Abstract. In this paper we give the Euler theorem and Dupin indicatrix for surfaces at a

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1 MATEMATIQKI VESNIK 65, 2 (2013), June 2013 originalni nauqni rad research paper THE EULER THEOREM AND DUPIN INDICATRIX FOR SURFACES AT A CONSTANT DISTANCE FROM EDGE OF REGRESSION ON A SURFACE IN E 3 1 Derya Sağlam and Özgür Boyacıoğlu Kalkan Abstract In this paper we give the Euler theorem and Dupin indicatrix for surfaces at a constant distance from edge of regression on a surface in E Introduction Let k 1, k 2 denote principal curvature functions and e 1, e 2 be principal directions of a surface M, respectively Then the normal curvature k n (v p ) of M in the direction v p = (cos θ)e 1 + (sin θ)e 2 is k n (v p ) = k 1 cos 2 θ + k 2 sin 2 θ This equation is called Euler s formulae (Leonhard Euler, ) The generalized Euler theorem for hypersurfaces in Euclidean space E n+1 can be found in [8] In 1984, A Kılıç and HH Hacısalihoğlu gave the Euler theorem and Dupin indicatrix for parallel hypersurfaces in E n [12] Also the Euler theorem and Dupin indicatrix are obtained for the parallel hypersurfaces in pseudo-euclidean spaces E1 n+1 and Eν n+1 in the papers [4, 6, 7] In 2005 HH Hacısalihoğlu and Ö Tarakçı introduced surfaces at a constant distance from edge of regression on a surface These surfaces are a generalization of parallel surfaces in E 3 Because the authors took any vector instead of normal vector [15] Euler theorem and Dupin indicatrix for these surfaces are given [2] In 2010 we obtained the surfaces at a constant distance from edge of regression on a surface in E1 3 [14] In this paper we give the Euler theorem and Dupin indicatrix for surfaces at a constant distance from edge of regression on a surface in E AMS Subject Classification: 51B20, 53B30 Keywords and phrases: Euler theorem; Dupin indicatrix; edge of regression 242

2 The Euler theorem and Dupin indicatrix 243 Definition 11 [3, 9, 10, 11, 13] (i) Hyperbolic angle: Let x and y be timelike vectors in the same timecone of Minkowski space Then there is a unique real number θ 0, called the hyperbolic angle between x and y, such that x, y = x y cosh θ (ii) Central angle: Let x and y be spacelike vectors in Minkowski space that span a timelike vector subspace Then there is a unique real number θ 0, called the central angle between x and y, such that x, y = x y cosh θ (iii) Spacelike angle: Let x and y be spacelike vectors in Minkowski space that span a spacelike vector subspace Then there is a unique real number θ between 0 and π called the spacelike angle between x and y, such that x, y = x y cos θ (iv) Lorentzian timelike angle: Let x be a spacelike vector and y be a timelike vector in Minkowski space Then there is a unique real