The Indicatrix of the Surface in Four-Dimensional Galilean Space

This article discusses geometric quantities associated with the concept of surfaces and the indicatrix of a surface in four-dimensional Galileo space. In this case, the second orderly line in the plane is presented as a surface indicatrix. It is shown that with the help of the Galileo space movement, the second orderly line can be brought to the canonical form. The movement in the Galileo space is radically different from the movement in the Euclidean space. Galileo movements include parallel movement, axis rotation, and sliding. Sliding gives deformation in the Euclidean space. The surface indicatrix is deformed by the Galileo movement. When the indicatrix is deformed, the surface will be deformed. In the classification of three-dimensional surface points in the four-dimensional Galileo phase, the classification of the indicatrix of the surface at this point was used. This shows the cyclic state of surface points other than Euclidean geometry. The geometric characteristics of surface curves were determined using the indicatrix test. It is determined what kind of geometrical meaning the identified properties have in the Euclidean phase. It is shown that the Galilean movement gives surface deformation in the Euclidean sense. Deformation of the surface is indicated by the fact that the Gaussian curvature remains unchanged.


Introduction.
While investigating a surface in space, it is important to know its geometric dimensions: the first and the second fundamental forms, the total and mean curvatures, and also the principal curvatures. The principal curvatures of the surface are clarified by its indicatrix at a given point. In order to classify the surface around the given point, it is important to classify its surface at that point.
The first and the second fundamental forms, the total and mean curvatures, and also the principal curvatures of the surface in three-dimensional Galilean space are determined in the [3], [8]. In this space the Frenet formulas have been formed in the [4] studies. And in this article, we are going to subdivide the points of the surface into classes with the help of the indecatrix of the surface.

Main Results
While studying the surfaces, it is necessary to check them around the certain point. The surface can be monitored by making a surface indicatrix around the point of the surface.
We choose the point of the surface as the beginning of the coordinates, and at this point, we get the coordinate system such as OXYZ, in this case the surface indicatrix of the four-dimensional Galilean space ( 4 Г ) will be the following type of the second order surface [5]: (1) If we make a special OYZ plane Euclidean rotation of this surface indicatrix [6] by the Galilean motion It can be defined as a canonical form.
Here 1 2 , h h are the angles of the Galilean space.
Proof. At first we take the equation (3) and put it to the equation (2). In this case We will clarify 1 2 , h h . As it is clear in the form (6), it should be 22 If we put 1 2 , h h , which is clarified in (6), to (5), the following formula will appear: We will get the equality (4). The theorem has been proven.
Accordingly, in order to move from the indicatrix equation of the surface (2) to the canonical form (7), it is necessary to make a Galilean motion: Generally, with the help of the Galilean motion, a certain surface moves to this type of the surface. Since it hasn't been impacted on the form of the surface, we can possibly determine its classification by this movement.
To classify the surface at the given point in 4 Γ , we will make the classification of its indicatrix.
It is one or two-sheet hyperboloid. If we consider this surface is 13 1 2 xz a = which is 0 y = plane with its cross section on OXZ plane, it can be the cyclic form in three-dimensional Galilean space [3].
If we take x a = ± . If we rotation the indicatrix of the surface at a given point with the help of Galilean transformation, the surface will also be deformed.
Theorem. If we rotation the indicatrix of the surface at a given point with the help of Galilean transformation, the curvature of the surface at this point will be stayed the same.
Proof. To demonstrate a total curvature of the surface, it is important to indicate that the multiplication of the principal curvatures can also stay the same. The principal curvatures are equal to the reverse value of the semi-axes quadrate of the indicatrix at the given point of the surface. So to save the total curvature at the given point of the surface, the multiplication of the half axes of the indicatrix at the given point of the surface has to be saved.
Taking the indicatrix of the surface in Galilean space as the form (2), let's find its specific numbers.  (2), the equation will be as the form (5). Let's make the following equation to find the specific numbers of its matrix. λ λ λ λ λ λ = are saved. So the multiplication of the semi-axes, the multiplication of the principal curvatures and the total curvatures at the given point of the surface are equal. The theorem has been proven. The result. There are some surfaces in Euclidian space, which are not isometric and have the similar total curvatures at the given point.
Proof. As we know, the Galilean substitution does not save an isometry in Euclidian space. The surface moves to another surface, which is not isometric to itself. [2] It is evident from all above, that the Galilean substitution does not impact on the total curvature at the point of the surface. So in Euclidian space the Galilean substitution transforms the surface to another surface, which is not symmetric, but has similar total curvatures at the given point.

Conclusions
The motion determinant in Galilean space is equal to 1, however, it is not orthogonal and symmetric. If we consider the system of the coordinates in four-dimensional Galilean space as 1 2 3 4 OX X X X , here we call 1 2 3 OX X X as a hyperplane. In this motion, the points of the special plane of the surface are not changed. Any other point slides to the point 1 2 3 ( , , , ) t x h y h z h + + + on the hyperplane which goes by this point ( , , , ) t x y z . In this motion the isometry is not saved. Even if the is not saved, as it has been shown above, the full curvature is saved.