The right conoid hypersurfaces in the four-dimensional Euclidean space $\mathbb{E}^{4}$ are introduced. The matrices corresponding to the fundamental form, Gauss map, and shape operator of these hypersurfaces are calculated. By utilizing the Cayley--Hamilton theorem, the curvatures of these specific hypersurfaces are determined. Furthermore, the conditions for minimality are presented. Additionally, the Laplace--Beltrami operator of this family is computed, and some examples are provided

Güler, Erhan

Yıldız, Mustafa

English

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 38, No 4 (2023), 817–828https://doi.org/10.22190/FUMI230624053GOriginal Scientific PaperRIGHT CONOID HYPERSURFACES IN FOUR-SPACEErhan Güler and Mustafa YıldızFaculty of Sciences, Department of MathematicsBartın University, 74100 Bartın, TürkiyeAbstract. The right conoid hypersurfaces in the four-dimensional Euclidean space E4are introduced. The matrices corresponding to the fundamental form, Gauss map, andshape operator of these hypersurfaces are calculated. By utilizing the Cayley–Hamiltontheorem, the curvatures of these specific hypersurfaces are determined. Furthermore,the conditions for minimality are presented. Additionally, the Laplace–Beltrami oper-ator of this family is computed, and some examples are provided.Keywords: conoid hypersurfaces, Cayley–Hamilton theorem, four-dimensional space.1. IntroductionA ruled surfacer(u, v) = α (v) + uβ (v)= (0, 0, h (v)) + u (cos f (v) , sin f (v) , 0)is termed a right conoid in three-dimensional space E3 if it can be generated by thetranslation of a straight line that intersects a fixed straight line, while ensuring thatthe lines maintain a perpendicular relationship throughout the generation process.By considering the xy-plane as the perpendicular plane and selecting the z-axis asthe reference line, the parametric Eq. for the right conoid is given byr(u, v) = x(u, v)y(u, v)z(u, v) = u cos f (v)u sin f (v)h (v) .Received June 24, 2023, accepted: November 07, 2023Communicated by Cornelia-Livia BejanCorresponding Author: Erhan Güler (eguler@bartin.edu.tr)2010 Mathematics Subject Classification. Primary 53A05, 53A07; Secondary 53C42, 53C43© 2023 by University of Nǐs, Serbia | Creative Commons License: CC BY-NC-ND818 E. Güler and M. YıldızHelicoid, Whitney umbrella, Wallis’s conical edge, Plücker’s conoid, hyperbolicparaboloid are each examples of a right conoid surface. See Berger and Gostiaux[2], Do Carmo [4], Gray [5], Kreyszig [6] for details.The aim of this study is to investigate the properties of the right conoid hyper-surfaces in the four-dimensional Euclidean space E4. Specifically, we aim to computethe matrices associated with the fundamental form, Gauss map, and shape operatorof these hypersurfaces. By employing the Cayley–Hamilton theorem, our objectiveis to determine the curvatures of these particular hypersurfaces. Additionally, weaim to establish the conditions for minimality within this context. Moreover, weseek to unveil the connection the Laplace–Beltrami operator of that kind hypersur-faces.In