number θ 0, called the Lorentzian timelike angle between x and y, such that x, y = x y sinh θ Definition 12 Let M and M f be two surfaces in E1 3 and N p be a unit normal vector of M at the point P M Let T p (M) be tanjant space at P M and {X p, Y p } be an orthonormal bases of T p (M) Let Z p = d 1 X p +d 2 Y p +d 3 N p be a unit vector, where d 1, d 2, d 3 R are constant numbers and ε 1 d ε 2 d 2 2 ε 1 ε 2 d 2 3 = ±1 If a function f exists and satisfies the condition f : M M f, f(p ) = P + rz p, r constant, M f is called as the surface at a constant distance from the edge of regression on M and M f denoted by the pair (M, M f ) If d 1 = d 2 = 0, then we have Z p = N p and f(p ) = P + rn p In this case M and M f are parallel surfaces [14] Theorem 13 [14] Let the pair (M, M f ) be given in E1 3 For any W χ(m), we have f (W ) = W + rd W Z, where W = 3 i=1 w i, W = 3 i=1 x w i and i x i P M, w i (P ) = w i (f(p)), 1 i 3 Let (φ, U) be a parametrization of M, so we can write that φ : U E1 3 M (u,v) P =φ(u,v) In this case {φ u p, φ v p } is a basis of T M (P ) Let N p is a unit normal vector at P M and d 1, d 2, d 3 R be a constant numbers then we may write that Z p = d 1 φ u p + d 2 φ v p + d 3 N p Since M f = {f(p ) f(p ) = P + rz p }, a parametric representation of M f is ψ(u, v) = φ(u, v) + rz(u, v) Thus we may write M f = {ψ(u, v) ψ(u, v) = φ(u, v) + r(d 1 φ u (u, v) + d 2 φ v (u, v) + d 3 N(u, v)), d 1, d 2, d 3, r are constant, ε 1 d ε 2 d 2 2 ε 1 ε 2 d 2 3 = ±1, }

3 244 D Sağlam, ÖB Kalkan If we take rd 1 = λ 1, rd 2 = λ 2, rd 3 = λ 3 then we have M f = {ψ(u, v) ψ(u, v) = φ(u, v) + λ 1 φ u (u, v) + λ 2 φ v (u, v) + λ 3 N(u, v), λ 1, λ 2, λ 3 are constant} Let {φ u, φ v } is basis of χ(m f ) If we take φ u, φ u = ε 1, φ v, φ v = ε 2 and N, N = ε 1 ε 2, then ψ u = (1 + λ 3 k 1 )φ u + ε 2 λ 1 k 1 N, ψ v = (1 + λ 3 k 2 )φ v + ε 1 λ 2 k 2 N is a basis of χ(m f ), where N is unit normal vector field on M and k 1, k 2 are principal of M [14] Theorem 14 [14] Let the pair (M, M f ) be given Let {φ u, φ v } (orthonormal and principal vector fields on M) be basis of χ(m) and k 1, k 2 be principal curvatures of M The matrix of the shape operator of M f with respect to the basis {ψ u = (1 + λ 3 k 1 )φ u + ε 2 λ 1 k 1 N, ψ v = (1 + λ 3 k 2 )φ v + ε 1 λ 2 k 2 N} of χ(m f ) is [ ] S f µ1 µ = 2 µ 3 µ 4 where µ 1 = (1 + λ 3k 2 ) A 3 { } k 1 ελ 1 u (λ2 2k2 2 ε 1 (1 + λ 3 k 2 ) 2 ) + k 1 A 2 µ 2 = ελ2 1λ 2 k 1 k 2 (1 + λ 3 k 2 ) k 1 A 3 u µ 3 = ελ 1λ 2 2k 1 k 2 (1 + λ 3 k 1 ) k 2 A 3 v µ 4 = (1 + λ { } 3k 1 ) k 2 A 3 ελ 2 v (λ2 1k1 2 ε 2 (1 + λ 3 k 1 ) 2 ) + k 2 A 2 and A = ε (ε 1 λ 2 1 k2 1 (1 + λ 3k 2 ) 2 + ε 2 λ 2 2 k2 2 (1 + λ 3k 1 ) 2 ε 1 ε 2 (1 + λ 3 k 1 ) 2 (1 + λ 3 k 2 ) 2 ) Definition 15 [6] Let M be a pseudo-euclidean surface in E 3 1 and p is nonumbilic point in M A function k n which is defined in the following form k n : T p M R, k n (X p ) = 1 X p 2 S(X p), X p is called a normal curvature function of M at p Definition 16 [7] Let M be a pseudo-euclidean surface in E1 3 shape operator of M Then the Dupin indicatrix of M at the point p is and S be D p = { X p S(X p ), X p = ±1, X p T p M }

