HYPERSURFACE FAMILIES WITH SMARANDACHE CURVES IN GALILEAN 4-SPACE

In this paper, we study the hypersurface families with Smarandache curves in 4-dimensional Galilean space G4 and give the conditions for di¤erent Smarandache curves to be parameter and the curve which generates the Smarandache curves is geodesic on a hypersurface in G4: Also, we investigate three types of marching-scale functions for one of these hypersurfaces and construct an example for it.


PRELIMINARIES
In physics, geodesics which are de…ned as a parallel transport of a tangent vector in a linear (a¢ ne) connection on the manifold M are very important for general relativity. Because, the geodesic equation which is given with a set of initial conditions is very useful in theoretical foundations of general relativity. Also, in general, relativity gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress-energy tensor. For example, the path of a planet orbiting a star is the projection of a geodesic of the curved four-dimensional spacetime geometry around the star onto three-dimensional space.
Furthermore, as an alternative de…nition of a geodesic line can be de…ned as the shortest curve connecting two points on a manifold. A curve ( ) on a hypersurface ' ( ; ; ) is geodesic i¤ the normal N ( ) of the curve ( ) and the normal ( ; 0 ; 0 ) of the hypersurface ' ( ; ; ) at any point on the curve ( ) are parallel to each other and a curve ( ) on the hypersurface ' ( ; ; ) is asymptotic i¤ the normal N ( ) of the curve ( ) and the normal ( ; 0 ; 0 ) of the hypersurface ' ( ; ; ) at any point on the curve ( ) are perpendicular.
The problem of constructing a family of surfaces from a given spatial geodesic curve …rstly has been studied by Wang et al. in 2004 and in that study, the authors have derived a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve [15]. After this study, in 2008 the generalization of the Wangs'assumption to more general marching-scale functions has been given by Kasap et al [7]. By using these studies, the problem of …nding a surface pencil from a given spatial asymptotic curve has been investigated in [4] and the necessary and su¢ cient condition for the given curve to be the asymptotic curve for the parametric surface has been stated in [1]. Also, the problem of …nding a hypersurface family from a given asymptotic curve in R 4 has been handled in [5].
Surfaces with common geodesic and family of surface with a common null geodesic in Minkowski 3-space have been studied in [8] and [13], respectively.
The Galilean space G 3 is a Cayley-Klein space equipped with the metric of signature (0; 0; +; +). The absolute …gure of the Galilean space consists of an ordered triple f!; f; Ig in which ! is the ideal (absolute) plane, f is the line (absolute line) in ! and I is the …xed elliptic involution of f .
If If the curve is both an asymptotic and parameter (isoparametric) curve on ', then it is called isoasymptotic on the hypersurface '. Similarly, if the curve is both a geodesic and parameter (isoparametric) curve on the hypersurface ', then it is called isogeodesic on the hypersurface '.
On the constructions of surface families with common geodesic and asymptotic curves in Galilean Space G 3 and an approach for hypersurface family with common geodesic curve in the Galilean Space G 4 have been handled in [9], [10] and [11], respectively.
On the other hand, the geometry of Smarandache curves has been very popular topic for di¤erential geometers, recently. Let ( ) be an admissible curve in G 4 and fT; N; B 1 ; B 2 g be its moving Frenet frame. Then T N; kT +N +B1+B2k , respectively. The problem of constructing a family of surfaces from a given some special Smarandache asymptotic curves in Euclidean 3-space has been analyzed in [14] and surfaces using Smarandache asymptotic curves in Galilean space have been studied in [2].
From (10), we have where we denote @' T N ( ; ; ) Thus, if we use (11) in (8), by obtaining the normal of the hypersurface (10), we can state the following theorem: where L 1 Here we must note that, from (12), we have So, from (2), (11), (12) and (13), the normal of the hypersurface (10) for = 0 and = 0 is obtained as where Also, from the de…nition of a given curve ( ) on the hypersurface ' ( ; ; ) to be geodesic, it must be Here, by taking the N B 2 -Smarandache curve of ( ) instead of the curve ( ) in (9), let us de…ne a parametric hypersurface ' N B2 ( ; ; ) which is given with the aid of the N B 2 -Smarandache curve of ( ) and the Frenet vectors of the curve ( ) as follows If N B 2 -Smarandache curve of the curve ( ) is a isoparametric curve on a hypersurface ' N B2 ( ; ; ) in G 4 for = 0 and = 0 ; then from (8), the normal of this hypersurface is where L 1 L 2 ; T 1 0 T 2 and M 1 0 M 2 : Also, from the de…nition of a given curve ( ) on the hypersurface ' ( ; ; ) to be geodesic where N B 2 -Smarandache curve r N B2 of the curve ( ) is isoparametric, it must be N B1 2 ( ; 0 ; 0 ) 6 = 0 and N B1 3 ( ; 0 ; 0 ) = N B1 4 ( ; 0 ; 0 ) = 0: So, using these conditions with (19) in (18), the proof completes.
For the purposes of simpli…cation and better analysis, Wang et al. have studied the case when the marching-scale functions can be decomposed into two factors in Euclidean 3-space. The factor-decomposition form possesses an evident advantage: the designer can select di¤erent sets of functions to adjust the shape of the surface until they are grati…ed with the design, and the resulting surface is guaranteed to belong to the isogeodesic surface pencil with the curve as the common geodesic [15]. Also in [3] and [11], the three types of the marching-scale function which have three parameters have been studied in 4-dimensional Galilean and Euclidean spaces,    Now, let we construct an example for this hypersurface family.