Section 2., a detailed explanation of the fundamental principles and conceptsunderlying four-dimensional Euclidean geometry is provided.Section 3. is dedicated to the presentation of the curvature formulas applicableto hypersurfaces in E4.In Section 4., a comprehensive definition of right conoid hypersurfaces is offered,emphasizing their distinctive properties and characteristics.In Section 5., the focus shifts to the discussion of the Laplace–Beltrami oper-ator for a smooth function in E4, and the application of the previously examinedhypersurfaces in its computation.In the last section, we present a conlusion.2. PreliminariesIn this paper, we use the following notations, formulas, Eqs., etc.Let M be an oriented hypersurface in En+1 with its shape operator S, positionvector x. Consider a local orthonormal frame field {e1, e2, . . . , en} consisting of prin-cipal directions of M coinciding with the principal curvature ki for i = 1, 2, . . . , n.Let the dual basis of this frame field be {f1, f2, . . . , fn}.We let sj = σj(k1, k2, . . . , kn), where σj denotes the j-th elementary symmetricfunction defined byσj(a1, a2, . . . , an) =∑1≤i1<i2<...<ij≤nai1ai2 . . . aij .We consider the notationrji = σj(k1, k2, . . . , ki−1, ki+1, ki+2, . . . , kn).According to the given definition, we have r0i = 1 and sn+1 = sn+2 = · · · = 0. Thefunction sk is referred to as the k-th mean curvature of the oriented hypersurfaceM .The mean curvature H = 1ns1 is also defined, and the Gauss–Kronecker curvatureof M is K = sn. If sj ≡ 0, the hypersurface M is known as j-minimal.Right Conoid Hypersurfaces in Four-Space 819In Euclidean (n + 1)-space, getting the curvature formulas Ki, i = 0, 1, . . . , n,(See [1], [3], and [7] for details.), we have the following characteristic polynomialEq. PS(λ) = 0 of S:(2.1)n∑k=0(−1)k skλn−k = det(S − λIn) = 0.Here, In indicates the identity matrix. Hence, we reveal the curvature formulas as(ni)Ki = si.In this paper, we have identified a vector with its transpose. Let x = x(u, v, w)be an immersion from M3 ⊂ E3 to E4.Definition 2.1. An inner product of two vectors φ1 = (φ11, φ12, φ13, φ14), φ2 =(φ21, φ22, φ23, φ24) of E4 is determined by〈φ1, φ2〉= φ11φ22 + φ12φ22 + φ13φ23 + φ14φ24.Definition 2.2. A triple vector product of φ1 = (φ11, φ12, φ13, φ14), φ2 = (φ21, φ22, φ23, φ24),φ3 = (φ31, φ32, φ33, φ34) in E4 is defined byφ1 × φ2 × φ3 = dete1 e2 e3 e4φ11 φ12 φ13 φ14φ21 φ22 φ23 φ24φ31 φ32 φ33 φ34 .Definition 2.3. The matrix (gij)−1·(hij) determines the shape operator matrixS of hypersurface x in Euclidean 4-space E4, where, (gij)3×3 and (hij)3×3 describethe first and the second fundamental form matrices, respectively, and gij = ⟨xi, xj⟩ ,hij = ⟨xij ,G⟩ , i, j = 1, 2, 3, xu = ∂x∂u when i = 1, xuv =∂2x∂u∂v when i = 1 and j = 2,etc., ek denotes the natural base elements of E4, and(2.2) G = xu × xv × xw∥xu × xv × xw∥determines the Gauss map of the hypersurface x.3. Curvatures in Four-SpaceIn this section, we reveal the curvature formulas of any hypersurface x = x(u, v, w)in E4.Theorem 3.1. A hypersurface x in E4 has the following curvature