Nonnull Curves with Constant Weighted Curvature in Lorentz-Minkowski Plane with Density

In this paper, the parametric expressions of spacelike and timelike curves with constant weighted curvature for some cases of a and b in Lorentz-Minkowski plane with density e are obtained.


Introduction
In 2003, Gromov has introduced the notions of weighted mean curvature of an n -dimensional hypersurface and weighted curvature of a curve on manifolds with density as respectively [5]. Here, H is the mean curvature and η is the normal vector field of an n -dimensional hypersurface; κ is the curvature and N is the normal vector of the curve.
After these definitions, the differential geometry of the curves and hypersurfaces on manifolds with density in Euclidean, Minkowski and Galilean spaces has been started to be an important topic for geometers, physicists, economists and etc. For instance, in 2006 the authors have defined the weighted Gaussian curvature and they have given a generalization of Gauss-Bonnet formula for 2-dimensional differentiable manifold with density in [4]. In [11][12][13], F.Morgan has studied the manifolds with density, provided the generalizations of theorems of Myers and others to Riemannian manifolds with density and studied the Perelman's proof of the Poincare conjecture, respectively.
The classification of constant weighted curvature curves in a plane with a log-linear density has been done in [7] and some other results, such as Fenchel's type theorem for the class of simple, closed, convex curves and the fact "the plane with density e x contains no isoperimetric region" have been proved in [14]. In [10], Lopez has studied the minimal surfaces in Euclidean 3-space with a log-linear density φ(x, y, z) = αx + βy + γz, where α, β and γ are real numbers not all-zero. Also, Belarbi et al. have studied the surfaces in R 3 with density and they have given some results in a Riemannian manifold M with density in [1] and [2], respectively. Furthermore, ruled and translation minimal surfaces in R 3 with density e z ; helicoidal surfaces in R 3 with density e −x 2 −y 2 and weighted minimal affine translation surfaces in Euclidean space with density have been studied in [6,18,19], respectively. Also, some types of surfaces have been studied by geometers in other spaces such as Minkowski 3-space and Galilean 3-space with density. For instance, a helicoidal surface of type I + with prescribed weighted mean curvature and Gaussian curvature in Minkowski 3-space and weighted minimal translation surfaces in Minkowski 3-space with density e z have been constructed in [15] and [16], respectively. In [17], weighted minimal translation surfaces in the Galilean 3-space with log-linear density has been classified and in [8], weighted minimal and weighted flat surfaces of revolution in Galilean 3-space with density e ax 2 +by 2 +cz 2 have been investigated. Now, we'll recall some basic notions about curves in Lorentz-Minkowski plane.
Let L 2 be Lorentz-Minkowski plane defined as a space to be usual 2-dimensional vector space consisting of vectors {(x 0 , x 1 ) : x 0 , x 1 ∈ R} but with a linear connection ∇ corresponding to its Minkowski metric g given by g(x, y) = −x 0 y 0 + x 1 y 1 . Here, there are three categories of vector fields, namely, In general, the type into which a given vector field X falls is called the causal character of X .
In the present study, we'll deal with spacelike and timelike curves in Lorentz-Minkowski plane with density and the aim of this study is to investigate the spacelike and timelike curves with constant weighted curvature in Lorentz-Minkowski plane with density e ax+by .

Spacelike Curves with Constant Weighted Curvature in Lorentz-Minkowski Plane with Density
e ax+by In this section, we obtain the weighted curvature κ φ of a spacelike curve α in Lorentz-Minkowski plane with density e ax+by and investigate the spacelike curves with constant weighted curvature for some cases of not all zero constants a and b.
The weighted curvature κ φ (u) of a spacelike curve α(u) = (x(u), y(u)) with arc-length paremeter in Lorentz-Minkowski plane with density e ax+by is obtained as where N (u) = (y ′ (u), x ′ (u)). Since α(u) = (x(u), y(u)) is a spacelike curve with arc-length papameter, we can take So, from (2.1) and (2.2), the constant weighted curvature κ φ (u) can be written as where λ ∈ R. Now, let us obtain the spacelike curves with constant weighted curvature λ according to some cases of constants a and b.

The Case of " a = b"
In this case, the equation (2.3) can be rewritten as Using the definition of the hyperbolic functions in (2.4), we have Now, we can find the spacelike curves by solving the equation (2.5) according to cases of constant λ.

The Case of "
In this case, the equation (2.3) can be rewritten as . (2.24) Now, we can obtain the spacelike curves by solving the equation (2.24) according to cases of constant λ.

Solving equation (2.24) for
and so, Thus, from (2.2) and (2.34), we have By integrating both sides of (2.36), we get Thus, we can write By integrating both sides of (2.48), we get and from (2.49), we can write (2.50) So, from (2.2) and (2.50), we have

Timelike curves with constant weighted curvature in Lorentz-Minkowski plane with density
e ax+by In this section, we obtain the weighted curvature κ φ of a timelike curve β in Lorentz-Minkowski plane with density e ax+by and investigate the timelike curves with constant weighted curvature for some cases of not all zero constants a and b with the same procedure in the previous section.

(3.2)
So, from (3.1) and (3.2), the constant weighted curvature κ φ can be written as where λ ∈ R. Now, let us obtain the timelike curves with constant weighted curvature λ according to some cases of constants a and b.

The case of " a = b"
In this case, the equation (3.3) can be rewritten as Thus, we have the following Theorems which can be obtained with the same procedure in Subsection 2.1.

Theorem 3.1 The timelike curve β(u) with vanishing weighted curvature in Lorentz-Minkowski plane with
density e a(x+y) is
So, we have the following Theorems which can be obtained with the same procedure in Subsection 2.2.

Conclusion and future work
In this study, we have obtained the weighted curvature κ φ of spacelike and timelike curves in Lorentz-Minkowski plane with density e ax+by and we have given the parametric expressions of the spacelike and timelike curves with constant weighted curvature for the cases of " a = b", " a ̸ = 0, b = 0 " and " a = 0, b ̸ = 0 ". Also, we have construct some examples of obtained curves for different values of constants a and b.
We hope that, this study will bring a new viewpoint and break fresh ground to geometers who are dealing with the curves in a plane with density. And in the near future, spacelike and timelike curves in Lorentz-Minkowski plane with different densities can be investigated by geometers.