The General Solutions of Frenet s System in the Equiform Geometry of the Galilean, Pseudo-Galilean, Simple Isotropic and Double Isotropic Space 1

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1 International Mathematical Forum, Vol. 6, 2011, no. 17, The General Solutions of Frenet s System in the Equiform Geometry of the Galilean, Pseudo-Galilean, Simple Isotropic and Double Isotropic Space 1 Zlatko Erjavec, Blaženka Divjak and Damir Horvat University of Zagreb, Faculty of Organization and Informatics Pavlinska 2, HR Varaždin, Croatia zlatko.erjavec@foi.hr blazenka.divjak@foi.hr damir.horvat1@foi.hr This paper is dedicated to Professor B. Pavković Abstract. In this paper several Cayley-Klein geometries and associated equiform geometries are described. Equiform geometry of Cayley-Klein space is obtained by requesting that similarity group of the space preserves angles between planes and lines, respectively. Further, invariants of curves in different equiform geometries are introduced. Then, the general solutions of Frenet s system in the equiform geometry of the Galilean, pseudo-galilean, simple isotropic and double isotropic space are given. The generalizations of given solutions to the sequences of functions are considered, as well. Mathematics Subject Classification: 53A35 Keywords: equiform geometry, Galilean space, pseudo-galilean space, simple isotropic space, double isotropic space, Frenet s system 1. Introduction The general solution of the Frenet s system of differential equations in the pseudo-galilean, Galilean, simple isotropic and double isotropic space are obtained in [3], [8], [9] and [10], respectively. The analogous general result in 1 This paper is partially supported by Croatian MSF project

2 838 Z. Erjavec, B. Divjak and D. Horvat Euclidean space is still unknown. This problem was considered in [6] where solutions in particular cases are obtained. The equiform geometry of Cayley-Klein space is defined by requesting that similarity group of the space preserves angles between planes and lines, respectively. The equiform geometries of aforementioned spaces are described in [5], [11] and [12]. Additional value of this paper is a fact that we form a repository of four Cayley-Klein geometries: the Galilean, pseudo-galilean, simple isotropic and double isotropic. Even these geometries have been investigated in more than hundred years, in the last two decades they have been put on the side of research. Recently, they have become interesting again since their importance for other fields, like soliton theory (see [13]), have been rediscovered. Therefore, we find it useful to make broader introduction to considered geometry at the beginning of each section. Then follows the main part of a section, introduction of associated equiform geometry and construction of general solution of Frenet s system in a given space. At the end of each section, the solution of the Frenet s system is generalized to sequences of functions, in the following way. At each point of a space curve two sequences of scalar functions are prescribed, such that the first functions these two sequences are the equiform curvature and the equiform torsion of the curve, respectively. Also, at each point of curve the sequence of the trihedrons (T i,n i,b i ) is associated, such that the first trihedron is the Frenet s trihedron of the curve and the following are defined by recursion formulas. For all trihedrons of considered sequence, the formulas analogue to Frenet s are valid. 2. The General Solution of Frenet s system in the Equiform Geometry of the Galilean Space 2.1. Introduction. The Galilean geometry is one of the real Cayley-Klein geometries. It has projective signature (0, 0, +, +) according to [7]. The absolute of the Galilean geometry is an ordered triple {ω, f, I} where ω is the ideal (absolute) plane, f a line in ω and I is the fixed eliptic involution of the points of f. The geometry of the Galilean space G 3 has been explained in details in the habilitation [14]. Particularly, the scalar product of two vectors a =(a 1,a 2,a 3 ) and b =(b 1,b 2,b 3 )ing 3 is defined by a b = { a 1 b 1, a 1 0orb 1 0, a 2 b 2 + a 3 b 3, a 1 = 0 and b 1 =0.

3 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I For an admissible curve c : I G 3, I R parameterized by the arc of length ds=dx, given by (2.1) c(x) = ( x, y(x),z(x) ), the curvature κ(x) and the torsion τ(x) are defined by (2.2) (2.3) κ(x) = y (x) 2 + z (x) 2, τ(x) = det ( c (x), c (x), c (x) ) and the associated trihedron is given by κ 2 (x) t(x) = ( 1,y (x),z (x) ), (2.4) n(x) = 1 ( 0,y (x),z (x) ), κ(x) b(x) = 1 ( 0, z (x),y (x) ). κ(x) For these vector fields the following Frenet s formulas hold (2.5) t = κ n, n = τ b, b = τ n. The general solution of the Frenet s system in Galilean space is given in [8] Equiform differential geometry of curves in the Galilean space. The equiform differential geometry of curves in the Galilean space G 3 has been studied in [11]. Let us recall some basic definitions from this work. The equiform curvature and the equiform torsion of an admissible curve are defined by (2.6) K = ρ, T = ρτ = τ κ, where ρ is the radius of curvature of the curve. The associated trihedron is given by (2.7) T = ρ t, N = ρ n, B = ρ b. The formulas analogous to the Frenet s in the equiform geometry of the Galilean space have the following form dt = K T + N, (2.8) dn = K N + T B, db = T N + K B, where σ is an equiform invariant parameter defined by σ = ds ρ.

