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Shape Operator Via Darboux Frame Curvatures and Its Applications

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Shape Operator Via Darboux Frame Curvatures and Its Applications INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 12 NO. 2 PAGE 182–187 (2019) Shape Operator via Darboux Frame Curvatures and Its Applications Bahar Uyar Düldül∗ and Mustafa Düldül (Communicated by Kazım Ilarslan)˙ ABSTRACT We present the shape operator’s matrix of a surface along a surface curve. By using the obtained matrix, we give a short proof of the Beltrami-Enneper theorem. Also, we give a new method for determining the well-known geodesic curves of a plane, a sphere and a circular cylinder by using the relation between the ordinary curvatures of the geodesic and the curvatures of the surface. Keywords: Shape operator; Darboux frame; geodesic curve; curvature. AMS Subject Classification (2010): Primary: 53A04; Secondary: 53A05. 1. Introduction In Euclidean 3-space, the shape of a regular curve is measured by its curvature functions, called curvature and torsion. If the curve lies on a regular surface, it has also curvatures with respect to the surface itself, called normal curvature, geodesic curvature and geodesic torsion. These functions determine important properties for surface curves. When these curvatures vanish along the curve, such curves are called as asymptotic curve, geodesic curve, and line of curvature, respectively. Similar to curves, we can measure the shape of a surface. To do this, we need to determine the shape operator of the surface. We can compute the shape operator of a surface by using its definition or the Weingarten equations. Using Weingarten equations yield the shape operator depending on the first and second fundamental form coefficients of the surface. The purpose of this paper is to give a new method for determining the shape operator of a surface. Our new method is based on the Darboux frame of a surface curve. By using the Darboux frame curvatures of the curve, we obtain the shape operator of a surface along the curve depending on the normal curvature and geodesic torsion of the surface curve. We then give the mean, Gaussian and principal curvatures depending on these curvatures. As an application, we give a short proof of the Beltrami-Enneper theorem, and we obtain the relation between the ordinary curvatures of the surface curve and the curvatures of the surface. Also, we give a new and easy method for determining the well-known geodesic curves of a plane, sphere and circular cylinder. 2. Preliminaries Let us consider a unit speed curve β lying on a regular surface M in Euclidean 3-space E3. Along β, we denote the unit tangent vector of the curve with T , the unit normal vector of the surface with U, and V = U × T . In this case, we call the frame fT ; V; Ug as Darboux frame. This frame has its own Frenet formulas given by 8 0 T = κgV + κnU < 0 V = −κgT + τgU ; 0 : U = −κnT − τgV where κg is the geodesic curvature, κn is the normal curvature and τg is the geodesic torsion of β [2]. Received : 14-January-2019, Accepted : 04-July-2019 * Corresponding author B. Uyar Düldül, M. Düldül Definition 2.1. Let M be a regular surface oriented with the unit normal U. The shape operator S : TP (M) ! T (M) M P 2 M S(X ) = −DU X 2 T (M) T (M) P of at the point is defined by P XP for P P , where P denotes the tangent plane of the surface at P . 3 Definition 2.2. Let M be a regular surface in E , P 2 M be a point and SP denotes the shape operator of M 1 at P . The Gaussian and mean curvatures of M at P are defined by K(P ) = det SP and H(P ) = 2 trace(SP ), respectively. Also, the eigenvalues of SP are called the principal curvatures of M at P . 