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The Principal Curvatures and the Third Fundamental Form of Dini Type

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The Principal Curvatures and the Third Fundamental Form of Dini Type THE PRINCIPAL CURVATURES AND THE THIRD FUNDAMENTAL FORM OF DINI-TYPE HELICOIDAL HYPERSURFACE IN 4-SPACE ERHAN GÜLER Abstract. We consider the principal curvatures and the third fundamental form of Dini-type 4 helicoidal hypersurface D(u; v; w) in the four dimensional Euclidean space E . We …nd the 4 Gauss map e of helicoidal hypersurface in E . We obtain characteristic polynomial of shape operator matrix S. Then, we compute principal curvatures ki=1;2;3; and the third fundamental form matrix III of D. 1. Introduction Theory of surfaces and hypersurfaces have been studied by many geometers for years such as [1 26]. In the rest of this paper, we identify a vector (a,b,c,d) with its transpose (a,b,c,d)t: Let : I be a curve in a plane in E4, and let ` be a straight line in for an open interval ! I R. A rotational hypersurface in E4 is de…ned as a hypersurface rotating a curve (i.e. pro…le curve) around a line (i.e. axis) `. Suppose that when a pro…le curve rotates around the axis `, it simultaneously displaces parallel lines orthogonal to the axis `, so that the speed of displacement is proportional to the speed of rotation. Resulting hypersurface is called the helicoidal hypersurface with axis ` and pitches a; b R 0 . Let ` be a line spanned by the vector (0; 0; 0; 1)t.2 Thenf orthogonalg matrix cos v cos w sin v cos v sin w 0 sin v cos w cos v sin v sin w 0 M(v; w) = 0 1 ; v; w R, sin w 0 cos w 0 2 B 0 0 0 1 C B C …xes the vector `. The matrix@ M can be found by solving the followingA equations simultaneously; t t M:` = `; M :M = M:M = I4; det M = 1: When the axis of rotation is `, there is an Euclidean 4 transformation by which the axis is ` transformed to the x4-axis of E . Parametrization of the pro…le curve is given by (u) = (u; 0; 0;' (u)) ; where ' (u): I R R is a di¤erentiable func- tion for all u I. So, the helicoidal hypersurface is given by H(u;! v; w) = M: t + (av + bw) :`t, 2 where u I; v; w [0; 2] ; a; b R 0 : Clearly, we write helicoidal hypersurface as follows 2 2 2 nf g H(u; v; w) = (u cos v cos w; u sin v cos w; u sin w; '(u) + av + bw) : Date: December 9, 2020. 2000 Mathematics Subject Classi…cation. Primary 53A35; Secondary 53C42. Key words and phrases. four dimensional, Dini-type helicoidal hypersurface, Gauss map, principal curvatures, the third fundamental form. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1 2 E. GÜLER In this paper, we study the principal curvatures and the third fundamental form of the Ulisse Dini-type helicoidal hypersurface in Euclidean 4-space E4. We give some basic notions of four dimensional Euclidean geometry in section 2. In section 3, we give Ulisse Dini-type helicoidal hypersurface, and calculate its principal curvatures, and the third fundamental form in section 4. In addition, we give a conclusion in the last section. 2. Preliminaries In this section, we introduce the fundamental form matrices I, II, III, the shape operator ma- trix S; the Gaussian curvature K, and the mean curvature H of a hypersurface M = M(u; v; w) in the Euclidean 4-space E4. Let M be an isometric immersion of a hypersurface M 3 in the E4. The inner product of !x = (x1; x2; x3; x4); !y = (y1; y2; y3; y4), and the vector product of !x ; !y ; !z = (z1; z2; z3; z4) on E4 are de…ned by x y = x1y1 + x2y2 + x3y3 + x4y4; ! ! e1 e2 e3 e4 x1 x2 x3 x4 !x !y !z = det 0 1 ; y1 y2 y3 y4 B z1 z2 z3 z4 C B C @ A respectively. A hypersurface M in 4-space has the …rst and the second fundamental form matrices EFA LMP I = FGB ;II = MNT ; 0 ABC 1 0 PTV 1 @ A @ A respectively. Here, E = Mu