formulas, K0 =1 by definition,(3.1) 3K1 =c2c3, 3K2 = −c1c3, K3 =c0c3,where c3λ3+c2λ2+c1λ+c0 = 0 describes the characteristic polynomial Eq. PS(λ) =0 of the shape operator matrix S, c3 = det (gij), c0 = det (hij), and (gij)3×3,(hij)3×3 denote the first, and the second fundamental form matrices, respectively.820 E. Güler and M. YıldızProof. The matrix (gij)−1·(hij) describes the shape operator matrix S of hyper-surface x in Euclidean 4-space E4. We reveal the characteristic polynomial Eq.det(S − λI3) = 0 of S. Thus, we obtain the curvaturesK0 = 1,3K1 = k1 + k2 + k3 = −c2c3,3K2 = k1k2 + k1k3 + k2k3 =c1c3,K3 = k1k2k3 = −c0c3,Definition 3.1. A hypersurface x is called j-minimal if Kj = 0, where j = 1, 2, 3.Theorem 3.2. A hypersurface x = x(u, v, w) in E4 has the following relationK0IV− 3K1III+ 3K2II−K3I = O3,where I, II, III, IV determines the fundamental form matrices, O3 represents the zeromatrix having order 3 of the hypersurface.Proof. Regarding n = 3 in (2.1), it runs.4. Right Conoid HypersurfacesIn this section, we define the right conoid hypersurface (RCH ), then find itsdifferential geometric properties in Euclidean 4-space E4.In E4, we consider a ruled hypersurfacex(u, v, w) = α (v, w) + uβ (v, w)= (0, 0, 0, h (v, w))+u (cos f (v) cos g (w) , sin f (v) cos g (w) , sin g (w) , 0) .Then, we present the following.Definition 4.1. A right conoid hypersurface is an immersion x : M3 ⊂ E3 −→E4 with the reference line x4, parametrized by(4.1) x(u, v, w) =x1(u, v, w)x2(u, v, w)x3(u, v, w)x4(u, v, w) =u cos f (v) cos g (w)u sin f (v) cos g (w)u sin g (w)h (v, w) .Here, u ∈ R−{0}, f = f (v) , g = g (w) , h = h (v, w) denote the differentiablefunctions, and 0 ≤ f, g < 2π.Right Conoid Hypersurfaces in Four-Space 821Taking the first derivatives of RCH determined by Eq. (4.1) with respect tou, v, w, respectively, we obtain the first fundamental form matrix(4.2) (gij) = 1 0 00 u2f2v cos2 g (w) + h2v hvhw0 hvhw u2g2w + h2w ,and fv =∂f∂v , f2v =∂2f∂v2 , etc.. Hence, g = det (gij) = u2W, whereW = f2v(u2g2w + h2w)cos2 g (w) + h2vg2w.Using the Gauss map formula (2.2) , we obtain the following Gauss map of the RCHdetermined by Eq. (4.1):(4.3) G = 1W1/2−fvhw cos f (v) sin g (w) cos g (w)− hvgw sin f (v)−fvhw sin f (v) sin g (w) cos g (w) + hvgw cos f (v)fvhw cos2 g (w)−ufvgw cos g (w) .By taking the second derivatives w.r.t. u, v, w, of RCH described by Eq. (4.1) , andby using the Gauss map given by Eq. (4.3), we find the second fundamental formmatrixh11 = 0, h12 =fvgwhv cos gW, h13 =fvgwhw cos gW,h21 =fvgwhv cos gW,h22 =u(f3vhw sin g cos g + gw (hvfvv − fvhvv))cos gW,(4.4)h23 = −ufvgw (hvgw sin g + hvw cos g)W, h31 =fvgwhw cos gW,h32 = −ufvgw (hvgw sin g + hvw cos g)W,h33 = −ufv (hwgww − gwhww) cos gW,and fuu =∂2f∂u2 , fuv =∂2f∂u∂v , etc.. By using (4.2) and (4.4), we compute the following822 E. Güler and M. Yıldızshape operator matrix S of (4.1). S = (sij)3×3 has the following componentss11 = 0,s12 =hvgwfv cos gW,s13 =hwgwfv cos gW,s21 =hvg3wfv cos gW2,s22 =1uW2[hwfv(g2wh2v + f2v(h2w + u2g2w)cos2 g)sin g+gw(fvhvhwhvw +(h2w + u2g2w)(hvfvv − fvhvv) + u2g2whvfvv)cos g],s23 =fvuW2[(−gwhw (hwhvw + hvhww)− u2g3whvw + hvh2wgww)cos g−hvg2w(h2w + u2g2w)sin g],s31 =1W2hwgwf3v cos3 g,s32 =1uW2[−u2f3v gwhvw cos3 g−hvgw (fvhvhvw − fvhwhvv + hvhwfvv) cos g−hvfv(g2wh2v + f2v(u2g2w + h2w)cos2 g)sin g],s33 =fvuW2[u2f2v (gwhww − hwgww) cos3 g+hv (gw (hwhvw + hvhww)− hvhwgww) cos g + g2wh2vhw sin g].Finally, using (3.1), with (4.2), (4.4), respectively, we find the curvatures of theRCH defined by Eq. (4.1) as follows.Theorem 4.1. Let x be a RCH determined by Eq. (4.1) in E4. x contains thefollowing curvatures, K0 = 1, by definition,K1 = −13uW2[u2f3v (gwhww − hwgww) cos3 g+2fvg2wh2vhw sin g + f3vhw(h2w + u2g2w)sin g cos2 g+(gwfv(2hvhwhvw + h2vhww − hvv(h2w + u2g2w))+hv(gwfvv(h2w + 2u2g2w)− fvhvhwgww)) cos g],Right Conoid Hypersurfaces in Four-Space 823K2 = −fv3W4[f5v g2wh2w(h2w + u2g2w)cos6 g−hwf5v(h2w + u2g2w)(gwhww − hwgww) sin g cos5 g+f3v g2whv(2gw(h2w + u2g2w)hvw + hvhw (hwgww − gwhww))sin g cos3 g+f2v gw(gwfvh2vw(h2w + u2g2w)+ fvg3wh2v(2h2w + u2g2w)−fvhwhvvgww(h2w + u2g2w)+gwhww(fvhvv(h2w + u2g2w)− hvfvv(h2w + 2u2g2w))+hvhwfvvgww(h2w + 2u2g2w)) cos4 g + h2vg4wf3v(h2w + u2g2w)sin2 g cos2 g+h2vg3w(fvgwh2vw − gwhwhvwfvv + fvg3wh2v + fvgwhvvhww−fvhwgwwhvv − 2gwhvfvvhww + 2hvhwfvvgww) cos2 g + fvg6wh4v sin2 g+h3vg5w (2fvhvw − hwfvv) sin g cos g],K3 =f2v g2w cos2 guW5[h3wf5v(h2w + u2g2w)sin g cos4 g+f2v (fv(h2w + u2g2w) (2gwhvhwhvw + gwh2vhww − gwh2whvv − h2vhwgww)+h2whvgwfvv(h2w + 2u2g2w)) cos3 g+hwh2vg2wf3v(3h2w + 2u2g2w)sin g cos2 g+h2vg2w(2fvgwhvhwhvw + fvgwh2vhww − fvgwh2whvv−fvh2vhwgww + gwhvh2wfvv) cos g + 2fvg4wh4vhw sin g].Here, K1 represents the mean curvature, K3 denotes the Gauss–Kronecker curva-ture.Proof. By using the Cayley–Hamilton theorem, we reveal the following character-istic polynomial Eq. PS(λ) = 0 of RCH defined by Eq. (4.1):K0λ3 − 3K1λ2 + 3K2λ−K3 = 0,whereK0 = 1,3K1 = s22 + s33,3K2 = −s12s21 − s13s31 + s22s33 − s23s32,K3 = −s12s21s33 + s12s31s23 + s32 (s21s13 + s22s23)− s22 (s13s31 + s23s32) .The curvatures Ki of x are obtained by the above Eqs..Theorem 4.2. Let x be a RCH described by Eq. (4.1) in E4. x has the followingprincipal curvaturesk1 =s12s21s33 − s12s31s23 − s21s13s32 + s13s22s31s12s21 + s13s31= k2,k3 = 0.824 E. Güler and M. YıldızProof. By using Eq. det(S − kI3) = 0, it is clear.For the sake of brevity, we use the following notationsΓ = −gwhvΨfvv + fvh2vhwgww + gwfvΦhvv −Θ,Ω = gwhww − hwgww,Φ = h2w + u2g2w,Ψ = h2w + 2u2g2w,Θ = fvgwhv (2hwhvw + hvhww) .Corollary 4.1. Let x be a RCH defined by Eq. (4.1) in E4. x is 1-minimal iff thefollowing partial differential Eq. appearsu2f3vΩcos3 g + f3vhwΦsin g cos2 g + 2fvg2wh2vhw sin g+(Θ + hv (gwfvvΨ− fvhvhwgww)− fvgwhvvΦ) cos g = 0,where u,W ≠ 0.Corollary 4.2. Let x be a RCH determined by Eq. (4.1) in E52. x is 2-minimal iffthe following partial differential Eq. occurs,f5v g2wh2wΦcos6 g − f5vhwΦΩsin g cos5 g+f3v g2whv (2gwΦhvw − hvhwΩ) sin g cos3 g+f2v gw(fvgwh2vwΦ+ g3wfvh2v(2h2w + u2g2w)+ hwgww (hvfvvΨ− fvhvvΦ)+gwhww (−hvfvvΨ+ fvhvvΦ)) cos4 g + f3v g4wh2vΦsin2 g cos2 g+h2vg3w(gwfv(g2wh2v + h2vw)+ (1− 2hvfvv) Ω− gwhwhvwfvv) cos2 