1 Introduction

ROTATIONAL SURFACES IN ISOTROPIC SPACES SATISFYING WEINGARTEN CONDITIONS

Alper Osman Öğrenmiş

Department of Mathematics, Faculty of Science

Firat University, Elazig, 23119, Turkey

aogrenmis@firat.edu.tr

Abstract. In this paper, we study the rotational surfaces in the isotropic 3-space satisfying Weingarten conditions in terms of the relative curvature (analogue of the Gaussian curvature) and the isotropic mean curvature . In particular, we classify such surfaces of linear Weingarten type in

Keywords: Isotropic space; rotational surface; Weingarten surface.

Math. Subject Classification 2010: , , .

## 1 Introduction

The work of surfaces with special properties in the isotropic 3-space has important applications in several applied sciences, e.g., computer science, Image Processing, architectural design and microeconomics, see [3, 4, 6, 8], [29]-[31].

Differential geometry of isotropic spaces have been introduced by K. Strubecker [37], H. Sachs [32]-[34], D. Palman [27] and others.

I. Kamenarovic ([17, 18]), B. Pavkovic ([28]), Z. M. Sipus ([35]) and M.E. Aydin ([1, 2]) have studied some classes of surfaces in

On the other hand, let be a regular surface of a Euclidean 3-space . For general references on the geometry of surfaces see [12, 15].

Denote the Levi-Civita connection of and the normal vector field to Then the operator given by

 S(v)=−∇vN,

is called the shape operator, where is a tangent vector field to It measures how bends in different directions. The eigenvalues of are called the principal curvatures donoted by and

The arithmetic mean of the principal curvatures are called the mean curvature, The Gaussian curvature is defined by

A surface in is called a Weingarten surface (W-surface) if it satisfies the following non-trivial functional relation

 ϕ(κ1,κ2)=0

for a smooth function of two variables. The above relation implies the following

 δ(K,H)=0,

which is the equivalent to the vanishing of the corresponding Jacobian determinant, i.e. for a coordinate pair on

If fulfills the following condition

 c1H+c2K=c3, ci∈R, (c1,c2,c3)≠(0,0,0), i=1,2,3,

then it is called a linear Weingarten surface (LW-surface). In the particular case (resp. ) the LW-surfaces are indeed the surfaces with constant Gaussian curvature (resp. mean curvature). These phenomenal surfaces have been stuied by many geometers in various ambient spaces, see [14, 20], [22]-[24], [26], [38].

The motivation of the present paper is to study Weingarten surfaces, in particular Weingarten rotational surfaces, in the isotropic 3-space which is one the Cayley–Klein spaces.

Most recently, M.E. Aydin ([2]) classified the helicoidal surfaces in , which are natural generalization of the rotational surfaces, with constant curvature and analyzed some special curves on such surfaces.

In the present paper, we provide that the rotational surfaces in are evidently Weingarten ones. Then we classified LW-rotational surfaces in satisfying the following relation

 K=m0H+n0, m0,n0∈R,

in which is the relative curvature and isotropic mean curvature.

## 2 Preliminaries

The isotropic 3-space is obtained from the 3-dimensional projective space with the absolute figure which is an ordered triple , where is a plane in and are two complex-conjugate straight lines in (see [35]). The homogeneous coordinates in are introduced in such a way that the absolute plane is given by and the absolute lines by The intersection point of these two lines is called the absolute point. The group of motions of is a six-parameter group given in the normal form (in affine coordinates)   by

 (2.1)

for

Such affine transformations are called isotropic congruence transformations or i-motions.

Consider the points and The isotropic distance, so-called i-distance of two points and is defined by

 ∥p1−p2∥i=((y1−x1)2+(y2−x2)2)12.

The i-metric is degenerate along the lines in direction, and such lines are called isotropic lines.

Planes, circles and spheres. There are two types of planes in ([29]-[31]).

(1) Non-isotropic planes are planes non-parallel to the direction. In these planes we basically have an Euclidean metric: This is not the one we are used to, since we have to make the usual Euclidean measurements in the top view. An i-circle (of elliptic type) in a non-isotropic plane is an ellipse, whose top view is an Euclidean circle. Such an i-circle with center and radius is the set of all points  with

(2) Isotropic planes are planes parallel to the axis. There, induces an isotropic metric. An i-circle (of parabolic type) is a parabola with parallel axis and thus it lies in an isotropic plane

An i-circle of parabolic type is not the iso-distance set of a fixed point, but it may be seen as a curve with constant isotropic curvature: A curve in an isotropic plane P (without loss of generality we set ) which does not possess isotropic tangents can be written as graph . Then, the i-curvature of at is given by the second derivative . For an i-circle of parabolic type is quadratic and thus is constant.

There are also two types of isotropic spheres. An i-sphere of the cylindrical type is the set of all points with . Speaking in an Euclidean way, such a sphere is a right circular cylinder with parallel rulings; its top view is the Euclidean circle with center and radius . The more interesting and important type of spheres are the i-spheres of parabolic type,

 z=A2(x2+y2)+Bx+Cy+D, \ \ A≠0.

From an Euclidean perspective, they are paraboloids of revolution with parallel axis. The intersections of these i-spheres with planes are i-circles. If is non-isotropic, then the intersection is an i-circle of elliptic type. If is isotropic, the intersection curve is an i-circle of parabolic type.

Curvature theory of surfaces. A surface immersed in is called admissible if it has no isotropic tangent planes. We restrict our framework to admissible regular surfaces. For such a surface , the coefficients of its first fundamental form are calculated with respect to the induced metric.

