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A New Method for Finding the Shape Operator of a Hypersurface in Euclidean 4-Space

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A New Method for Finding the Shape Operator of a Hypersurface in Euclidean 4-Space Filomat 32:17 (2018), 5827–5836 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1817827U University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat A New Method for Finding the Shape Operator of a Hypersurface in Euclidean 4-Space Bahar Uyar D ¨uld¨ula aYıldız Technical University, Education Faculty, Department of Mathematics and Science Education, Istanbul,˙ Turkey Abstract. In this paper, by taking a Frenet curve lying on a parametric or implicit hypersurface and using the extended Darboux frame field of this curve, we give a new method for calculating the shape operator’s matrix of the hypersurface depending on the extended Darboux frame curvatures. This new method enables us to obtain the Gaussian and mean curvatures of the hypersurface depending on the geodesic torsions of the curve and the normal curvatures of the hypersurface. 1. Introduction In differential geometry, the shape of a geometric object (curve, surface, etc.) has always been a point of interest. In Euclidean space, along a space curve we have only one direction to move. This direction is determined by the tangent vector of the curve. The curvature of the curve measures the rate of change of its tangent vector, and together with the torsion of the curve they give us some information to its shape. However, along a surface we have different directions to move. These directions constitute a plane at a point, called the tangent plane of the surface at that point. When we move along a given direction, the tangent plane and thereby the normal vector of this plane, that is, the unit normal vector of the surface will change its direction except for some special cases. It is well-known that the rate of change of the unit normal vector of a surface has been measured by the shape operator. That’s why computing the shape operator of a surface is important in surface theory. The calculation of the shape operator of a surface in Euclidean 3-space is well-known [3, 4]. The shape operator’s matrix of a surface can be expressed by depending on the first and second fundamental form coefficients of the surface by using the Weingarten equations [3]. In this paper, we give a method which enables us to compute the shape operator of a hypersurface in Euclidean 4-space. We consider a Frenet curve lying on a hypersurface, and use the extended Darboux frame of the Frenet curve in Euclidean 4-space while constructing this new method. Our method includes two cases in which one can obtain directly the shape operator’s matrix of a hypersurface along a Frenet curve. The elements of obtained new matrix are composed of the geodesic torsions of first and second order of the curve and the normal curvatures of the hypersurface. 2010 Mathematics Subject Classification. Primary 53A07; Secondary 53A04, 53A55 Keywords. Shape operator; hypersurface; Darboux frame Received: 15 February 2018; Accepted: 20 November 2018 Communicated by Ljubica Velimirovic´ Email address: [email protected] (Bahar Uyar Duld¨ ul)¨ B. Uyar D¨uld¨ul / Filomat 32:17 (2018), 5827–5836 5828 2. Preliminaries Definition 2.1. Let be an orientable hypersurface with the unit normal vector field N. The shape operator M N 4 : χ( ) χ( ) of is defined by (X) = X for X χ( ), where is the Riemannian connection of E , [5].S M ! M M S −D 2 M D 4 Definition 2.2. Let be an orientable hypersurface in E and P be a point of . Then K(P) = det P and 1 M M S H(P) = trace( P) are called the Gaussian and the mean curvatures of at P, respectively, [5]. 