Research Article
Timelike Tangent Developable Surfaces and Bonnet Surfaces
Soley Ersoy and Kemal Eren
Department of Mathematics, Faculty of Arts and Science, Sakarya University, 54187 Sakarya, Turkey
Correspondence should be addressed to Soley Ersoy; sersoy@sakarya.edu.tr
Received 20 November 2015; Accepted 3 January 2016
Academic Editor: Chun-Gang Zhu
Copyright © 2016 S. Ersoy and K. Eren. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 sufcient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the
aid of this criterion. Tis 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 fat
helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.
1. Introduction
Surfaces which admit a one-parameter family of isomet-
ric deformations preserving the mean curvature are called
Bonnet surfaces. In 1867, Bonnet proved that any surface
with constant mean curvature in R
3
(which is not totally
umbilical) is a Bonnet surface [1]. Cartan obtained some
detailed results for Bonnet surfaces in [2]. Lawson extended
Bonnet’s results to any surface with constant mean curvature
in Riemannian 3-manifold of constant curvature. Also, it
was proved that any Bonnet surface of nonconstant mean
curvature depends on six arbitrary constants [3]. Character-
ization for isometric deformation preserving the principal
curvatures of surfaces was obtained by the aid of diferential
forms by Chern in [4]. Te geometric characterizations of
helicoidal surfaces of constant mean curvature, helicoidal
surfaces as Bonnet surfaces, and tangent developable surfaces
as Bonnet surfaces were studied by Roussos in [5], [6] and
[7], respectively. Roussos obtained a characterization for
isometric deformation preserving the mean curvature by
using the method of Chern. Soyuc ¸ok gave the necessary and
sufcient condition of a surface to be Bonnet surface, which
is to have a special system of isothermal parameters [8].
Moreover, Soyuc ¸ok proved that 3-dimensional hyperspace
in 4-dimensional space is Bonnet surface if and only if
hypersurface has orthogonal net [9]. Ba˘ gdatlı and Soyuc ¸ok
studied hypersurfaces preserving the mean curvature and
proved that a hypersurface in R
+1
is Bonnet surface if and
only if hypersurface has orthogonal A-net [10]. On the other
hand, Chen and Li studied 3-dimensional Minkowski space
and classifed timelike Bonnet surfaces [11].
2. Preliminaries
Let be a timelike surface in 3-dimensional Minkowski
space with nondegenerate metric tensor =−
2
1
+
2
2
+
2
3
,
where {
1
,
2
,
3
} is a system of the canonical coordinates
in R
3
. Let Φ:→ R
3
1
be a timelike immersion that
admits a nontrivial isometry preserving the mean curvature.
Nontriviality means that the immersion in the family is not
in the form of ∘Φ, where : R
3
1
→ R
3
1
is an immersion
of R
3
1
. Tese kinds of surfaces are called timelike Bonnet
surfaces by Chen and Li in [11]. Suppose that {
1
,
2
,
3
} is a
local orthonormal frame at the point ∈, where
1
is a
timelike tangent vector,
2
is a spacelike unit tangent vector,
and
3
is a spacelike unit normal vector feld at ∈.
3
can
be regarded as a map
3
:→
2
1
, where
2
1
= { ∈ R
3
1
:
⟨,⟩ = 1} is the de Sitter space. Let
, 1≤≤3, be dual
1-forms of
defned by
(
)=⟨
,
⟩=
, 1≤,≤3,
and let
, 1≤,≤3, be connection forms; then
=
1
1
+
2
2
,
1
=
2
1
2
+
3
1
3
,
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2016, Article ID 6837543, 7 pages
http://dx.doi.org/10.1155/2016/6837543