4 840 Z. Erjavec, B. Divjak and D. Horvat 2.3. The general solution of Frenet s system in the equiform geometry of Galilean space. We shall now find all vector fields T, N, B and all functions K, T : I R assigned to a curve c such that the formulas analogous to Frenet s hold, i. e. (2.9) db = K T + N, = K N + T B, = T N + K B. The trihedron {T, N, B } is supposed to be orthogonal in the sense of the geometry of the space G 3. Clearly, the vector fields T, N, B can be written in the form (2.10) T = a 11 T + a 12 N + a 13 B, N = a 21 T + a 22 N + a 23 B, B = a 31 T + a 32 N + a 33 B, where a ij : I R, i,j =1, 2, 3 are unknown functions which should be determined in a such a way that for the vector fields (2.10) the Frenet s formulas (2.9) hold. If we derive the vector fields (2.10), using the Frenet formulas (2.8), we obtain (2.11) db = (a 11 + Ka 11)T +(a 11 + a 12 + Ka 12 Ta 13 )N +(T a 12 + a 13 + Ka 13)B, = (a 21 + Ka 21)T +(a 21 + a 22 + Ka 22 Ta 23 )N +(T a 22 + a 23 + Ka 23)B, = (a 31 + Ka 31)T +(a 31 + a 32 + Ka 32 Ta 33 )N +(T a 32 + a 33 + Ka 33)B. If we insert (2.10) in (2.9) it follows (2.12) db = (K a 11 + a 21 )T +(K a 12 + a 22 )N +(K a 13 + a 23 )B, = (K a 21 + T a 31 )T +(K a 22 + T a 32 )N +(K a 23 + T a 33 )B, = ( T a 21 + K a 31 )T +( T a 22 + K a 32 )N +( T a 23 + K a 33 )B.

5 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I By comparing (2.11) and (2.12) we find the following system of differential equations for the unknown functions a ij, K, T, (2.13) a 11 + Ka 11 = a 21 + K a 11, a 11 + a 12 + Ka 12 Ta 13 = a 22 + K a 12 a 13 + Ka 13 + T a 12 = a 23 + K a 13 a 21 + Ka 21 = K a 21 + T a 31, a 21 + a 22 + Ka 22 Ta 23 = K a 22 + T a 32, a 23 + Ka 23 + T a 22 = K a 23 + T a 33, a 31 + Ka 31 = K a 31 T a 21, a 31 + a 32 + Ka 32 Ta 33 = K a 32 T a 22, a 33 + Ka 33 + T a 32 = K a 33 T a 23, The system (2.13) is called the Frenet s system for curves in the equiform geometry of the Galilean space. The general solution of the Frenet s system for curves in the equiform geometry of the Galilean space G 3 is given by the following Theorem. Theorem 2.1. Let c : I G 3, I R be an admissible curve of class at least C 4 and K, T its equiform curvature and torsion, respectively. Further, let f,g : I R be differentiable functions of class at least C 2 such that 1+ f Tg 0. Then the general solution of the Frenet system for curves in the equiform geometry of the Galilean space has the form a 11 =1, a 12 = f, a 13 = g, a 21 =0, a 22 = cos ϕ, a 23 = sin ϕ, a 31 =0, a 32 = sin ϕ, a 33 = cos ϕ, (2.14) K = K, T = T + ϕ, where ϕ : I R is a differentiable function given by ϕ = Arctan g + T f 1+f Tg. Proof. Notice that the norms of the vector fields T, N, B are equal to ρ by definition. The vector fields t, n and b associated to the general solution of Frenet s formulas related to the group of similarity were orthonormal, therefore it was expected that the fields t, n and b are orthonormal, too. Here in the equiform geometry, we do not request orthonormality, but we want the vector fields T, N, B to be orthogonal in galilean sense and to have the same length as T, N and B. Since we want that T has the same length as T in G 3 it obviously has the form T = T + fn + gb, where f and g are functions of the equiform invariant parameter σ.

