К О Р О Т К I П О В I Д О М Л Е Н Н Я

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1 К О Р О Т К I П О В I Д О М Л Е Н Н Я UDC E. Ö. Canfes, A. Özdeğer Istanbul Techn. Univ., Kadir Has Univ., Istanbul, Turkey A CHARACTERIZATION OF TOTALLY UMBILICAL HYPERSURFACES OF A SPACE FORM BY GEODESIC MAPPING ХАРАКТЕРИСТИКА ТОТАЛЬНО ОМБIЛIЧНИХ ГIПЕРПОВЕРХОНЬ ПРОСТОРОВОЇ ФОРМИ ЗА ДОПОМОГОЮ ГЕОДЕЗИЧНИХ ВIДОБРАЖЕНЬ The idea of considering the second fundamental fm of a hypersurface as the first fundamental fm of another hypersurface has found very useful applications in Riemannian and semi-riemannian geometry, specially when trying to characterize extrinsic hyperspheres and ovaloi. Recently, T. Adachi and S. Maeda gave a characterization of totally umbilical hypersurfaces in a space fm by circles. In this paper, we give a characterization of totally umbilical hypersurfaces of a space fm by means of geodesic mapping. Iдея використання другої фундаментальної форми гiперповерхнi як першої фундаментальної форми iншої гiперповерхнi знайшла дуже важливi застосування у рiмановiй та напiврiмановiй геометрiї, зокрема при описi зовнiшнiх гiперсфер та овалоїдiв. Нещодавно T. Adachi та S. Maeda навели характеристику тотально омбiлiчних гiперповерхонь у просторовiй формi за допомогою кiл. У цiй роботi ми наводимо характеристику тотально омбiлiчних гiперповерхонь просторової форми за допомогою геодезичних вiдображень. 1. Introduction. Let M n and M n be two hypersurfaces of the space fm M n+1 [3 5] and let g, g and ḡ be the respective positive definite metric tenss. Denote by, and the cresponding connections induced by g, g and ḡ. In this paper, we choose the first fundamental fm of M n as g = e 2σ ω, 1.1 where ω is the second fundamental fm of M n which is supposed to be positive definite and σ is a differentiable function defined on M n. Let {x i }, {x i } and {y α } be the respective codinate systems in M n, M n and M n+1 and let f be a one-to-one differentiable mapping of M n upon M n defined by x i = f i x 1, x 2,..., x n, i = 1, 2,..., n, 1.2 in which f i are smooth functions defined on M n and have a non-vanishing Jacobian. Then, it is clear that the cresponding points of M n and M n are represented by the same set of codinates and that the codinate vects crespond. Let R, R and R be the covariant curvature tenss of Mn+1, M n and M n respectively and let K be the Riemannian curvature of Mn+1. We then have 1 Rβγδɛ = Kḡβδḡγɛ ḡβɛḡγδ. 1.3 On the other hand, under the condition 1.3 the Codazzi equations 1 In the sequel, Latin indices i, j, k,... run from 1 to n, while the Greek indices α, β, γ will run from 1 to n + 1. c E. Ö. CANFES, A. ÖZDEĞER,