g+fvg6wh4v sin2 g + h3vg5w (2fvhvw − hwfvv) sin g cos g = 0,where fv,W ̸= 0.Corollary 4.3. Let x be a HRF given by Eq. (4.1) in E52. x is 3-minimal iff thefollowing partial differential Eq. holdsh3wf5vΨsin g cos4 g+f2v (gwhvh2wfvvΨ+ fvΦ(gwhw (2hvhvw − hwhvv) + h2vΩ)) cos3 g+f3v g2wh2vhw(3h2w + 2u2g2w)sin g cos2 g+h2vg2w(2fvgwhvhwhvw + fvh2vΩ+ h2wgw (hvfvv − fvhvv)) cos g+2fvg4wh4vhw sin g = 0,where fv, gw, cos g, u,W ≠ 0.Note that the solutions for h in Corollary 4.1, Corollary 4.2, and Corollary 4.3are open problems.Right Conoid Hypersurfaces in Four-Space 8255. Right Conoid Hypersurfaces with ∆x = Qx in E4In this section, our focus is on the Laplace–Beltrami operator of a smooth func-tion in E4. We will proceed to compute it utilizing the RCH defined by Eq. (4.1).Definition 5.1. The Laplace–Beltrami operator of a smooth function ϕ = ϕ(x1, x2, x3) |D(D ⊂ R3) of class C3 depends on the first fundamental form (gij) of a hypersurfacex, is defined by(5.1) ∆ϕ =1g1/24∑i,j=1∂∂xi(g1/2gij∂ϕ∂xj),where(gij)= (gkl)−1and g = det (gij) .Therefore, we give the following.Theorem 5.1. The Laplace–Beltrami operator of the RCH x denoted by Eq. (4.1)is given by ∆x = 3K1G, where K1 describes the mean curvature, G represents theGauss map of x.Proof. The Laplace–Beltrami operator of the RCH given by Eq. (4.1) is determinedby∆x =1g1/2[∂∂u(g1/2g11∂x∂u)+∂∂v(g1/2g22∂x∂v)+∂∂v(g1/2g23∂x∂w)+∂∂w(g1/2g32∂x∂v)+∂∂w(g1/2g33∂x∂w)],(5.2)whereg11 = 1, g12 = 0, g13 = 0,(5.3)g21 = 0, g22 =u2g2w + h2wg, g23 = −hvhwg,g31 = 0, g32 = −hvhwg, g33 =u2f2v cos2 g + h2vg,and g = u2(f2v(u2g2w + h2w)cos2 g (w) + h2vg2w). By taking the derivatives of thefunctions determined by Eqs. (5.3) in (5.2), w.r.t. u, v, w, resp., we obtain ∆x =826 E. Güler and M. Yıldız(∆x1,∆x2,∆x3,∆x4) with components∆x1 = −1uW2[−u2f4vhwΩcos f sin g cos4 g − f4vh2wΦcos f sin2 g cos3 g−u2f3v gwhvΩsin f cos3 g − 2f2v g2wh2vh2w cos f cos g sin2 g−f3v gwhvhwΦsin f sin g cos2 g + fvhwΓ cos f cos2 g sin g−2fvg3wh3vhw sin f sin g + hvgwΓ sin f cos g],∆x2 = −1uW2[−u2f4vhwΩsin f sin g cos4 g − h2wf4vΦsin f sin2 g cos3 g+u2f3v gwhvΩcos f cos3 g + fvhwΓ cos2 g sin f sin g+f3v gwhvhwΦcos f cos2 g sin g − 2f2v g2wh2vh2w sin f cos g sin2 g−hvgwΓ cos f cos g + 2fvg3wh3vhw cos f sin g],∆x3 = −1uW2[fvhw(u2f3vΩcos5 g + f3vhwΦcos4 g sin g − Γ cos3 g+2fvg2wh2vhw cos2 g sin g)],∆x4 = −1uW2[ufvgw(−u2f3vΩcos4 g − f3vhwΦcos3 g sin g+Γcos2 g − 2fvg2wh2vhw cos g sin g)].Definition 5.2. The hypersurface x is called harmonic if each componets of ∆x iszero.Example 5.1. Substituting f (v) = v, g (w) = w, h (v, w) = w on a RCH defined by Eq.(4.1) in E4, we have(5.4) G = 1(u2 + 1)1/2(− cos v sinw,− sin v sinw, cosw,−u) ,S =0 0 1(u2+1)1/20 tanwu(u2+1)1/201(u2+1)3/20 0 ,the principal curvatures are given by k1 =1u2+1= −k2, k3 = tanwu(u2+1)1/2, and the curva-tures are determined byK1 =tanw3u (u2 + 1)3/2,K2 = −13 (u2 + 1)2,K3 = −tanwu (u2 + 1)5/2.Right Conoid Hypersurfaces in Four-Space 827Then,∆x =tanwu (u2 + 1)2(− cos v sinw,− sin v sinw, cosw,−u) .Finally, the hypersurface is non-minimal and non-harmonic.Example 5.2. By taking