6 842 Z. Erjavec, B. Divjak and D. Horvat Similarly, since T, N, B have to be orthogonal in G 3, we have N = cos ϕ N + sin ϕ B, B = sin ϕ N + cos ϕ B, where ϕ is also a function of σ. Comparing these formulas with (2.10) we obtain a 11 =1, a 12 = f, a 13 = g, a 21 =0, a 22 = cos ϕ, a 23 = sin ϕ, a 31 =0, a 32 = sin ϕ, a 33 = cos ϕ, and if we put this in the first and the fifth of the equations (2.13) we get K = K, T = T + ϕ. From the second and the third equation of the system (2.13) it follows ϕ = Arctan g + T f 1+f Tg. It is easy to verify that the obtained solution (2.14) satisfies all equations of the Frenet s system (2.13). Hence the theorem is proved. We can generalize the Theorem 2.1 in the following way. Let c : I G 3, I R be an admissible curve of class at least C 4, K 1 = K(σ), T 1 = T (σ) its equiform curvature and equiform torsion respectively. We define sequences of functions K i, T i : I R recursively by the formulas K i+1 = K i, T i+1 = T i + ϕ i, where ϕ i = Arctan g i + T if i 1+f i T i =1, 2, 3... ig i and f i,g i : I R are arbitrary differentiable functions of class at least C 2. Further, let F i = {T i, N i, B i } be a sequence of orthogonal trihedrons in G 3 defined by Then the following theorem holds. T i+1 = T i + f i N i + g i B i, N i+1 = cos ϕ i N i + sin ϕ i B i, B i+1 = sin ϕ i N i + cos ϕ i B i. Theorem 2.2. For derivations of the vector fields of the trihedrons F i and the functions K i, T i the following Frenet s type formulas hold dt i = K i T i + N i, dn i = K i N i + T i B i db i = T i N i + K i B i.

7 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I Proof. The Theorem 2.2 can be proved by induction using the Theorem 2.1. By the formulas (2.7) and (2.8) we know that in the equiform geometry of the Galilean space, there is the trihedron {T, N, B} which is orthogonal and for which Frenet s formulas hold. If we denote the trihedron {T, N, B} by {T 1, N 1, B 1 }, then the proposition P (2) follows directly from the proof of the Theorem 2.1 replacing {T N, B } by {T 2, N 2, B 2 }. Further, we want to show that P (i) implies P (i +1). Let suppose that the trihedron {T i, N i, B i } is orthogonal and the Frenet s formulas hold. The orthogonality of the trihedron {T i+1, N i+1, B i+1 } follows directly from orthogonality of the trihedron {T i, N i, B i } by construction so we only have to prove that the Frenet s formulas are fulfilled. dt i+1 d = (T i + f i N i + g i B i )= = K i T i + N i + f i N i + f i (K i N i + T i B i )+g i B i + g i ( T i N i + K i B i )= = K i (T i + f i N i + g i B i ) + (1 + f i T i g i ) N i +(g i + T i f i ) B i = = K i (T i + f i N i + g i B i ) + (cos ϕ i N i + sin ϕ i B i )= = K i+1 T i+1 + N i+1 dn i+1 d = (cos ϕ i N i + sin ϕ i B i )= = sin ϕ i ϕ i N i + cos ϕ i (K i N i + T i B i ) + cos ϕ i ϕ i B i + sin ϕ i ( T i N i + K i B i )= = K i (cos ϕ i N i + sin ϕ i B i )+(T i + ϕ i ) ( sin ϕ i N i + cos ϕ i B i )= = K i+1 N i+1 + T i+1 B i+1 db i+1 d = ( sin ϕ i N i + cos ϕ i B i )= = cos ϕ i ϕ i N i sin ϕ i (K i N i + T i B i ) sin ϕ i ϕ i B i + cos ϕ i ( T i N i + K i B i )= = (T i + ϕ i ) (cos ϕ i N i + sin ϕ i B i )+K i ( sin ϕ i N i + cos ϕ i B i )= = T i+1 N i+1 + K i+1 B i+1 3. The General Solution of Frenet s system in the Equiform Geometry of the Pseudo-Galilean Space 3.1. Introduction. The pseudo-galilean geometry is one of the real Cayley- Klein geometries. It has projective signature (0, 0, +, ) according to [7]. The absolute of the pseudo-galilean geometry is an ordered triple {ω, f, I} where ω is the ideal (absolute) plane, f a line in ω and I is the fixed hyperbolic involution of the points of f. Note that the construction of the Galilean geometry was similarly, but here is used a hyperbolic involution instead of elliptic. The geometry of the pseudo-galilean space G 1 3 has been explained in details in the dissertation [2]. Particularly, the scalar product of two vectors a =