4 The Euler theorem and Dupin indicatrix The Euler theorem for surfaces at a constant distance from edge of regression on a surface in E 3 1 Theorem 21 Let M f be a surface at a constant distance from edge of regression on a M in E1 3 Let k 1 and k 2 denote principal curvature function of M and let {φ u, φ v } be orthonormal basis such that φ u and φ v are principal directions on M Let Y p T p M and we denote the normal curvature by kn(f f (Y p )) of M f in the direction f (Y p ) Thus where k f n(f (Y p )) = y 1 = Y p, φ u, y 2 = Y p, φ v, µ 1y ε 1 ε 2 µ 2y 1 y 2 + µ 3y 2 2 λ 1 y2 1 2ε 1ε 2 λ 1 λ 2 k 1 k 2 y 1 y 2 + λ 2 y2 2 (21) λ i = ε i (1 + λ 3 k i ) 2 ε 1 ε 2 λ 2 i k 2 i, (i = 1, 2), µ 1 = ε 1 µ 1 (1 + λ 3 k 1 ) 2 λ 1 k 1 (ε 1 ε 2 µ 1 λ 1 k 1 + µ 2 λ 2 k 2 ), µ 2 = ε 2 µ 2 (1 + λ 3 k 2 ) 2 λ 2 k 2 (µ 1 λ 1 k 1 + ε 1 ε 2 µ 2 λ 2 k 2 ) + ε 1 µ 3 (1 + λ 3 k 1 ) 2 λ 1 k 1 (ε 1 ε 2 µ 3 λ 1 k 1 + µ 4 λ 2 k 2 ), µ 3 = ε 2 µ 4 (1 + λ 3 k 2 ) 2 λ 2 k 2 (µ 3 λ 1 k 1 + ε 1 ε 2 µ 4 λ 2 k 2 ) (22) Proof Let f (Y p ) T f(p) M f Then k f n(f (Y p )) = 1 f (Y p ) 2 S f (f (Y p )), f (Y p ) (23) Let us calculate f (Y p ) and S f (f (Y p )) Since φ u and φ v are orthonormal we have Y p = ε 1 Y p, φ u φ u + ε 2 Y p, φ v φ v = ε 1 y 1 φ u + ε 2 y 2 φ v Further without lost of generality, we suppose that Y p is a unit vector Then f (Y p ) = ε 1 y 1 f (φ u ) + ε 2 y 2 f (φ v ) = ε 1 y 1 ψ u + ε 2 y 2 ψ v (24) On the other hand we find that S f (f (Y p )) = ε 1 y 1 S f (ψ u ) + ε 2 y 2 S f (ψ v ) = ε 1 y 1 (µ 1 (1 + λ 3 k 1 )φ u + µ 2 (1 + λ 3 k 2 )φ v + (µ 1 ε 2 λ 1 k 1 + µ 2 ε 1 λ 2 k 2 )N) + ε 2 y 2 (µ 3 (1 + λ 3 k 1 )φ u + µ 4 (1 + λ 3 k 2 )φ v + (µ 3 ε 2 λ 1 k 1 + µ 4 ε 1 λ 2 k 2 )N) (25) Thus using equations (24) and (25) in equation (23) we obtain (21) Corollary 22 Let M f be a surface at a constant distance from edge of regression on M in E 3 1 Let k 1 and k 2 denote principal curvature function of M and let {φ u, φ v } be orthonormal basis such that φ u and φ v are principal directions on M Let us denote the angle between Y p T p M and φ u, φ v by θ 1 and θ 2 respectively Thus the normal curvature of M f in the direction f (Y p )

