INFINITESIMAL BENDING OF CURVES ON THE RULED SURFACES MARIJA NAJDANOVIĆ

In this paper we study infinitesimal bending of curves that lie on the ruled surfaces in Euclidean 3-dimensional space. We obtain an infinitesimal bending field under whose effect all bent curves remain on the same ruled surface as the initial curve. Specially, we consider infinitesimal bending of the curves which belong to the cylinder as well as to the hyperbolic paraboloid and find corresponding infinitesimal bending fields. We examine the variation of the curvature of a curve under infinitesimal bending on the hyperbolic paraboloid. Some examples are visualized using program packet Mathematica.


INTRODUCTION
Infinitesimal bending of curves and surfaces is a special part of the surface bending theory which presents one of the main consisting parts of the global differential geometry.The problems of infinitesimal bending have physical applications (in the study of elasticity, for example) and have a long history.
Historically, the first result of the surface bending theory belongs to Cauchy.Later, bending theory was developed thanks to the works of leading mathematicians of the considered area like Blaschke, Cohn-Vossen, A. D. Alexandrov, A. V. Pogorelov, I. N. Vekua, V. T. Fomenko, I. Kh.Sabitov, I. I. Karatopraklieva, V. A. Alexandrov and many others.
Infinitesimal bending is determined by the stationary of arc length with appropriate precision.A concept of infinitesimal bending dealt first with infinitesimal bending of surfaces and then with the same problem in the theory of curves and manifolds.
First we shall give some basic facts, definitions and theorems according to (Velimirović, 2001a& Velimirović, 2009).
Let us consider continuous regular curve C ⊂ R 3 , given with the equation included in a family of the curves where u is a real parameter and we get C for = 0 (C = C 0 ).
Theorem 2. (Efimov, 1948) Necessary and sufficient condition for z(u) to be an infinitesimal bending field of a curve C is to be dr where • stands for the scalar product in R 3 .
The next theorem is related to determination of the infinitesimal bending field of a curve C. Theorem 3. (Velimirović, 2001a) Infinitesimal bending field for the curve C (1) is where p(u) and q(u), are arbitrary integrable functions and vectors n 1 (u) and n 2 (u) are respectively unit principal normal and binormal vector fields of the curve C. As infinitesimal bending field can be written in the form Corresponding author: marijamath@yahoo.com

MATHEMATICS, COMPUTER SCIENCE AND MECHANICS
where p(t), q(t) are arbitrary integrable functions, or in the form where P i (t), i = 1, 2, i Q(t) are arbitrary integrable functions, too.An interesting problem is the infinitesimal bending of a plane curve which stays in the plane after bending.This problem was considered in the paper (Velimirović, 2001a).It was found corresponding infinitesimal bending field, i. e. it was proved the next theorem.
Theorem 4. Infinitesimal bending field that plane curve under infinitesimal bending includes in a family of planes curves where i and j are unit vectors in the direction of Cartesian axes.
Also, in the same paper it was proved that the area of the region determined by a plane curve being infinitesimally bent staying plane is stationary.
An interesting question considered in the paper (Velimirović et al., 2010) is about infinitesimal bending of a spherical curve but so that all bent curves are on the same sphere.It was proved that there isn't infinitesimal bending of a spherical curve belonging to the sphere.
In this paper we confront the question: Is it possible to infinitesimally bend a curve C which lies on the ruled surface S , but so that all bent curves of the family C stay on S ?The answer is affirmative and in the sequel we shall give an explicit formula for such an infinitesimal bending field.

DETERMINATION OF INFINITESIMAL BENDING FIELD
Theorem 5. Let a ruled surface S be given by 9) with a directrix ρ = ρ(u) and generatrices in the direction of the vector e(u), and let a curve be on the surface S .Then infinitesimal bending field which given curve leaves on the surface S is where uρ u • e + v 0 and c is a constant. Proof.Let be a curve on the surface S and be an infinitesimal bending of the curve C determined with the field z.As the family of the curves C , ∈ (−1, 1) belongs to the surface S , the field z must be in the form where z 1 (t) is a real continuous differentiable function.
Having in mind that z is an infinitesimal bending field, the condition (3) must be satisfied, i. e.
i. e. (ρ As it is valid e = 1, we conclude that must be satisfied e . Using this fact, we obtain homogenous linear differential equation whose solution is c is a constant and uρ u • e + v 0. Putting ( 16) into (12), we obtain (11).
Let us note that if the directrix C : ρ = ρ(u) is at the same time the striction line of the ruled surface S , then it is valid ė • ρ = 0 (see Gray (1998)), therefore e(u) is the field of the infinitesimal bending of curve C which that curve includes in a family of curves on the ruled surface S , In addition, each curve of the family C is "parallel" to the curve C, i. e. the cut of the each generatrix between C and C is of the same length.Indeed, due to e(u 1 ) = e(u 2 ) = 1.Note that for the directrix of the ruled surface can be taken every curve on the surface which is cut or touched by the generatrices, and therefore, the directrix can be the striction line, if there is one.
The ruled surfaces are not the only surfaces on which it is possible to infinitesimally bend curves.We will show that in the following example.
Example 6.Let S be the rotation surface in R 3 of class C ∞ , given as the graph of the function Note that S is not a ruled surface (in fact it is not ruled in any open neighborhood of any point of the unit circle (cos θ, sin θ, 0)).Consider the curve C ⊂ S given by where Then the family of curves given by belongs to the surface S and presents infinitesimal bending of the curve C determined with the field z = (0, g(t), 0).