8 844 Z. Erjavec, B. Divjak and D. Horvat (a 1,a 2,a 3 ) and b =(b 1,b 2,b 3 )ing 1 3 is defined by { a 1 b 1, a 1 0orb 1 0, a b = a 2 b 2 a 3 b 3, a 1 = 0 and b 1 =0. For an admissible curve c : I G 1 3, I R parameterized by the arc of length ds=dx, given by (3.1) c(x) = ( x, y(x),z(x) ), the curvature κ(x) and the torsion τ(x) are defined by (3.2) κ(x) = y (x) 2 z (x) 2, τ(x) = det ( c (x), c (x), c (x) ) (3.3) κ 2 (x) respectively and the associated trihedron is given by t = ( 1,y (x),z (x) ), (3.4) n = 1 ( 0,y (x),z (x) ), κ(x) b = 1 ( 0,εz (x),εy (x) ), κ(x) where ε =+1or 1, chosen by criterion det(t, n, b) =1, i.e. it means y 2 (x) z 2 (x) = ε ( y 2 (x) z 2 (x) ). For these vector fields the following Frenet s formulas hold (3.5) t = κ n, n = τ b, b = τ n. The general solution of the Frenet s system in the pseudo-galilean space G 1 3 is given in [3] and details of the theory of curves in [4]. Let us mention that uniqueness in the fundamental theorem for theory of curves in G 1 3 does not hold, contrary to the situation in the Galilean geometry where the uniqueness is fulfilled Equiform differential geometry of curves in the pseudo-galilean space. The equiform differential geometry of pseudo-galilean space G 1 3 has been studied in [5]. Let us recall some basic definitions from this work. The equiform curvature and the equiform torsion of an admissible curve are defined by (3.6) K = ρ, T = ρτ = τ κ, where ρ is the radius of curvature of the curve c. The associated trihedron is given by (3.7) T = ρ t, N = ρ n, B = ρ b.

9 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I The formulas analogous to the Frenet s in the equiform geometry of the pseudo-galilean space have the following form dt (3.8) dn db = K T + N, = K N + T B, = T N + K B, where σ is an equiform invariant parameter defined by σ = ds ρ The general solution of Frenet s system in the equiform geometry of the pseudo-galilean space. We shall now find all vector fields T, N, B and all functions K, T : I R assigned to a curve c such that the formulas analogous to Frenet s hold, i. e. = K T + N, (3.9) = K N + T B, db = T N + K B. The trihedron {T, N, B } is supposed to be orthogonal in the sense of the geometry of the space G 1 3. The vector fields T, N, B can be written in the form (3.10) T = a 11 T + a 12 N + a 13 B, N = a 21 T + a 22 N + a 23 B, B = a 31 T + a 32 N + a 33 B. where a ij : I R, i,j = 1, 2, 3 are unknown functions which should be determined in a such a way that for the vector fields (3.10) the Frenet s formulas (3.9) hold. If we derive the vector fields (3.10), using the Frenet s formulas (3.8), we obtain (3.11) db = (a 11 + Ka 11 )T +(a 11 + a 12 + Ka 12 + T a 13 )N +(T a 12 + a 13 + Ka 13 )B, = (a 21 + Ka 21)T +(a 21 + a 22 + Ka 22 + T a 23 )N +(T a 22 + a 23 + Ka 23)B, = (a 31 + Ka 31)T +(a 31 + a 32 + Ka 32 + T a 33 )N +(T a 32 + a 33 + Ka 33)B.

10 846 Z. Erjavec, B. Divjak and D. Horvat (3.12) If we insert (3.10) in (3.9), it follows db = (K a 11 + a 21 )T +(K a 12 + a 22 )N +(K a 13 + a 23 )B, = (K a 21 + T a 31 )T +(K a 22 + T a 32 )N +(K a 23 + T a 33 )B, = (T a 21 + K a 31 )T +(T a 22 + K a 32 )N +(T a 23 + K a 33 )B. By comparing (3.11) and (3.12) we find the following system of differential equations for the unknown functions a ij, K, T (3.13) a 11 + Ka 11 = a 21 + K a 11, a 11 + a 12 + Ka 12 + T a 13 = a 22 + K a 12 a 13 + Ka 13 + T a 12 = a 23 + K a 13 a 21 + Ka 21 = K a 21 + T a 31, a 21 + a 22 + Ka 22 + T a 23 = K a 22 + T a 32, a 23 + Ka 23 + T a 22 = K a 23 + T a 33, a 31 + Ka 31 = K a 31 + T a 21, a 31 + a 32 + Ka 32 + T a 33 = K a 32 + T a 22, a 33 + Ka 33 + T a 32 = K a 33 + T a 23. The system (3.13) is called the Frenet s system for curves in the equiform geometry of the pseudo-galilean space G 1 3. The general solution of the Frenet s system for curves in the equiform geometry of the pseudo-galilean space G 1 3 is given by the following theorem. Theorem 3.1. Let c : I G 1 3, I R be an admissible curve of class at least C 4, K, T its equiform curvature and torsion respectively, and f,g : I R differentiable functions of class at least C 2 such that 1+f + T g 0. Then the general solution of the Frenet s system for curves in the equiform geometry of the pseudo-galilean space G 1 3 has the form a 11 =1, a 12 = f, a 13 = g, a 21 =0, a 22 = cosh ϕ, a 23 = sinh ϕ, a 31 =0, a 32 = sinh ϕ, a 33 = cosh ϕ, (3.14) K = K, T = T + ϕ, where ϕ : I R is a differentiable function given by ϕ = Arth g + T f 1+f + T g. Proof. Due to the same reason as in the Galilean space, we want T has the same length as T and therefore it has the form where f and g are functions of σ. T = T + fn + gb,