Mu;F = Mu Mv;G = Mv Mv;A = Mu Mw;B = Mv Mw;C = Mw Mw; L = Muu e; M = Muv e; N = Mvv e; P = Muw e; T = Mvw e; V = Mww e; and e is the Gauss map Mu Mv Mw e = : Mu Mv Mw k k 1 Hence, I :II gives the shape operator matrix of M s s s 1 11 12 13 S = s21 s22 s23 ; det I 0 1 s31 s32 s33 @ A where det I = (EG F 2)C A2G + 2ABF B2E; THE PRINCIPAL CURVATURES AND THE THIRD FUNDAMENTAL FORM 3 2 s11 = ABM CFM AGP + BF P + CGL B L; 2 s12 = ABN CFN AGT + BF T + CGM B M; 2 s13 = ABT CFT AGV + BF V + CGP B P; 2 s21 = ABL CFL + AF P BP E + CME A M; 2 s22 = ABM CFM + AF T BT E + CNE A N; 2 s23 = ABP CFP + AF V BV E + CTE A T; 2 s31 = AGL + BF L + AF M BME + GP E F P; 2 s32 = AGM + BF M + AF N BNE + GT E F T; 2 s33 = AGP + BF P + AF T BT E + GV E F V: Therefore, using II:S, we get the third fundamental form matrix 1 III = ; det I 0 1 where @ A = A2M 2 + 2ABLM + 2AF MP 2GALP B2L2 + 2BF LP 2EBMP F 2P 2 2CF LM + CGL2 + CEM 2 + GEP 2; = ABM 2 CFM 2 B2LM A2MN F 2PT + CMNE BNP E BMT E + GP T E + ABLN CF LN + CGLM +AF NP AGMP + BF MP + AF MT AGLT + BF LT; = BF P 2 AGP 2 B2LP A2MT F 2PV + CMTE BMV E BP T E + GP V E + ABMP + ABLT CFMP +CGLP CF LT + AF MV AGLV + BF LV + AF P T; = A2N 2 + 2ABMN + 2AF NT 2GAMT B2M 2 + 2BF MT 2EBNT F 2T 2 2CFMN + CGM 2 + CEN 2 + GET 2; = AF T 2 B2MP A2NT F 2TV BT 2E + CNTE BNV E + GT V E + ABNP + ABMT CFNP + CGMP CFMT + AF NV AGMV + BF MV AGP T + BF P T; = A2T 2 + 2ABP T + 2AF T V 2GAP V B2P 2 + 2BF P V 2EBT V F 2V 2 2CFPT + CGP 2 + CET 2 + GEV 2: 3. The Principal Curvatures and the Third Fundamental Form of the Dini-Type Helicoidal Hypersurface We consider Dini-type helicoidal hypersurface sin u cos v cos w sin u sin v cos w D(u; v; w) = ; (1) 0 sin u sin w 1 B cos u + log tan u + av + bw C B 2 C @ A 4 E. GÜLER where u R 0 and 0 v; w 2: Using the …rst di¤erentials of (1) with respect to u; v; w; we get the2 …rstnf quantitiesg cot2 u a cot u cos u b cot u cos u I = a cot u cos u sin2 u cos2 w + a2 ab ; 0 b cot u cos u ab sin2 u + b2 1 and then, we have det I @= b2 + 1 cos2 w + a2 sin2 u cos2 u: The GaussA map of (1) is given by cos ucos v cos2 w + a sin v b cos v sin w cos w 1 cos u sin v cos2 w + a cos v b sin v sin w cos w eD = 0 1 ; (2) pW (cos u sin w + b cos w) cos w B sin u cos w C B C where W = b2 + 1 cos2 w +@a2: Using the second di¤erentials of the (1) withA respect to u; v; w; with (2) ; we have the second quantities of the (1) cot u cos w a cos u cos w b cos u cos w 1 II = a cos u cos w (b sin w cos u cos w) sin u cos2 w a sin u sin w : 1=2 W 0 b cos u cos w a sin u sin w sin u cos u cos w 1 1 @ A Computing I :S; we obtain the shape operator matrix of (1) 2 2 3 sin u cos w a cos w a (b cos w+cos u sin w)+b(b +1) cos w W 1=2 cos u W 1=2 cos u W 3=2 cos u 2 b sin w cos u cos w a(b +1) sin w S = 0 0 1 : (3) W 1=2 sin u W 3=2 sin u a2(b sin w cos u cos w) (b2+1) cos u cos3 w B 0 a sin w C B W 1=2 sin u W 3=2 sin u C @ A Theorem 1. Let D : M 3 E4 be an immersion given by (1). Then, characteristic polynomial of the (3) of the (1) !is given by X3 + pX2 + qX + r = 0: where cos2 u cos3 w + b2 cos2 u cos3 w + W cos2 u cos w W cos w sin2 u + a2 cos2 u cos w 8 bW cos u sin w a2b cos u sin w 9 p = < =; W 3=2 cos u sin u : cos3 u cos4 w + a2 cos3 u cos2 w + b2 cos3 u cos4 w ; a2 cos u sin2 w cos u cos4 w sin2 u 8 2 2 2 2 2 9 > W cos u cos w sin u a cos u cos w sin u > > 2 4 2 3 2 3 > > b cos u cos w sin u b cos u cos w sin w > <> b cos2 u cos3 w sin w +bW cos w sin2 u sin w => 2a2b cos2 u cos w sin w + a2b cos w sin2 u sin w q = > >; > 2 2 > > W cos u sin u > : a2 sin2 w cos2 u cos4 w a2 cos2 u cos2 w ; b2 cos2 u cos4 w + b cosu cos3 w sin w cos w 8 +b3cos u cos3 w sin w + 2a2b cos u cos w sin w 9 r = < = : W 5=2 sin u cos u : ; THE PRINCIPAL CURVATURES AND THE THIRD FUNDAMENTAL FORM 5 Proof.
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