ROTATIONAL SURFACES GENERATED BY PLANAR CURVES IN E WITH DENSITY

In this paper, we obtain the parametric expressions of curves which have zero weighted curvature in a plane with density eax+by and create the Smarandache curves of the obtaining curves. Also, we construct the rotational surfaces which are generated by planar curves with vanishing weighted curvature and give some characterizations for them.


Introduction
Differential geometers have studied the curves in a plane for a long time and the differential geometry of curves has been interested widely in different spaces such as Euclidean, Minkowski, Galilean, pseudo Galilean and etc.The Frenet vectors and curvature of a curve α(u) = (x(u), y(u), 0) in the plane are (see [15] and [17]) x (t) 2 + y (t) 2  (x (u), y (u), 0), x (u) 2 + y (u) 2  (−y (u), x (u), 0), (1.1) B = ∓(0, 0, 1), Here we must note that, the sign of "∓" which is stated in the above formulas of Frenet vectors and curvature will be taken as "+" throughout this study.
Recently, new curves have been obtained by using the Frenet frames of curves.One of these new curves is Smarandache curve and if we denote T N -Smarandache curve as β(u), T B-Smarandache curve as δ(u) and N B-Smarandache curve as γ(u), then they are given by respectively.Since the Smarandache curves play an important role in Smarandache geometry, many geometers have been interested in these curves.More information about Smarandache curves can be found in [1], [2], [18], [19].
Also, lots of studies have been done about rotational surface in Euclidean space E 3 and Lorentz-Minkowski space E 3 1 in recent years (see [5], [6], [7], [12] and etc).A rotational surface in Euclidean space is generated by rotating of an arbitrary curve about an arbitrary axis.In this sense, let α(u) = (x(u), y(u), 0) be a regular plane curve in the xy−plane which is defined on a open invertal I ⊂ R. A rotational surface M in E 3 is defined by the following three cases: If the axes of revolution are x-axis, y-axis and z-axis, then the rotational surfaces are given by On the other hand, after Morgan's article [14], manifolds with density have been a very popular topic in many areas for geometers, physicists, economicists.For example, in physics, an object may have differing internal densities so in order to determine the object's mass it is necessary to integrate volume weighted with density.In differential geometry, let we consider a bounded curve on the closed Euclidean half plane (boundary on the x-axis) and the surfaces of revolution formed by rotating that curve about x-axis.Areas and arc lengths on that surface correspond to areas and arc lengths on the half plane with a weighting of 2πy.In government and economics, it is often necessary to consider aggregate properties of groups and subgroups of people.For large groups, these aggregate properties can be determined by integrating over the members of the group, the differing individual properties (much like different densities).
In 2003, Gromov [8] firstly has described κ ϕ weighted curvature of curves and H ϕ weighted mean curvature of n−dimensional hypersurfaces on a manifold with density e ϕ as and respectively.Here, N is the normal vector, κ is the curvature of the curve; η is the normal vector field, H is the mean curvature of the hypersurfaces and N , ∇ϕ = dϕ dN , η , ∇ϕ = dϕ dη .Furthermore, Corwin [4] has defined weighted Gaussian curvature of a surface in a manifold with density by where G is Gaussian curvature of the surface and ∆ is the Laplacian operator.For another characterizations of manifolds with density, we refer to [3], [9], [10], [11], [13], [16], [20] and etc.

Curves with Vanishing Weighted Curvature in a Plane with Density e ax+by
In this section, firstly we construct the weighted curvature of a curve in a plane with positive density e ax+by .After that, we investigate the curves whose weighted curvatures are equal to zero according to the cases of a and b.Followingly, we create Smarandache curves of these curves as examples.
Weighted curvature κ ϕ of a curve α(u) = (x(u), y(u), 0) in a plane with density e ax+by is obtained as So, Corollary 2.1.If the curve α(u) = (x(u), y(u), 0) is a unit speed curve, then the weighted curvature κ ϕ of the curve α(u) in the plane with density e ax+by is Proposition 2.1.Weighted curvature κ ϕ of the curve α(u) = (x(u), y(u), 0) in the plane with density e ax+by vanishes if and only if Now, we will examine the weighted curvature of a curve in a plane with density e ax+by for different values of constants a and b.
First, let's assume that b = 0; i.e. the density of the plane is e ax .
Then from (2.1), the weighted curvature κ ϕ of the curve α(u) = (x(u), y(u), 0) in a plane with density e ax is obtained as .
Hence, we have Proposition 2.2.Weighted curvature κ ϕ of the curve α(u) = (x(u), y(u), 0) in the plane with density e ax is zero if and only if From (2.2), we have So, taking the sign of "∓" which has been stated in the last equation as "+", we have Theorem 2.1.Let α(u) be a curve with vanishing weighted curvature in a plane with density e ax .Then, it can be parametrized by where c 1 , c 2 ∈ .
In Figure 1, one can see the graph of curve (2.Example 2.1.From (1.1), the tangent, normal and binormal vectors of the curve (2.3) are obtained as respectively.Then, the T B-Smarandache curve δ(u) which is defined with the aid of the Frenet vectors of curve (2.3) can be written as respectively.
Note that, the T N -Smarandache curve β(u) and the N B-Smarandache curve γ(u) of the curve (2.3) can be obtained with the similar procedure of above.Now, taking x(u) = sin u, c 1 = 3 in (2.4), we get Figure 2 shows the graph of this curve for some different values of a.   and the graph of this curve for some different values of a can be seen in Figure 4 .Then from (2.1), the weighted curvature κ ϕ of the curve α(u) = (x(u), y(u), 0) in the plane with density e by is obtained as . So, we get Proposition 2.3.Weighted curvature κ ϕ of the curve α(u) = (x(u), y(u), 0) in the plane with density e by is zero if and only if From (2.9), the results are similar to the curves with vanishing weighted curvature in the plane with density e ax .