11 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I Similarly, since T, N, B have to be orthogonal in G 1 3, we have N = cosh ϕn + sinh ϕb, B = sinh ϕn + cosh ϕb, where ϕ is also a function of σ. Comparing these formulas with (3.10) we obtain a 11 =1, a 12 = f, a 13 = g, a 21 =0, a 22 = cosh ϕ, a 23 = sinh ϕ, a 31 =0, a 32 = sinh ϕ, a 33 = cosh ϕ, and if we put these in the first and the fifth equation of (3.13) we get K = K, T = T + ϕ. From the second and the third equation of the system (3.13) it follows ϕ = Arth g + T f 1+f + T g. It is easy to verify that (3.14) satisfies all equations of the Frenet s system. Hence the theorem is proved. We can generalize the Theorem 3.1 in the following way. Let c : I G 1 3, I R be an admissible curve of class at least C 4, K 1 = K(σ), T 1 = T (σ) its equiform curvature and equiform torsion respectively. We define sequences of functions K i, T i : I R recursively by the formulas K i+1 = K i, T i+1 = T i + ϕ i, where ϕ i = Arth g i + T if i 1+f i + T, i =1, 2, 3... ig i and f i,g i : I R are arbitrary differentiable functions of class at least C 2. Furthermore, let F i = {T i, N i, B i } be a sequence of orthogonal trihedrons in G 1 3 defined by T i+1 = T i + f i N i + g i B i, N i+1 = cosh ϕ i N i + sinh ϕ i B i, B i+1 = sinh ϕ i N i + cosh ϕ i B i, Then the following theorem is true. Theorem 3.2. For derivations of the vector fields of the trihedra F i and the functions K i, T i the following Frenet s type formulas hold dt i = K i T i + N i, dn i = K i N i + T i B i db i = T i N i + K i B i.

12 848 Z. Erjavec, B. Divjak and D. Horvat Proof. The Theorem 3.2 can be proved by induction by using the Theorem 3.1, in analogous way to the proof of the Theorem The General Solution of Frenet s system in the Equiform Geometry of the Simple Isotropic Space 4.1. Introduction. The simple isotropic geometry is one of the real Cayley- Klein geometries. The absolute of the simple isotropic geometry is an ordered triple {ω, f 1,f 2 } where ω is the ideal (absolute) plane and f 1, f 2 couple of complex conjugate lines in ω. The geometry of simple isotropic space I (1) 3 has been explained in details in [15]. Particularly, the scalar product of two vectors a =(a 1,a 2,a 3 ) and b =(b 1,b 2,b 3 )ini (1) 3 is defined by a b = { a 1 b 1 + a 2 b 2, a i 0orb i 0, (i =1, 2), a 3 b 3, a i = b i =0, (i =1, 2). For an admissible curve c : I I (1) 3, I R parameterized by the arc of length, given by (4.1) c(s) = ( x(s),y(s),z(s) ), c(s) = ( x(s),y(s) ) the curvature κ(s) and the torsion τ(s) are defined by (4.2) κ(s) = det ( c (s), c (s) ), τ(s) = det ( c (s), c (s), c (s) ) (4.3), κ 2 (s) respectively and the associated trihedron is given by (4.4) t = ( x (s),y (s),z (s) ), n = 1 ( x (s),y (s),z (s) ), κ(s) b = (0, 0, 1). For these vector fields the following Frenet s formulas hold (4.5) t = κ n, n = κ t + τ b, b = 0. The general solution of the Frenet s system in the simple isotropic space is given in [9] Equiform differential geometry of the simple isotropic space. The equiform differential geometry of the simple isotropic space I (1) 3 has been studied in [12]. Let us recall some basic definitions from this work.

