In the present paper, we investigate differential geometric properties soliton surface $M$ associated with Betchov-Da Rios equation. Then, give derivative formulas of Frenet frame unit speed curve $\Phi=\Phi(s,t)$ for all $t$. Also, discuss linear map Weingarten type in tangent space that generates two invariants: $k$ and $h$. Moreover, obtain necessary sufficient conditions equation to be a mi...

We study some Weingarten spacelike hypersurfaces in a de Sitter space S 1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product H(1−coth ̺)× S(1 − tanh ̺), 1 < k < n − 1 in S 1 (1), the hyperbolic cylinder H(1 − coth ̺) × S(1 − tanh ̺) or spherical cylinder H(1 − coth ̺)×...

We introduce the embedded Weingarten map as a geometric invariant of piecewise smooth surfaces. It is given by a (3 × 3)-matrix and provides complete curvature information in a continuous way. Thus, it is the appropriate tool for the C-analysis of subdivision surfaces near extraordinary points. We derive asymptotic expansions and show that the convergence of the sequence of embedded Weingarten ...

This paper extends parts of the results from [17] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like S2 or the torus S1 × S1. Then shape space is either the manifold of submanifolds of Rn of type M , or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M . We investigate almost ...

This paper presents the concept of the "participant perspective" as an approach to the study of spoken language. It discusses three aspects of this concept and shows that they can offer helpful tools in spoken language research. Employing the participant perspective provides us with an alternative to many of the approaches currently in use in the study of spoken language in that it favours smal...

We define discrete flat surfaces in hyperbolic 3-space H from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in H, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfa...

In this paper we study the linear Weingarten equation defined by fully non-linear PDEadivDu1+|Du|2+bdetD2u(1+|Du|2)2=ϕ(11+|Du|2) in a domain Ω⊂R2, where ϕ∈C1([−1,1]) and a,b∈R. We approach existence of radial solutions when Ω is disk small radius, giving an affirmative answer PDE elliptic type. hyperbolic case show that no solution exists, while parabolic find explicitly all solutions. prove un...

In this work we study spacelike and timelike surfaces of revolution in Minkowski space E1 that satisfy the linear Weingarten relation aH+bK = c, where H and K denote the mean curvature and the Gauss curvature of the surface and a, b and c are constants. The classification depends on the causal character of the axis of revolution. We will give a first integral of the equation of the generating c...