The normal field of is always the isotropic vector The coefficients of the second fundamental form of are calculated with respect to the normal field of (for details, see [33], p. 155).

The relative curvature (so called isotropic Gaussian curvature) and isotropic mean curvature are defined by

 K=LM−N2EG−F2, \ \ H=EN−2FM+GL2EG−F2. (2.2)

## 3 LW-rotational surfaces in I3

Let us consider the i-motions given by then the Euclidean rotations in the isotropic space is given by in affine coordinates

 ⎧⎪⎨⎪⎩x′=c1+xcosc2−ysinc2,y′=c3+xsinc2+ycosc2,z′=z,

where

Definition 3.1. Let  be a curve lying in the isotropic plane given by  where . By rotating the curve  around axis, we obtain that the rotational surface in is of the form

 X(u,v)=(ucosv,usinv,g(u)). (3.1)

Similarly when the profile curve  lies in the isotropic plane, then the rotational surface in  is given by

 X(u,v)=(−usinv,ucosv,g(u)). (3.2)

Remark 3.1. The rotational surfaces given by and are locally isometric and thus we only consider the ones of the form .

Let be the rotational surface given by in  Then the nonzero components of first fundamental form of are calculated by induced metric from as follows

 E=1, G=u2. (3.3)

The nonzero components of second fundamental form of are

 L=g′′, N=ug′, (3.4)

where and From and we get

 K=1ug′g′′, H=1ug′+g′′, (3.5)

which yields that the curvatures and depend only on the variable , namely In the sequel, we have the following result.

Theorem 3.1. Rotational surfaces in are Weingarten surfaces.

We are also able to investigate the LW-rotational surfaces in  with the relation

 K=m0H+n0, m0,n0∈R. (3.6)

If in then those reduce to ones with constant relative curvature. Thus we aim to obtain the LW-rotational surfaces in  with

The following result classifies the LW-rotational surfaces satisfying

Theorem 3.2. Let be a LW-rotational surface in . Then one of the following holds

(i)  is of the form

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩X(u,v)=(ucosv,usinv,g(u)),g(u)=m02u2±u2√c1+m20u2±c2ln∣∣∣2m0(m0u+√c1+m20u2)∣∣∣,c1,c2∈R∖{0};

(ii) is an elliptic paraboloid from the Euclidean perspective, i.e.

(iii)  is given by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩X(u,v)=(ucosv,usinv,g(u)),g(u)=m02u2±u2√c1+(m20+n0)u2±±c1m20+n0ln∣∣∣2((m20+n0)u+√m20+n0√c1+(m20+n0)u2)∣∣∣,c1∈R, c1<0.

Proof. Assume is a LW-rotational surface in  having the relation Then, from it follows

 1ug′g′′=m0g′+g′′u+n0. (3.7)

We have two cases:

Case a. Hence we can rewrite as

 g′′(g′−m0u)−m0g′=0. (3.8)

If in then and vanish which is not possible. Then we have

 g′′−m0g′g′−m0u=0. (3.9)

By solving we obtain

 g(u)=m02u2±u2√e2c1+m20u2±e2c12m0ln∣∣2m0(m0u+√e2c1+m20u2)∣∣,

which gives the statement (i) of the theorem.

Case b. Then we have from

 g′′(g′−m0u)−m0g′=n0u. (3.10)

When then and This implies the statement (ii) of the theorem.

Otherwise, we conclude from that

 g′′−m0g′g′−m0u=n0ug′−m0u. (3.11)

After solving we derive

 g(u)=m02u2±u2√−e2c3+(m20+n0)u2∓±e2c3m20+n0ln∣∣∣2((m20+n0)u+√m20+n0√−e2c3+(m20+n0)u2)∣∣∣,

Therefore the proof is completed.

Example 3.1. Consider the elliptic paraboloid in  from the Euclidean perspective given by

 X(u,v)=(ucosv,usinv,0.25u2), (u,v)∈[0,2π].

Then and We plot it as in Fig. 1.

 \FRAMEitbpF1.6734in1.4909in0inFigure Fig 1. {LW-rotational surface with% }m0=0.5, n0=−0.25

## 4 Rotational surfaces in I3 with H/K=const.

The authors in [7] introduced a new kind of curvature for the hypersurfaces of Euclidean spaces, called by amalgamatic curvature and explored its geometric meaning by proving an inequality related to the absolute mean curvature of the hypersurface. In the particular case , the amalgamatic curvature is indeed the harmonic ratio of the principal curvatures of any given surface, i.e., the ratio of the Gaussian curvature and the mean curvature.

By considering this argument, we can consider the rotational surfaces in satisfying Thus the statement (i) of Theorem 3.2 is indeed a classification of the rotational surfaces in satisfying

Therefore, we have the following trivial result.

Corollary 4.1. Let be a LW-rotational surface in satisfying Then it is of the form

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩X(u,v)=(ucosv,usinv,g(u)),g(u)=m02u2±u2√c1+m20u2±c2ln∣∣∣2m0(m0u+√c1+m20u2)∣∣∣,c1,c2∈R∖{0}; (4.1)

Example 4.1. Take  and  in  Then we obtain a rotational surface in with given by

 X(u,v)=(ucosv,usinv,u2+u√1+u2+ln∣∣2(u+√1+u2)∣∣),

where , Then it can be plotted as in Fig. 2.

 \FRAMEitbpF1.0309in1.8766in0inFigure Fig 2. {Rotational surface with }H/K=1

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