13 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I The equiform curvature and the equiform torsion of an admissible curve is defined by (4.6) K = ρ, T = ρτ = τ κ, where ρ is the radius of curvature of the curve r. The associated trihedron is given by (4.7) T = ρ t, N = ρ n, B = ρ b. The formulas analogous to the Frenet s in the equiform geometry of the simple isotropic space have the following form dt = K T + N, (4.8) dn = T + K N + T B, db = K B, where σ is an equiform invariant parameter defined by σ = ds ρ 4.3. The general solution of Frenet s system in the equiform geometry of the simple isotropic space. We shall now find all vector fields T, N, B and all functions K, T : I R assigned to a curve c such that the formulas analogous to Frenet s hold, i. e. = K T + N, (4.9) = T + K N + T B, db = K B. The trihedron {T, N, B } is supposed to be orthogonal in the sense of the geometry of the space I3 1. The vector fields T, N, B can be written in the form (4.10) T = a 11 T + a 12 N + a 13 B, N = a 21 T + a 22 N + a 23 B, B = B, where a ij : I R, i=1, 2, j=1, 2, 3 are unknown functions which should be determined in such a way that for the vector fields (4.10) the Frenet s formulas (4.9) hold. If we derive the vector fields (4.10), using the Frenet s formulas (4.8), we obtain

14 850 Z. Erjavec, B. Divjak and D. Horvat (4.11) = (a 11 + Ka 11 a 12 )T +(a 11 + a 12 + Ka 12 )N +(T a 12 + a 13 + Ka 13 )B, = (a 21 + Ka 21 a 22 )T +(a 21 + a 22 + Ka 22)N +(T a 22 + a 23 + Ka 23)B. (4.12) If we insert (4.10) in (4.9) it follows = (K a 11 + a 21 )T +(K a 12 + a 22 )N +(K a 13 + a 23 )B, = ( a 11 + K a 21 )T +( a 12 + K a 22 )N +( a 13 + K a 23 + T )B. By comparing (4.11) and (4.12) we find for the unknown functions a ij, K, T, the following system of differential equations a 11 + Ka 11 a 12 = a 21 + K a 11, a 11 + a 12 + Ka 12 = a 22 + K a 12 (4.13) a 13 + Ka 13 + T a 12 = a 23 + K a 13 a 21 + Ka 21 a 22 = a 11 + K a 21, a 21 + a 22 + Ka 22 = a 12 + K a 22, a 23 + Ka 23 + T a 22 = a 13 + K a 23 + T. The system (4.13) is called the Frenet s system for curves in the equiform geometry of the simple isotropic space. The general solution of the Frenet system for curves in the equiform geometry of the simple isotropic space I (1) 3 is given by the following theorem. Theorem 4.1. Let c : I I (1) 3, I R be an admissible curve of class at least C 4, functions K, T its equiform curvature and torsion respectively, and f,g : I R differentiable functions of class at least C 2. Then the general solution of the Frenet s system for curves in the equiform geometry of the simple isotropic space I (1) 3 has the form a 11 = α, a 12 = β, a 13 = f, a 21 = β, a 22 = α, a 23 = g, (4.14) K = K, T = f + g + αt, where for α, β R hold α 2 + β 2 =1, and g f = β T. Proof. Since T, N, B have to be orthogonal in I (1) 3 and they have the same length as T, N, B, it follows T = cos ϕ T + sin ϕ N + f B, N = sin ϕ T + cos ϕ N + g B, B = B. Comparing these formulas with (4.10) we obtain a 11 = cos ϕ, a 12 = sin ϕ, a 13 = f, a 21 = sin ϕ, a 22 = cos ϕ, a 23 = g.

15 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I Further, if we put these coefficients in the first equation of (4.13), knowing that because B = B hold K = K, we get ϕ = const. If we introduce notation α = cos ϕ and β = sin ϕ, then from the third and the sixth equation of the system (4.13) it follows T = f + g + α T, β = g f. T It is easy to verify that (4.14) satisfies all equations of the Frenet s system. Hence the theorem is proved. We can generalize the Theorem 4.1 in the following way. Let c : I I (1) 3, I R be an admissible curve of class at least C 4, functions K 1 = K(σ), T 1 = T (σ) its equiform curvature and equiform torsion respectively. We define sequences of functions K i, T i : I R recursively by the formulas where K i+1 = K i, T i+1 = α i T i + f i + g i, ( ) gi f 2 i α i = 1 R i =1, 2, 3... T i and f i,g i : I R are differentiable functions of class at least C 2 for which hold g i f i = β i T i. Further, let F i = {T i, N i, B i } be a sequence of orthogonal trihedrons in I (1) 3 defined by T i+1 = α i T i + β i N i + f i B i, N i+1 = β i T i + α i N i + g i B i, B i+1 = B i. Then the following theorem is true. Theorem 4.2. For derivations of the vector fields of the trihedra F i and the functions K i, T i the following Frenet s type formulas hold dt i = K dn i i T i + N i, = T db i i + K i N i + T i B i = K i B i. Proof. The Theorem 4.2 can be proved by induction by using the Theorem 4.1, in analogous way to the proof of the Theorem The General Solutions of Frenet s systems in the Equiform Geometry of the Double Isotropic Space 5.1. Introduction. The double isotropic geometry is one of the real Cayley- Klein geometries. The absolute of the double isotropic geometry is an ordered