5 246 D Sağlam, ÖB Kalkan (a) Let N p be a timelike vector then k f n(f (Y p )) = (b) Let N p be a spacelike vector µ 1 cos 2 θ 1 + µ 2 cos θ 1 cos θ 2 + µ 3 cos 2 θ 2 λ 1 cos2 θ 1 + λ 2 cos2 θ 2 2λ 1 λ 2 k 1 k 2 cos θ 1 cos θ 2 (b1) If Y p and φ u are timelike vectors in the same timecone then kn(f f µ 1 cosh 2 θ 1 + δ 2 µ 2 cosh θ 1 sinh θ 2 + µ 3 sinh 2 θ 2 (Y p )) = λ 1 cosh2 θ 1 + λ 2 sinh2 θ 2 2δ 2 λ 1 λ 2 k 1 k 2 cosh θ 1 sinh θ 2 (b2) If Y p and φ v are timelike vectors in the same timecone then kn(f f µ 1 sinh 2 θ 1 + δ 1 µ 2 sinh θ 1 cosh θ 2 + µ 3 cosh 2 θ 2 (Y p )) = λ 1 sinh2 θ 1 + λ 2 cosh2 θ 2 2δ 1 λ 1 λ 2 k 1 k 2 sinh θ 1 cosh θ 2 (b3) If Y p T p M is a spacelike vector and φ u is timelike vector then kn(f f µ 1 sinh 2 θ 1 δ 1 δ 2 µ 2 sinh θ 1 cosh θ 2 + µ 3 cosh 2 θ 2 (Y p )) = λ 1 sinh 2 θ 1 + λ 2 cosh2 θ 2 + 2δ 1 δ 2 λ 1 λ 2 k 1 k 2 sinh θ 1 cosh θ 2 (b4) If Y p T p M is a spacelike vector and φ v is timelike vector then kn(f f µ 1 cosh 2 θ 1 δ 1 δ 2 µ 2 cosh θ 1 sinh θ 2 + µ 3 sinh 2 θ 2 (Y p )) = λ 1 cosh2 θ 1 + λ 2 sinh2 θ 2 + 2δ 1 δ 2 λ 1 λ 2 k 1 k 2 cosh θ 1 sinh θ 2 where λ 1, λ 2, µ 1, µ 2 and µ 3 are given in (22) and δ i, (i = 1, 2) is 1 or 1 depending on y i is positive or negative, respectively Proof (a) Let N p be a timelike vector In this case θ 1 and θ 2 are spacelike angle then y 1 = Y p, φ u = cos θ 1 y 2 = Y p, φ v = cos θ 2 Substituting these equations in (21), we get k f n(f (Y p )) (b) Let N p be a spacelike vector (b1) If Y p and φ u are timelike vectors in the same timecone then there is a hyperbolic angle θ 1 and a Lorentzian timelike angle θ 2 Since the proof is obvious y 1 = cosh θ 1 and y 2 = δ 2 sinh θ 2 (b2) If Y p and φ v are timelike vectors in the same timecone then there is a Lorentzian timelike angle θ 1 and a hyperbolic angle θ 2 Thus y 1 = δ 1 sinh θ 1 and y 2 = cosh θ 2

6 The Euler theorem and Dupin indicatrix 247 (b3) If Y p T p M is a spacelike vector and φ u is timelike vector then there is a Lorentzian timelike angle θ 1 and a central angle θ 2 Thus y 1 = δ 1 sinh θ 1 and y 2 = δ 2 cosh θ 2 (b4) If Y p T p M is a spacelike vector and φ v is timelike vector then there is a central angle θ 1 and a Lorentzian timelike angle θ 2 Thus y 1 = δ 1 cosh θ 1 and y 2 = δ 2 sinh θ 2 As a special case if we take λ 1 = λ 2 = 0, λ 3 = r = constant, then we obtain that M and M f are parallel surfaces The following corollary is known the Euler theorem for parallel surfaces in E 3 1 Corollary 23 Let M and M r be parallel surfaces in E 3 1 Let k 1 and k 2 denote principal curvature function of M and let {φ u, φ v } be orthonormal basis such that φ u and φ v are principal directions on M Let Y p T p M and we denote the normal curvature by k r n(f (Y p )) of M r, in the direction f (Y p ) Thus k r n(f (Y p )) = ε 1k 1 (1 + rk 1 )y ε 2 k 2 (1 + rk 2 )y 2 2 ε 1 (1 + rk 1 ) 2 y ε 2(1 + rk 2 ) 2 y 2 2 Proof Since λ i = ε i (1 + rk i ) 2, (i = 1, 2), µ 1 = ε 1 k 1 (1 + rk 1 ), µ 2 = 0, µ 3 = ε 2 k 2 (1 + rk 2 ), from (21) we find kn(f r (Y p )) Corollary 