widetilde{H}$$ .

We classify biharmonic geodesic spheres in the Cayley projective plane. Our results completes the classification of all biharmonic homogeneous hypersurfaces in simply connected compact Riemannian symmetric spaces of rank 1. In addition we show that complex Grassmannian manifolds, and exceptional Lie groups $F_4$ and $G_2$ admit proper biharmonic real hypersurfaces.

The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrodinger equation (NLS). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schrodinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the \(\tau \) function of the 2-component KP hierarchy.

We study almost contact curves in normal almost contact metric 3-manifolds satisfying \({\triangle{H} = \lambda{H}}\) or \({\triangle^\bot {H} = \lambda{H}}\) . Moreover we study almost contact curve of type AW(k) in normal almost contact metric 3-manifolds. We give natural equations of planar biminimal curves.

We study biminimalcurves in 2-dimensional Riemannian man-ifolds of constant curvature. IntroductionElastic curves provide examples of classically known geometric variationalproblem. A plane curve is said to be an elastic curve if it is a critical point ofthe elastic energy, or equivalently a critical point of the total squared curvature[9].In this paper, we study another geometric variational problem of curves inRiemannian 2-manifolds of constant curvature. The Euler-Lagrange equationstudied in this paper is derived from the theory of biharmonic maps in Rie-mannian geometry.A smooth map φ : (M,g) → (N,h) between Riemannian manifolds is saidto be biharmonic if it is a critical point of the bienergy functional:E 2 (φ) =Z M |τ(φ)| 2 dv g ,where τ(φ) = tr ∇dφ is the tension field of φ. Clearly, if φ is harmonic, i.e.,τ(φ) = 0, then φ is biharmonic. A biharmonic map is said to be proper if it isnot harmonic.Chen and Ishikawa [3] studied biharmonic curves and surfaces in semi-Euclidean space (see also [6]). Caddeo, Montaldo and Piu [1] studied bihar-monic curves on Riemannian 2-manifolds. They showed that biharmonic curveson Riemannian 2-manifolds of non-positive curvature are geodesics. Proper bi-harmonic curves on the unit 2-sphere are small circles of radius 1/√2.Next, Loubeau and Montaldo introduced the notion of biminimal immersion[10].An isometric immersion φ : (M,g) → (N,h) is said to be biminimal if it is acritical point of the bienergy functional under all normal variations. Thus thebiminimality is weaker than biharmonicity for isometric immersions, in general.

Every 3-dimensional Riemannian manifold with 4-dimensional isometry group admits a normal almost contact structure compatible to the metric. In this paper we study affine biharmonic curves in model spaces of Thurston geometry except Sol.