16 852 Z. Erjavec, B. Divjak and D. Horvat triple {ω, f, U} where ω is the ideal (absolute) plane, f ω a (absolute) line and U f a point (the absolute point). The geometry of the double isotropic space I (2) 3 has been explained in [1]. Particularly, the scalar product of two vectors a =(a 1,a 2,a 3 ) and b =(b 1,b 2,b 3 ) in I (2) 3 is defined by a 1 b 1, a 1 0orb 1 0 a b = a 2 b 2, a 1 = b 1 = 0 and (a 2 0orb 2 0) a 3 b 3, a 1 = b 1 = a 2 = b 2 =0 For an admissible curve c : I I (2) 3, I R parameterized by the arc of length ds=dx, given by (5.1) c(x) = ( x, y(x),z(x) ), the curvature κ(x) and the torsion τ(x) are defined by (5.2) κ(x) = y (x), ( ) z (x) (5.3) τ(x) = y (x) respectively. Further, the associated trihedron is given by t = ( 1,y (x),z (x) ) ) (5.4), n = (0, 1, z y, b =(0, 0, 1). For these vector fields the following Frenet s formulas hold (5.5) t = κ n, n = τ b, b = 0. The general solution of the Frenet s system in the double isotropic space is given in [10] Equiform differential geometry of the double isotropic space. The equiform differential geometry of the double isotropic space I (2) 3 has been studied in [12]. Let us recall some basic definitions from this work. The equiform curvature and the equiform torsion of an admissible curve are defined by (5.6) K = ρ, T = ρτ = τ κ, where ρ is the radius of curvature of the curve c. The associated trihedron is given by (5.7) T = ρ t, N = ρ n, B = ρ b.

17 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I The formulas analogous to the Frenet s in the equiform geometry of the double isotropic space have the following form dt = K T + N, (5.8) dn = K N + T B, db = K B, where σ is an equiform invariant parameter defined by σ = ds ρ 5.3. The general solution of Frenet s system in the equiform geometry of the double isotropic space. We shall now find all vector fields T, N, B and all functions K, T : I R assigned to a curve c such that the formulas analogous to Frenet s hold, i. e. = K T + N, (5.9) = K N + T B, db = K B. The trihedron {T, N, B } is supposed to be orthogonal in the sense of the geometry of the space I (2) 3. The vector fields T, N, B can be written in the form (5.10) T = a 11 T + a 12 N + a 13 B, N = a 21 T + a 22 N + a 23 B, B = B, where a ij : I R, i =1, 2,j =1, 2, 3 are unknown functions that should be determined in a such a way that for the vector fields (5.10) the Frenet s formulas (5.9) hold. If we derive the vector fields (5.10), using the Frenet s formulas (5.8), we obtain (5.11) = (a 11 + Ka 11)T +(a 11 + a 12 + Ka 12)N +(T a 12 + a 13 + Ka 13)B, = (a 21 + Ka 21 )T +(a 21 + a 22 + Ka 22 )N +(T a 22 + a 23 + Ka 23 )B. If we insert (5.10) in (5.9) it follows (5.12) = (K a 11 + a 21 )T +(K a 12 + a 22 )N +(K a 13 + a 23 )B, = (K a 21 )T +(K a 22 )N +(K a 23 + T )B.