24 Let M and M r be parallel surfaces in E 3 1 Let k 1 and k 2 denote principal curvature function of M and let {φ u, φ v } be orthonormal basis such that φ u and φ v are principal directions on M Let us denote the angle between Y p T p M and φ u, φ v by θ 1 and θ 2 respectively Thus the normal curvature of M f in the direction f (Y p ) (a) Let N p be a timelike vector then k r n(f (Y p )) = k 1(1 + rk 1 ) cos 2 θ 1 + k 2 (1 + rk 2 ) cos 2 θ 2 (1 + rk 1 ) 2 cos 2 θ 1 + (1 + rk 2 ) 2 cos 2 θ 2 (b) Let N p be a spacelike vector (b1) If Y p and φ u are timelike vectors in the same timecone then k r n(f (Y p )) = k 1(1 + rk 1 ) cosh 2 θ 1 + k 2 (1 + rk 2 ) sinh 2 θ 2 (1 + rk 1 ) 2 cosh 2 θ 1 (1 + rk 2 ) 2 sinh 2 θ 2

7 248 D Sağlam, ÖB Kalkan (b2) If Y p and φ v are timelike vectors in the same timecone then k r n(f (Y p )) = k 1(1 + rk 1 ) sinh 2 θ 1 k 2 (1 + rk 2 ) cosh 2 θ 2 (1 + rk 1 ) 2 sinh 2 θ 1 + (1 + rk 2 ) 2 cosh 2 θ 2 (b3) If Y p T p M is a spacelike vector and φ u is timelike vector then k r n(f (Y p )) = k 1(1 + rk 1 ) sinh 2 θ 1 + k 2 (1 + rk 2 ) cosh 2 θ 2 (1 + rk 1 ) 2 sinh 2 θ 1 + (1 + rk 2 ) 2 cosh 2 θ 2 (b4) If Y p T p M is a spacelike vector and φ v is timelike vector then k r n(f (Y p )) = k 1(1 + rk 1 ) cosh 2 θ 1 k 2 (1 + rk 2 ) sinh 2 θ 2 (1 + rk 1 ) 2 cosh 2 θ 1 (1 + rk 2 ) 2 sinh 2 θ 2 3 The Dupin indicatrix for surfaces at a constant distance from edge of regression on surfaces in E 3 1 Theorem 31 Let M f be a surface at a constant distance from edge of regression on M in E1 3 Let k 1 and k 2 denote principal curvature functions of M and {φ u, φ v } be orthonormal bases such that φ u and φ v are principal directions on M Thus D f f(p) = { f (Y p ) T f(p) M f c 1 y1 2 + ε 1 ε 2 c 2 y 1 y 2 + c 3 y2 2 = ±1 }, where f (Y p ) = ε 1 y 1 (1 + λ 3 k 1 )φ u + ε 2 y 2 (1 + λ 3 k 2 )φ v + ε 1 ε 2 (y 1 λ 1 k 1 + y 2 λ 2 k 2 )N c 1 = ε 1 µ 1 (1 + λ 3 k 1 ) 2 λ 1 k 1 (ε 1 ε 2 µ 1 λ 1 k 1 + µ 2 λ 2 k 2 ), c 2 = ε 2 µ 2 (1 + λ 3 k 2 ) 2 λ 2 k 2 (µ 1 λ 1 k 1 + ε 1 ε 2 µ 2 λ 2 k 2 ) + ε 1 µ 3 (1 + λ 3 k 1 ) 2 λ 1 k 1 (ε 1 ε 2 µ 3 λ 1 k 1 + µ 4 λ 2 k 2 ), c 3 = ε 2 µ 4 (1 + λ 3 k 2 ) 2 λ 2 k 2 (µ 3 λ 1 k 1 + ε 1 ε 2 µ 4 λ 2 k 2 ) Proof Let f (Y p ) T f(p) M f Since D f f(p) = { f (Y p ) S f (f (Y p )), f (Y p ) = ±1 } the proof is clear According to this theorem the Dupin indicatrix of M f general will be a conic section of the following type: at the point f(p) in Corollary 32 Let M f be a surface at a constant distance from edge of regression on M in E 3 1 The Dupin indicatrix of M f at the point f(p) is: (a) an ellipse, if c 2 2 4c 1 c 3 < 0, (b) two conjugate hyperbolas, if c 2 2 4c 1 c 3 > 0, (c) parallel two lines, if c 2 2 4c 1 c 3 = 0