INFINITESIMAL BENDING OF CURVES ON THE CYLIN-DER
A cylinder is an example of the ruled surfaces.Let us find an infinitesimal bending field of a curve on the cylinder, that leaves the given curve on the cylinder after bending.
Theorem 7. Let the cylinder be given by the equation S : x 2 +y 2 = a 2 .Let C : r(t) : (α, β) → R 3 be a regular continuous curve on the cylinder S and z(t) be a vector field of class C 1 which given curve includes in the family of the curves C : r = r(t) + z(t), ∈ (−1, 1), on the cylinder S , under infinitesimal bending.a) If the curve C is in the plane z = const, then infinitesimal bending field is z(t) = z 3 (t)k, where k = (0, 0, 1) and z 3 (t) is an arbitrary real function of class C 1 .b) Otherwise, infinitesimal bending field is a constant vector z = ck, where c is a real constant.Bending is rigid and reduces to the translation along z-axis.
Example 8. Let us consider some curves on the cylinder S : r(u, v) = (3 cos u, 3 sin u, v) and their infinitesimal bending.These examples are visualized using the software package Mathematica (Gray, 1998).
For the circle C 1 : r(t) = (3 cos t, 3 sin t, 0), which is in the plane z = 0, we can take infinitesimal bending field z(t) = tk.On the figure Fig.
(1) we can see the curve C 1 and bent curves for = 0.1, 0.3, 0.5, 0.9.Obviously, the curve C 1 clefts under that infinitesimal bending and is included in a family of helices on the cylinder.

INFINITESIMAL BENDING OF CURVES ON THE HY-PERBOLIC PARABOLOID
A hyperbolic paraboloid is an example of doubly ruled surfaces.In the sequel we will find infinitesimal bending field of an arbitrary curve belonging to this surface.Theorem 9. Let a hyperbolic paraboloid be given by the equation S : r(u, v) = (u, v, uv) and continuous regular curve on it by C : r(t) = r(u(t), v(t)).Let z(t) be a vector field of class C 1 which given curve under infinitesimal bending includes in the family of the curves C : r = r(t) + z(t), ∈ (−1, 1), on the hyperbolic paraboloid S .Then the equations v + u(uv) .0 , and u + v(uv) .0, determine the field z(t).c is an arbitrary constant. Proof.Let be an infinitesimal bending of C determined by the field z(t) = (z 1 (t), z 2 (t), z 3 (t)), z 1 , z 2 , z 3 are real continuous differentiable functions.As the curves (25) are on the surface S , it must be satisfied the next condition By dividing with 0 we obtain Since the condition (26) must be valid for each ∈ (−1, 1) \ {0}, it must be and From ( 28) we get z 1 (t) = 0 or z 2 (t) = 0. We distinguish two cases.
It is easy to show that the fields ( 23) and ( 24) present infinitesimal bending fields of the curve C which that curve leaves on the hyperbolic paraboloid S , after bending.
Example 10.Let the curve C 3 : r(t) = (t, t, t 2 ), t ∈ (a, b) ⊆ R, be given on the surface S : r(u, v) = (u, v, uv).According to Theorem 9, after the necessary calculations, we obtain that infinitesimal bending fields of the curve C 3 which given curve leaves on the surface S , for c = 1, have the next form  Example 11.For the curve C 4 : r(t) = (t, t 2 , t 3 ) the corresponding infinitesimal bending fields are .
Graphical presentation of the family of bent curves C 4 under the field z 2 is given in Fig. (5).

VARIATION OF THE CURVATURE OF CURVES ON THE HYPERBOLIC PARABOLOID
Under infinitesimal bending, geometric magnitudes describing a curve are changing and this change is determined by the variation.We define the variation according to (Vekua (1959)).
Definition 12. Let A = A(u) be the magnitude that characterizes a geometric property on the curve C and A = A (u) the corresponding magnitude on the curve C being infinitesimal bending of the curve C, Coefficients δA, δ 2 A, . . ., δ n A, . . .are the first, the second, ..., the nth variation of the geometric magnitude A, respectively under infinitesimal bending C of the curve C.
Obviously, for the first variation is effective i. e. δA = lim Infinitesimal bending is a kind of deformation under which the coefficients of the first fundamental form don't get the variations of the first order, i. e. these variations are zero.The magnitudes expressed by the coefficients of the first fundamental form and derivatives of these coefficients also have no variation of the first order (for example Cristoffel's symbols, area of a region on the surface and other).However, coefficients of the second fundamental form have, generally speaking, variations different from zero.

Figure 1 .Figure 2 .
Figure 1.Infinitesimal bending of the circle C 1 on the cylinder.

Figure 3 .
Figure 3. Trivial infinitesimal bending of the circle C 2 on the cylinder.

Figure 4 .
Figure 4. Infinitesimal bending of the curve C 3 on the hyperbolic paraboloid.

Figure 5 .
Figure 5. Infinitesimal bending of the curve C 4 on the hyperbolic paraboloid.