Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces associated with a space curve

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature κ g of the curve equals the curvature κ . The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function k that is defined on the curve and satisfies | k | ≤ κ , and that the geodesic curvature of the curve relative to each such constructed surface satisfies κ g = k .

Inextensible Flows of Curves on Lightlike Surfaces

In this paper, we study inextensible flows of a curve on a lightlike surface in Minkowski three-space and give a necessary and sufficient condition for inextensible flows of the curve as a partial differential equation involving the curvatures of the curve on a lightlike surface. Finally, we classify lightlike ruled surfaces in Minkowski three-space and characterize an inextensible evolution of a lightlike curve on a lightlike tangent developable surface.

Hypersurfaces with Generalized 1-Type Gauss Maps

In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the Laplace operator, Δ, on the submanifold, it satisfies ΔG=fG+gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in E3. Second, we show that the Gauss map of any cylindrical surface in E3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E3, except planes. Finally, we show that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps.

Generalized null 2-type immersions in Euclidean space

We define generalized null 2-type submanifolds in them-dimensional Euclidean space 𝔼m. Generalized null 2-type submanifolds are a generalization of null 2-type submanifolds defined by B.-Y. Chen satisfying the conditionΔ H=f H+gCfor some smooth functionsf,gand a constant vectorCin 𝔼m, whereΔandHdenote the Laplace operator and the mean curvature vector of a submanifold, respectively. We study developable surfaces in 𝔼3and investigate developable surfaces of generalized null 2-type surfaces. As a result, all cylindrical surfaces are proved to be of generalized null 2-type. Also, we show that planes are the only tangent developable surfaces which are of generalized null 2-type. Finally, we characterize generalized null 2-type conical surfaces.

Timelike Tangent Developable Surfaces and Bonnet Surfaces

A criterion was given for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space by Chen and Li, 1999. In this study, we obtain a necessary and sufficient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the aid of this criterion. This is examined under the condition of the curvature and torsion of the base curve of the timelike developable surface being nonconstant. Moreover, we investigate the nontrivial isometry preserving the mean curvature for a timelike flat helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.

Inextensible Flows of Tangent Developable Surfaces of Biharmonic Curves in SL2?(R)

We present some results on the inextensible flows of tangent developable surfaces of biharmonic curves in the . Finally, we find out explicit parametric equations tangent developable surfaces of biharmonic curves in the . Keywords: Biharmonic curve;Inextensible flows. © 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v4i2.8987 J. Sci. Res. 4 (2), 365-371 (2012)

On inextensible flows of tangent developable of biharmonic B-Slant helices according to Bishop frames in the special 3-dimensional Kenmotsu manifold

In this paper, we study inextensible flows of tangent developable surfaces of biharmonic B-slant helices in the special three-dimensional Kenmotsu manifold K with η-parallel ricci tensor. We express some interesting relations about inextensible flows of this surfaces.