8 The Euler theorem and Dupin indicatrix 249 Corollary 33 Let M and M r be parallel surfaces in E 3 1 Let k 1 and k 2 denote principal curvature functions of M and {φ u, φ v } be orthonormal bases such that φ u and φ v are principal directions on M In this case D r f(p) = { f (Y p ) T f(p) M r ε 1 k 1 (1 + rk 1 )y ε 2 k 2 (1 + rk 2 )y 2 2 = ±1 } Hence the point f(p) of M r is: (a) an elliptic point, if ε 1 ε 2 k 1 k 2 (1 + rk 1 )(1 + rk 2 ) > 0, (b) a hyperbolic point, if ε 1 ε 2 k 1 k 2 (1 + rk 1 )(1 + rk 2 ) < 0, (c) a parabolic point, if k 1 k 2 (1 + rk 1 )(1 + rk 2 ) = 0 REFERENCES [1] N Aktan, A Görgülü, E Özüsağlam, C Ekici, Conjugate tangent vectors and asymptotic directions for surfaces at a constant distance from edge of regression on a surface, IJPAM 33 (2006), [2] N Aktan, E Özüsaglam, A Görgülü, The Euler theorem and Dupin indicatrix for surfaces at a constant distance from edge of regression on a surface, Int J Appl Math Stat 14 (2009), [3] M Bilici, M Çalışkan, On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space, Int Math Forum 4 (2009), [4] A C Çöken, The Euler theorem and Dupin indicatrix for parallel pseudo-euclidean hypersurfaces in pseudo-euclidean space in semi-euclidean space E n+1 ν, Hadronic J Suppl 16 (2001), [5] A C Çöken, Dupin indicatrix for pseudo-euclidean hypersurfaces in pseudo-euclidean space R n+1 v, Bull Cal Math Soc 89 (1997), [6] A Görgülü, A C Çöken, The Euler theorem for parallel pseudo-euclidean hypersurfaces in pseudo-euclidean space E n+1 1, J Inst Math Comp Sci (Math Ser) 6 (1993), [7] A Görgülü, A C Çöken, The Dupin indicatrix for parallel pseudo-euclidean hypersurfaces in pseudo-euclidean space in semi-euclidean space E n+1 1, Journ Inst Math Comp Sci (Math Ser) 7 (1994), [8] H H Hacısalihoğlu, Diferensiyel Geometri, Inönü Üniversitesi Fen Edeb Fak Yayınları, 1983 [9] M Kazaz, M Onder, Mannheim offsets of timelike ruled surfaces in Minkowski 3-space, arxiv: v3 [mathdg] [10] M Kazaz, H H Ugurlu, M Onder, M Kahraman, Mannheim partner D-curves in Minkowski 3-space E1 3, arxiv: v3 [mathdg] [11] M Kazaz, H H Ugurlu, M Onder, Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space, arxiv: v2 [mathdg] [12] A Kılıç, H H Hacısalihoğlu, Euler s Theorem and the Dupin representation for parallel hypersurfaces, J Sci Arts Gazi Univ 1 (1984), [13] O Neill, B, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, London, 1983 [14] D Sağlam, Ö Boyacıoğlu Kalkan, Surfaces at a constant distance from edge of regression on a surface in E1 3, Diff Geom Dyn Systems 12 (2010), [15] Ö Tarakcı, H H Hacısalihoğlu, Surfaces at a constant distance from edge of regression on a surface, Appl Math Comput 155 (2004), (received ; available online ) Afyonkarahisar Kocatepe University, Faculty of Art and Sciences, Department of Mathematics, Afyonkarahisar, TURKEY dryilmaz@akuedutr, bozgur@akuedutr