f (v) = v, g (w) = w, h (v, w) = v on a RCH determined byEq. (4.1) in E4, the Gauss map is determined byG = 1(u2 cos2 w + 1)1/2(− sin v, cos v, 0,−u cosw) .Then, the shape operator matrix is given byS =0 cosw(u2 cos2 w+1)1/20cosw(u2 cos2 w+1)3/20 − u sinw(u2 cos2 w+1)3/20 sinwu(u2 cos2 w+1)1/20 .The principal curvatures are determined by k1 =cos1/2(2w)u2 cos2 w+1= −k2, k3 = 0. The curvaturesare described byK1 = 0,K2 = −cos 2w3 (u2 cos2 w + 1)2,K3 = 0.Therefore, ∆x = (0, 0, 0, 0) . That is, the hypersurface is 1-minimal, 3-minimal, and har-monic.6. ConclusionThis research has focused on the study of right conoid hypersurfaces in the four-dimensional Euclidean space E4. The main objective was to analyze and understandthe geometric properties of these hypersurfaces.We computed the matrices associated with the fundamental form, Gauss map,and shape operator of the right conoid hypersurfaces. These matrices provide crucialinformation about the local geometry of the surfaces, including their curvaturesand tangent spaces. By employing the Cayley–Hamilton theorem, the curvatures ofthese specific hypersurfaces were determined. This theorem allowed for an effectivecomputation of the curvatures by expressing the characteristic polynomial of thematrices in terms of the matrices themselves. Moreover, the research presentedthe conditions for minimality in the context of right conoid hypersurfaces. Theseconditions define when a hypersurface can be considered minimal within this specificfamily. Additionally, the research explored the Laplace–Beltrami operator of theright conoid hypersurfaces.This research contributes to the understanding of right conoid hypersurfaces inE4, providing valuable insights into their geometric properties, curvatures, mini-mality conditions, and their relation to the Laplace–Beltrami operator.828 E. Güler and M. YıldızREFERENCES1. L. J. Alias and N. Gürbüz: An extension of Takashi theorem for the linearizedoperators of the highest order mean curvatures. Geom. Dedicata 121 (2006), 113–127.2. M. Berger and B. Gostiaux: Géométrie Différentielle: Variétés, Courbes et Sur-faces. Presses Univ. France, France, 1987.3. B. Y. Chen, E. Güler, Y. Yaylı, and H. H. Hacısalihoğlu: Differential geometryof 1-type submanifolds and submanifolds with 1-type Gauss map. Int. Elec. J. Geom.,16(1) (2023), 4–49.4. M. P. Do Carmo: Differential Geometry of Curves and Surfaces. Englewood Cliffs,Prentice Hall, USA, 1976.5. A. Gray: Modern Differential Geometry of Curves and Surfaces with Mathematica.2nd ed. Boca Raton, FL: CRC Press, 1997.6. A. Kreyszig: Differential Geometry. Dover Pub., New York, 1991.7. W. Kühnel: Differential Geometry. Curves-Surfaces-Manifolds. Third ed. Translatedfrom the 2013 German ed. AMS, Providence, RI, 2015.8. T. Levi–Civita: Famiglie di superficie isoparametriche nellordinario spacio euclideo.Rend. Acad. Lincei 26 (1937), 355–362.

RIGHT CONOID HYPERSURFACES IN FOUR-SPACE

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RIGHT CONOID HYPERSURFACES IN FOUR-SPACE

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