18 854 Z. Erjavec, B. Divjak and D. Horvat By comparing (5.11) and (5.12) we find the following system of differential equations for the unknown functions a ij, K, T, (5.13) a 11 + Ka 11 = a 21 + K a 11, a 11 + a 12 + Ka 12 = a 22 + K a 12, a 13 + Ka 13 + T a 12 = a 23 + K a 13, a 21 + Ka 21 = K a 21, a 21 + a 22 + Ka 22 = K a 22, a 23 + Ka 23 + T a 22 = K a 23 + T. The system (5.13) is called the Frenet s system for curves in the equiform geometry of the double isotropic space. The general solution of the Frenet s system for curves in the equiform geometry of the double isotropic space I (2) 3 is given by the following theorem. Theorem 5.1. Let c : I I (2) 3, I R be an admissible curve of class at least C 4, functions K, T its equiform curvature and torsion respectively, f : I R differentiable functions of class at least C 2 and α R. Then the general solution of the Frenet s system for curves in the equiform geometry of the double isotropic space I (2) 3 has the form a 11 =1, a 12 = α, a 13 = f, a 21 =0, a 22 =1, a 23 = α T + f, (5.14) K = K, T = T +(α T + f ). Proof. Since T has the same length as T then it has the form T = T + g N + f B, where g and f are functions of σ. Further, since T, N, B have to be orthogonal in I (2) 3, it follows N = N + h B, B = B, where h is also function of σ. Comparing these formulas with (5.10) we obtain a 11 =1, a 12 = g, a 13 = f, a 21 =0, a 22 =1, a 23 = h, and if we put these in the first and the second equation of (5.13) we get K = K, g = const. = α. From the third and the sixth equation of the system (5.13) it follows h = α T + f, T = T +(α T + f ). It is easy to verify that (5.14) satisfies all equations of the Frenet s system. Hence the theorem is proved.

19 Frenet s system solutions in equiform geometry of G 3, G 1 3, I1 3, I We can generalize the Theorem 5.1 in the following way. Let c : I I (2) 3, I R be an admissible curve of class at least C 4, K 1 = K(σ), T 1 = T (σ) its equiform curvature and equiform torsion respectively. We define sequences of functions K i, T i : I R recursively by the formulas K i+1 = K i, T i+1 = T i +(α i T i + f i ) i =1, 2, 3..., where f i : I R are arbitrary differentiable functions of class at least C 2 and α i R. Furthermore, let F i = {T i, N i, B i } be a sequence of orthogonal trihedrons in I (2) 3 defined by T i+1 = T i + α i N i + f i B i, N i+1 = N i +(α i T + f ) B i, B i+1 = B i. Now the following theorem is true. Theorem 5.2. For derivations of the vector fields of the trihedrons F i and the functions K i, T i the following Frenet s type formulas hold dt i = K i T i + N i, dn i = K i N i + T i B i, db i = K i B i. Proof. The Theorem 5.2 can be proved by induction by using the Theorem 5.1, in analogous way to the proof of Theorem 2.2. References [1] H. Brauner: Geometrie des Zweifach Isotropen Raumes I, J. Reine Angew. Math. 224 (1966), [2] B. Divjak: Geometrija pseudogalilejevih prostora, Ph. D. Thesis, University of Zagreb, [3] B. Divjak: The General Solution of the Frenet s System of Differential Equations for Curves in the Pseudo-Galilean Space G 1 3, Mathematical Communications Vol 2 (1997), [4] B. Divjak: Curves in Pseudo-Galilean Geometry, Annales Univ. Sci. Budapest 41 (1998), [5] Z. Erjavec, B. Divjak: Equiform Differential Geometry of Curves in the Pseudo- Galilean Space, Mathematical Communications Vol 13 (2008), [6] Z. Kurnik, V. Volenec: Über die Begleitenden Dreibeine der Raumkurve, Glasnik Mat. Vol 6 (26) (1971), [7] E. Molnar: The Projective Interpretation of the Eight 3-dimensional Homogeneous Geometries, Beiträge Algebra Geom. 38 (1997), [8] B. J. Pavković: The General Solution of the Frenet System of Differential Equations for Curves in the Galilean Space G 3, Rad JAZU 450 (1990), [9] B. J. Pavković: Allgemeine Lösung des Frenetschen System von Differentialgleichungen im isotropen und pseudoisotropen dreidimensionalen Raum, Glasnik Mat. Vol 10 (30) (1975),

20 856 Z. Erjavec, B. Divjak and D. Horvat [10] B. J. Pavković, I. Kamenarović: The General Solution of the Frenet s System in the Doubly Isotropic Space I (2) 3, Rad JAZU 428 (1987), [11] B. J. Pavković, I. Kamenarović: The Equiform Differential Geometry of Curves in the Galilean Space G 3, Glasnik Mat. Vol 22 (42) (1987), [12] B. J. Pavković: Equiform Geometry of Curves in the Isotropic Spaces I (1) 3 and I (2) 3, Rad JAZU (1986), [13] C. Rogers, W. K. Schief: Backlund and Darboux Transformations: Geometry and Modern applications in Soliton Theory, Cambridge University Press, [14] O. Röschel: Die Geometrie des Galileischen Raumes, Habilitationsschrift, Institut für Math. und Angew. Geometrie, Leoben, [15] H. Sachs: Isotrope Geometrie des Raumes, Vieweg, Braunschweig/ Wiesbaden, Received: October, 2010