3 S M 4 Definition 2.3. Let be an orientable hypersurface in E , P be a point of , and XP TP( ). The normal M M 2 M (XP); XP curvature at P of in the direction of XP is defined by κn(XP) = hS i,[3]. M XP; XP h i 4 4 P Definition 2.4. Let e1; e2; e3; e4 be the standard basis of R . The ternary product of the vectors u = uiei, f g i=1 P4 P4 v = viei, and w = wiei is defined by the vector [7] i=1 i=1 e1 e2 e3 e4 u1 u2 u3 u4 u v w = ; ⊗ ⊗ v1 v2 v3 v4 w1 w2 w3 w4 and satisfies the following properties: u v w x y (u v w) = u; y v; y w; y ; ⊗ ⊗ ⊗ ⊗ h i h i h i u; x v; x w; x h i h i h i x; u x; v x; w D E h i h i h i x y z; u v w = y; u y; v y; w : ⊗ ⊗ ⊗ ⊗ h i h i h i z; u z; v z; w h i h i h i 2.1. Curves on a parametric hypersurface in E4 Let be a regular hypersurface given by its parametric equation R = R(u ; u ; u ) and β : I R M 1 2 3 ⊂ !M be a Frenet curve with arc-length parametrization. Since is regular, R1 R2 R3 , 0 at every point of @R M ⊗ ⊗ , where Ri = . Then, the unit normal vector field of is given by M @ui M R R R N = 1 ⊗ 2 ⊗ 3 : (1) R R R jj 1 ⊗ 2 ⊗ 3jj Moreover, since the curve β(s) lies on , we may write β(s) = R u (s); u (s); u (s) . Then we have M 1 2 3 X3 β0(s) = Riui0; (2) i=1 X3 X3 β00(s) = Riui00 + Rijui0u0j; (3) i=1 i;j=1 X3 X3 X3 β000(s) = Riui000 + 3 Rijui00u0j + Rijkui0u0juk0 ; i=1 i;j=1 i;j;k=1 @2R @3R where Rij = , and Rijk = , 1 i; j; k 3. @uj@ui @uk@uj@ui ≤ ≤ B. Uyar D¨uld¨ul / Filomat 32:17 (2018), 5827–5836 5829 2.2. Curves on an implicit hypersurface in E4 Let be a regular hypersurface given by f (x; y; z; w) = 0, and β(s) = (x(s); y(s); z(s); w(s)) be a Frenet M f curve on . Since is regular, at every point of we have N(s) = r (s) for the unit normal vector field M M M f of along β. Besides, we have [1] jjr jj M f ; β0 = 0; hr i t f ; β00 = β0H f β0 ; (4) hr i − t d(H f ) t f ; β000 = 3β0H f β00 β0 β0 ; hr i − − ds where β0 = [x0 y0 z0 w0], β00 = [x00 y00 z00 w00], β000 = [x000 y000 z000 w000], f = [ fx fy fz fw], and r 2 3 6 fxx fxy fxz fxw 7 6 7 d H f h i 6 fyx fyy fyz fyw 7 @H f t @H f t H f = 6 7 ; = β0 β0 ; 6 fzx fzy fzz fzw 7 ds @x @w 46 57 ··· fwx fwy fwz fww 2 3 2 3 6 fxxx fxyx fxzx fxwx 7 6 fxxw fxyw fxzw fxww 7 @H 6 7 @H 6 7 f 6 fyxx fyyx fyzx fywx 7 f 6 fyxw fyyw fyzw fyww 7 = 6 7 ; ; = 6 7 : @x 6 fzxx fzyx fzzx fzwx 7 ··· @w 6 fzxw fzyw fzzw fzww 7 46 57 46 57 fwxx fwyx fwzx fwwx fwxw fwyw fwzw fwww 2.3. The extended Darboux frame(ED-frame) field in E4 Let E4 be an orientable hypersurface and β : I R be a Frenet curve in Euclidean 4-space E4. LetMT ⊂and N denote the unit tangent vector field of⊂ the!M curve β and the unit normal vector field of restricted to the curve β, respectively. Then the extended Darboux frame (ED-frame) field along β is denotedM by T; E; D; N [2], where f g β00 β00; N N E = − h i ; if N; T; β00 is linearly independent (Case 1); (5) β β ; N N f g jj 00 − h 00 i jj β000 β000; N N β000; T T E = − h i − h i ; if N; T; β00 is linearly dependent (Case 2) β β ; N N β ; T T f g jj 000 − h 000 i − h 000 i jj and D = N T E; in both cases: ⊗ ⊗ The differential equations for ED-frame field have the form 2 3 2 1 3 2 3 T0 6 0 κ1 0 κn 7 T 6 7 6 1 2 1 7 6 7 6 E 7 6 κ1 0 κ1 τ1 7 6 E 7 6 0 7 = 6 7 6 7 (Case 1), 6 D 7 6 − 2 2 7 6 D 7 6 0 7 6 0 κ1 0 τ1 7 6 7 4 5 46 − 1 2 57 4 5 N0 κn τ τ 0 N − − 1 − 1 2 3 2 3 2 3 T0 6 0 0 0 κn 7 T 6 7 6 2 1 7 6 7 6 E 7 6 0 0 κ1 τ1 7 6 E 7 6 0 7 = 6 7 6 7 (Case 2), 6 D 7 6 2 7 6 D 7 6 0 7 6 0 κ1 0 0 7 6 7 4 5 46 − 1 57 4 5 N0 κn τ 0 0 N − − 1 i i where κ1 and τ1 are the geodesic curvature and the geodesic torsion of order i,(i = 1; 2), respectively [2]. B. Uyar D¨uld¨ul / Filomat 32:17 (2018), 5827–5836 5830 3. Shape Operator’s Matrix of a Hypersurface Let us consider an orientable hypersurface in 4-dimensional Euclidean space and take a Frenet curve β(s) lying on this hypersurface.
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