2 584 E. Ö. CANFES, A. ÖZDEĞER and the Gauss equation k ω ij j ω ik + R βγδɛ N β yγ x i y δ x j y ɛ x k = 0 R ijkl = R y β y γ y δ y ɛ βγδɛ x i x j x k x l + ω ikω jl ω il ω jk transfm, respectively, into and k ω ij j ω ik = R ijkl = Kg ik g jl g il g jk + ω ik ω jl ω il ω jk 1.5 in which N β are the components of the unit nmal vect field of M n [4]. 2. Relation between the connections and. It is well-known that the connection coefficients of a Riemannian space whose metric tens is g are given by [5] Γ l ij = 1 2 glh i g jh + j g ih h g ij, k = x k. 2.1 Replacing g in 2.1 by the metric tens g of M n given by 1.1 and doing the necessary calculations we first find the connection coefficients Γ l ij of M n as Γ l ij = 1 2 e2σ g lk j ω ik + i ω jk k ω ij + j σδ l i + i σδ l j k σg lk g ij. 2.2 On the other hand, f the covariant derivative of the second fundamental tens ω of M n we have [3, 4] i ω jk = i ω jk Γ h ijω hk Γ h ik ω jh. 2.3 Changing the indices i, j and k cyclically we obtain two me equations: j ω ki = j ω ki Γ h ijω hk Γ h kj ω ih, 2.4 k ω ij = k ω ij Γ h ki ω hj Γ h kj ω ih. 2.5 Subtracting 2.5 from the sum of 2.3 and 2.4 and using the Codazzi equations 1.4, we obtain In view of 2.6, 2.2 becomes i ω jk = i ω jk + j ω ik k ω ij 2ω hk Γ h ij. 2.6 Γ l ij = Γ l ij + δ l i j σ + δ l j i σ g lk g ij k σ e2σ g lk i ω jk is the desired relation connecting the connection coefficients of M n and M n. 3. Geodesic mappings of M n upon M n. If the map f defined by 1.2 transfms every geodesic in M n into a geodesic in M n, f is called a geodesic mapping of M n into M n. M n and M n will be in geodesic crespondence if and only if the respective connection coefficients Γ h ij and Γ h ij of M n and M n are related by [3] Γ i jk = Γi jk + δi jψ k + δ i k ψ j, 3.1 where ψ k are the components of some 1-fm which is known to be a gradient. We first prove the following lemma which will be needed in our subsequent wk.

3 A CHARACTERIZATION OF TOTALLY UMBILICAL HYPERSURFACES OF A SPACE FORM Lemma 3.1. Let M n and M n be hypersurfaces of the space fm M n+1 and let the metric tens of M n be defined by 1.1. If M n and M n are in geodesic crespondence, then the 1-fm ψ k is the gradient of 2σ. Proof. Since is a metric connection we have so that with the help of 1.1 and 3.1 we obtain 0 = k g ij = k g ij g lj Γl ik g li Γl jk 0 = 2ω ij k σ + k ω ij 2ψ k ω ij ψ i ω kj ψ j ω ki. 3.2 Interchanging the indices j and k in 3.2 we find Subtracting 3.3 from 3.2 and putting in 3.3 we conclude that 0 = 2ω ik j σ + j ω ik 2ψ j ω ik ψ i ω kj ψ k ω ji. 3.3 φ k = ψ k 2 k σ 3.4 ω ij φ k ω ik φ j = in which the Codazzi equations 1.4 have been used. We note that, since ψ k is a gradient, it follows from 3.4 that φ k is also a gradient. Multiplying 3.5 by e 2σ and using 1.1 we obtain φ k g ij φ j g ik = 0 3.6, multiplying 3.6 by g ij and summing with respect to i and j we find f n > 1 that φ k = Combination of 3.4 and 3.7 yiel ψ k = 2 k σ. We next prove the following theem. Theem 3.1. The hypersurface M n of a space fm M n+1 will be totally umbilical if and only if M n can be geodesically mapped upon M n. Proof. Sufficiency. Let γ be a geodesic through the point p M n which is defined by x i = x i s, s being the arc length of γ. Then, the nmal curvature, say κ n, of M n in the direction of γ, i.e., in the direction of dxi, is [4] Multiplying 3.2 by dxi dx k 2κ n k σ dxk + kω ij dxk 2 κ n = ω ij. 3.8 and summing with respect to i, j, k and using 3.8 we obtain ψ k dx k κ n ψ i κ n ψ j κ n =