3 Definition 2.3. Let M be a regular surface in E , P 2 M be a point, and XP 2 TP (M) be a unit tangent vector. Then κn(XP ) = hS(XP ); XP i is called the normal curvature at P of M in the direction of XP . If κn(XP ) is constant 0 for all XP 2 TP (M), then P is called umbilic point on M. If κn(β ) = 0 along a regular curve β lying on M, then β is called an asymptotic curve on M. Lemma 2.1. Let M be a regular surface parametrized by X(u; v) and β(s) = X(u(s); v(s)) be a unit speed curve lying on M. Then, the normal curvature of the curve is obtained by [4] 0 2 0 0 0 2 κn = L(u ) + 2Mu v + N(v ) ; (2.1) and the geodesic torsion of the curve is obtained by [3] 1 0 2 0 0 0 2 τg = p (EM − FL)(u ) + (EN − GL)u v + (FN − GM)(v ) ; (2.2) EG − F2 where E; F; G and L; M; N denote the first and second fundamental form coefficients of the surface, respectively. Lemma 2.2. Let M be a regular surface given by f(x; y; z) = 0 and β(s) = (x(s); y(s); z(s)) be a unit speed curve lying on M. Then, the normal curvature of the curve is obtained by [4] f (x0)2 + f (y0)2 + f (z0)2 + 2(f x0y0 + f y0z0 + f x0z0 κ = − xx yy zz xy yz xz n p 2 2 2 (2.3) fx + fy + fz and the geodesic torsion of the curve is obtained by [3] 1 0 0 0 τg = f(a3fy − a2fz)x + (a1fz − a3fx)y + (a2fx − a1fy)z g ; (2.4) jjrfjj where n a = 1 f x0 + f x0 + f x0 − 1 f 2 f x0 + f x0 + f x0 i jjrfjj xixi i xixj j xixk k jjrfjj3 xi xixi i xixj j xixk k 0 0 0 0 0 0 o +fxi fxj fxj xi xi + fxj xj xj + fxj xk xk + fxi fxk fxkxi xi + fxkxj xj + fxkxk xk with x1 = x; x2 = y; x3 = z (i; j; k = 1; 2; 3 cyclic). 3. Shape operator’s matrix along a surface curve In this section, we consider a regular curve lying on an oriented surface in Euclidean 3-space and compute the shape operator’s matrix of the surface along the curve. The following two Lemmas will be needed in Proposition 3.1. Lemma 3.1. Let M be a regular surface parametrized by X(u; v); β(s) = X(u(s); v(s)) be a unit speed curve lying on M, and fT ; V; Ug denotes the Darboux frame field of β. Then, the normal curvature of M in the direction of V can be obtained by 1 0 2 0 0 0 2 κn(V) = λ1(u ) + 2λ2u v + λ3(v ) ; (3.1) EG − F2 where λ1 = F(FL − EM) + E(EN − FM); λ2 = F(GL − FM) + E(FN − GM); λ3 = G(GL − FM) + F(FN − GM) are computed along β. 183 www.iejgeo.com Shape Operator via Darboux Frame Curvatures 0 0 0 Xu × Xv Xu × Xv Proof. Since T = β (s) = Xuu + Xvv and U = = p , we have jjXu × Xvjj EG − F2 Xu × Xv 0 0 V = U × T = p × Xuu + Xvv EG − F2 1 0 0 0 0 = p − Fu − Gv Xu + Eu + Fv Xv: (3.2) EG − F2 Thus, we obtain the desired result by using κn(V) = hS(V); Vi. Lemma 3.2. Let M be a regular surface given by its implicit equation f(x1; x2; x3) = 0; β(s) = (β1(s); β2(s); β3(s)) be a unit speed curve lying on M, and fT ; V; Ug denotes the Darboux frame field of β. Then, the normal curvature of M in the direction of V can be obtained by 3 −1 X n 2 2 0 2 κn(V) = q fjjfk + fkkfj − 2fjkfjfk (βi) (3.3) 2 2 2 3 (f1 + f2 + f3 ) i=1 2 0 0 o +2 fjkfifk + fikfkfj − fifjfkk − fijfk βiβj ; where i; j; k = 1; 2; 3 (cyclic) and @f @2f fr = ; frs = ; r; s = 1; 2; 3 @xr @xs@xr are computed along β. 0 0 0 0 rf 1 Proof. Since T = β (s) = β (s); β (s); β (s) and U = = f1; f2; f3 , we have 1 2 3 jjrfjj jjrfjj 1 0 0 0 0 0 0 V = U × T = f2β − f3β ; f3β − f1β ; f1β − f2β : p 2 2 2 3 2 1 3 2 1 f1 + f2 + f3 The desired result follows from (2.3). Proposition 3.1. Let M be a regular surface, β be a unit speed curve lying on M, and fT ; V; Ug denotes the Darboux frame field of β. Then, the shape operator’s matrix of M along β with respect to the basis fT ; Vg is given by κ (T ) τ S = n g : (3.4) τg κn(V) Proof. We may write U 0 S(T ) = −∇T = −U = κn(T )T + τgV: (3.5) Let S(V) = aT + bV: Then, we obtain 0 a = hS(V); T i = hV;S(T )i = hV; −U i = τg; b = hS(V); Vi = κn(V); i.e. S(V) = τgT + κn(V)V: (3.6) Thus, the assertion is clear from (3.5) and (3.6). www.iejgeo.com 184 B. Uyar Düldül, M. Düldül 4. Applications of Main Result If we use Proposition 3.1, the following results can be given: Corollary 4.1. Let β be a unit speed curve lying on an oriented surface M, and fT ; V; Ug denotes the Darboux frame field of β. Then, the curvatures of the surface along β depending on the Darboux frame curvatures are given by 2 Gaussian curvature : K = κn(T )κn(V) − τg (4.1) 1 Mean curvature : H = κn(T ) + κn(V) (4.2) 2 r ! 2 1 2 Principal curvatures : k1;2 = κn(T ) + κn(V) ± κn(T ) − κn(V) + 4τ (4.3) 2 g Corollary 4.2.
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