Rotational Surfaces Generated By Planar Curves with Vanishing Weighted Curvature in E 3 With Density
In this section, we obtain the parametric representations of rotational surfaces which are generated by curves with vanishing weighted curvature in a plane with density e ax+by and give their mean and Gaussian curvatures.Also, we draw the graphics of these curves, surfaces, the Gaussian and mean curvatures functions' graphics and the variations of Gaussian and mean curvatures on these surfaces.

Rotational Surfaces Generated By Curve (2.3).
First, we suppose that the axis of revolution is x-axis.Then, from (1.2) and (2.3), the rotational surface M can be parametrized by , respectively.
We know that, minimal surfaces are defined as surfaces with zero mean curvature and a surface for which the Gaussian curvature vanishes everywhere is called a flat surface.Also, a point p on a regular surface M ⊂ R 3 is said to be elliptic, parabolic and hyperbolic, if the Gaussian curvature G(p) > 0, G(p) = 0 and G(p) < 0, respectively.
In Figure 5, one can see the curve (2.3) which generates the rotational surface and the rotational surface Also, the Gaussian and mean curvatures of this surface are calculated as follows   In Figure 8, the Gaussian and mean curvatures functions' graphics and the variations of Gaussian and mean curvatures on rotational surface (3.2) can be seen above and below, respectively.

Rotational Surfaces Generated By Curve (2.6).
First, let the axis of revolution be the x-axis.Then, by following the similar procedure with previous subsection, from (1.2) and (2.6), the rotational surface M is obtained as      From above, it is obvious that the mean and Gaussian curvatures of this surface are zero.Figure 15

Conclusion and Future Work
In the present paper, we've obtained the weighted curvature of a curve in a plane with density e ax+by , where a, b ∈ R not all zero constants and investigated the curves with vanishing weighted curvature according to the cases a = 0, b = 0 and a = 0, b = 0. Later, we've constructed the Smarandache curves of these curves and obtained the tangent, normal and binormal vectors and curvature of them.Also, we've constructed the rotational surfaces which are generated by curves with vanishing weighted curvature in the plane with density e ax+by and give some characterizations by obtaining their mean and Gaussian curvatures.We've drawn these curves and surfaces with the aid of Mathematica.
We think that, this study will provide a different perspective to the researchers who're dealing with the weighted spaces.As a problem, these curves and surfaces can be studied in different spaces such as Minkowski space, Galilean space and pseudo Galilean space and etc.Also, these structures can be studied in Euclidean or other spaces with different densities and in the near future, the authors will deal with these open problems in detail.

Figure 1 .
Figure 1.Curve (2.3) with vanishing weighted curvature in a plane with density e ax

Figure 3
Figure 3 shows the graph of curve (2.6) for y(u) = arccos(e u ), c 1 = 0, c 2 = 5 and some different values of a.

Theorem 3 . 1 .
The rotational surface (3.1) which is generated by curve (2.3) with vanishing weighted curvature isn't minimal and flat.Theorem 3.2.i) All of the points of the surface (3.1) are elliptic, if ac 2 > 0;

Figure 5 .
Figure 5.The curve (2.3) which generates the rotational surface and the rotational surface (3.1)

Theorem 3 . 3 .
The rotational surface (3.2) which is generated by curve (2.3) with vanishing weighted curvature isn't minimal and flat.Theorem 3.4.i) If a and x(u) have different signs, then the points which satisfy this condition on the surface (3.2) are elliptic; ii) none of the points of the surface (3.2) are parabolic; iii) if a and x(u) have same signs, then the points which satisfy this condition on the surface (3.2) are hyperbolic.

Figure 7 .
Figure 7.The curve (2.3) which generates the rotational surface and the rotational surface (3.2)

Figure 9 .
Figure 9.The curve (2.3) which generates the rotational surface and the rotational surface (3.3)

(3. 4 )Theorem 3 . 5 .Theorem 3 . 6 .
X(u, v) = c 2 − ln (cos(c 1 + ay(u))) a , y(u) cos v, y(u) sin v .Hence, the Gaussian and mean curvatures of this surface are G = a sin(2(c 1 + ay(u))) 2y(u) and H = sin(c 1 + ay(u)) + ay(u) cos(c 1 + ay(u)) 2y(u) , respectively.The rotational surface (3.4) which is generated by curve (2.6) with vanishing weighted curvature isn't minimal and flat.For 2kπ < c 1 + ay(u) < π 2 + 2kπ, k ∈ N, i) if a and y(u) have same signs, then the points which satisfy this condition on the surface (3.4) are elliptic; ii) if a and y(u) have different signs, then the points which satisfy this condition on the surface (3.4) are hyperbolic.For 3π 2 + 2kπ < c 1 + ay(u) < 2π + 2kπ, k ∈ N, i) if a and y(u) have different signs, then the points which satisfy this condition on the surface (3.4) are elliptic;ii) if a and y(u) have same signs, then the points which satisfy this condition on the surface (3.4) are hyperbolic.Also, the points of the surface (3.4) which satisfy the condition "c 1 + ay(u) = 2kπ, k ∈ N " are parabolic.Next, taking y(u) = arccos(e u ), c 1 = 0, c 2 = 5 and a = 1, then the curve (2.6) which generates the rotational surface and the rotational surface (3.4) can be seen in Figure11.

Figure 11 .
Figure 11.The curve (2.6) which generates the rotational surface and the rotational surface (3.4)

Figure 13 .
Figure 13.The curve (2.6) which generates the rotational surface and the rotational surface (3.5)

Figure 15 .
Figure 15.The curve (2.6) which generates the rotational surface and the surface of revolution (3.6)