4 586 E. Ö. CANFES, A. ÖZDEĞER Since ψ k is a gradient, there exists a differentiable function ψ such that ψ k = k ψ. On the other hand, differentiating 3.8 covariantly in the direction of γ and using the Frenet s fmula [3] dx i dx k k = κ gη i, 1 where κ g is the geodesic curvature and 1 η is the unit principal nmal vect field of γ relative to M n, we find that k ω ij dxk = dκ n 2κ gω ij 1 η i dxj Using 3.10 in 3.9 and remembering that γ is a geodesic κ g = 0 in M n, we get [ κn x i + 2 σ x i 4 ψ ] dx i x i κ n = 0, along γ. On the other hand, by 1.1 and 3.11, we find [ ] dx i x i ln κ n + 2σ 4ψ = = g ij = e 2σ ω ij = e 2σ ω ij from which it follows that κ n > 0. From 3.1 it follows that, 2 = e 2σ κ n 2, ln κ n + 2σ 4ψ = const = C along γ. By Lemma 3.1, ψ = 2σ + C 2, C 2 = Const and therefe 3.12 gives κ n = ce 6σ, 3.13 where c is an arbitrary positive constant. From 3.13 it follows that the lines of curvature of M n are indeterminate at all points of M n. Consequently, M n is totally umbilical. Necessity. Assume that M n is a totally umbilical hypersurface of M n+1 which means that ω ij = = H n g ij where H is the mean curvature of M n. In this case, 1.1 becomes so that M n and M n are confmal. From 1.5 it follows that R ijkl = g ij = ρ 2 g ij ρ 2 = e 2σ H, 3.14 n K + H2 n 2 g ik g jl g il g jk showing that M n has the constant curvature K + H2. So H is constant. n2

5 A CHARACTERIZATION OF TOTALLY UMBILICAL HYPERSURFACES OF A SPACE FORM We will show that M n can also be geodesically mapped upon M n. Since M n is confmal to M n, their connection coefficients are related by [6] Γ h ij = Γ h ij + δj h ρ i + δi h ρ j g ij ρ h ρ i = i ρ, ρ h = g th ρ t To show that this confmal mapping between M n and M n is also a geodesic mapping, accding to 3.15 and 3.1 we have to find a 1-fm ψ k such that Transvecting 3.16 by g ij we get Γ h ij + δ h j ψ i + δ h i ψ j = Γ h ij + δ h j ρ i + δ h i ρ j g ij ρ h δ h j ψ i ρ i + δ h i ψ j ρ j + g ij ρ h = g ih ψ i ρ i + g jh ψ j ρ j + nρ h = 0 Multiplying 3.17 by g hj and summing f h we obtain Then, by 3.14 we find that ψ j = 2g ih ψ i ρ i + nρ h = ψ j + n 2ρ j = 0. 2 n 2 H j e σ, H > 0. n With this choice of ψ j the confmal mapping mentioned above becomes also a geodesic mapping. Theem 1.1 is proved. In the special case where σ = 0 throughout M n, i.e., when g = ω, we may mention below some properties of M n which is in geodesic crespondence with M n. 1. From Lemma 3.1 and the relation 3.1 we conclude that any geodesic mapping of M n upon M n is connection preserving. 2. By 3.13 it follows that M n has constant nmal curvature along each geodesic through a point p M n. 3. The underlying geodesic mapping is a homothety. 1. Verpot S. The geometry of the second fundamental fm: curvature properties and variational aspects: Ph. D Thesis. Katholieke Univ. Leuven, Adachi T., Maeda S. Characterization of totally umbilic hypersurfaces in a space fm by circles // Czechoslovak Math. J , 1. P Gerretsen J. Lectures on tens calculus and differential geometry. Groningen: P. Nodhoff, Weatherburn C. E. An Introduction to Riemannian geometry and the tens calculus. Cambridge Univ. Press, Yano K., Kon M. Structures on manifol. Wlci., Chen B. Y. Geometry of submanifol. New Yk: Marcel Dekker Inc., Received