Example 3. Let ( ) be a curve which is parametrized by
From (4) and (5), it is easy to obtain that  in G 4 . Di¤ erent projections from four-space to three-spaces of the hypersurface (29) for = =2 can be seen in the Fig.1: From now on, we'll give the parametric hypersurfaces given by di¤erent Smarandache curves of curve ( ) and their normal vector …elds. Also, we'll state the theorems which give us the conditions for which ( ) is a geodesic curve where Smarandache curves of the curve ( ) is isoparametric on these hypersurfaces. One can prove these theorems and investigate the conditions for di¤erent types of marching-scale functions with the similar methods given in the above case. CASE 3.
Here, by taking the T B 1 -Smarandache curve of ( ) instead of the curve ( ) in (9), let us de…ne a parametric hypersurface ' T B1 ( ; ; ) which is given with the aid of the T B 1 -Smarandache curve of ( ) and the Frenet vectors of the curve ( ) as follows If T B 1 -Smarandache curve of the curve ( ) is a isoparametric curve on a hypersurface ' T B1 ( ; ; ) in G 4 for = 0 and = 0 ; then from (8), the normal of this hypersurface is Thus, For the curve ( ) to be a geodesic where T B 2 -Smarandache curve r T B2 of the curve ( ) is isoparametric on the hypersurface ' T B1 ( ; ; ) in G 4 , the following conditions must hold:

CASE 5.
Here, by taking the N B 1 -Smarandache curve of ( ) instead of the curve ( ) in (9), let us de…ne a parametric hypersurface ' N B1 ( ; ; ) which is given with the aid of the N B 1 -Smarandache curve of ( ) and the Frenet vectors of the curve ( ) as follows If N B 1 -Smarandache curve of the curve ( ) is a isoparametric curve on a hypersurface ' N B1 ( ; ; ) in G 4 for = 0 and = 0 ; then from (8), the normal of this hypersurface is So, we can state the following Theorem:  (8), the normal of this hypersurface is where T N B1 1 ( ; 0 ; 0 ) = 0;  (8), the normal of this hypersurface is

CASE 9.
Here, by taking the T B 1 B 2 -Smarandache curve of ( ) instead of the curve ( ) in (9), let us de…ne a parametric hypersurface ' T B1B2 ( ; ; ) which is given with the aid of the T B 1 B 2 -Smarandache curve of ( ) and the Frenet vectors of the curve ( ) as follows