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Extension of Eigenvalue Problems on Gauss Map of Ruled Surfaces Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 September 2018 doi:10.20944/preprints201809.0407.v1 Peer-reviewed version available at Symmetry 2018, 10, 514; doi:10.3390/sym10100514 EXTENSION OF EIGENVALUE PROBLEMS ON GAUSS MAP OF RULED SURFACES MIEKYUNG CHOI AND YOUNG HO KIM* Abstract. A finite-type immersion or smooth map is a nice tool to classify subman- ifolds of Euclidean space, which comes from eigenvalue problem of immersion. The notion of generalized 1-type is a natural generalization of those of 1-type in the usual sense and pointwise 1-type. We classify ruled surfaces with generalized 1-type Gauss map as part of a plane, a circular cylinder, a cylinder over a base curve of an infinite type, a helicoid, a right cone and a conical surface of G-type. 1. Introduction Nash's embedding theorem enables us to study Riemannian manifolds extensively by regarding a Riemannian manifold as a submanifold of Euclidean space with sufficiently high codimension. By means of such a setting, we can have rich geometric information from the intrinsic and extrinsic properties of submanifolds of Euclidean space. Inspired by the degree of algebraic varieties, B.-Y. Chen introduced the notion of order and type of submanifolds of Euclidean space. Furthermore, he developed the theory of finite- type submanifolds and estimated the total mean curvature of compact submanifolds of Euclidean space in the late 1970s ([3]). In particular, the notion of finite-type immersion is a direct generalization of eigen- value problem relative to the immersion of a Riemannian manifold into a Euclidean space: Let x : M ! Em be an isometric immersion of a submanifold M into the Eu- clidean m-space Em and ∆ the Laplace operator of M in Em. The submanifold M is said to be of finite-type if x has a spectral decomposition by x = x0 + x1 + ::: + xk, where x0 is a constant vector and xi are the vector fields satisfying ∆xi = λixi for some λi 2 R (i = 1; 2; :::; k). If the eigenvalues λ1; λ2; :::; λk are different, it is called k-type. Since this notion was introduced, many works have been made in this area (see [3, 5]). This notion of finite-type immersion was naturally extended to that of pseudo-Riemannian manifolds in pseudo-Euclidean space and it was also applied to 2010 Mathematics Subject Classification. 53B25, 53B30. Key words and phrases. ruled surface, pointwise 1-type Gauss map, generalized 1-type Gauss map, conical surface of G-type. * supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (2016R1A2B1006974). 1 © 2018 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 September 2018 doi:10.20944/preprints201809.0407.v1 Peer-reviewed version available at Symmetry 2018, 10, 514; doi:10.3390/sym10100514 2 MIEKYUNG CHOI AND YOUNG HO KIM* smooth maps, particularly the Gauss map defined on submanifolds of Euclidean space or pseudo-Euclidean space ([1, 2, 3, 10, 11]). Regarding the Gauss map of finite-type, B.-Y. Chen and P. Piccini ([6]) initiated to study submanifolds of Euclidean space with finite-type Gauss map and classified compact surfaces with 1-type Gauss map, that is, ∆G = λ(G + C), where C is a constant vector and λ 2 R. Since then, quite a few works on ruled surfaces and ruled submanifolds with finite-type Gauss map in Euclidean space or pseudo-Euclidean space have been established ([1, 2, 3, 4, 7, 8, 9, 12, 13, 14, 15]). However, some surfaces including a helicoid and a right cone in Euclidean 3-space have an interesting property concerning the Gauss map: The helicoid in E3 parame- terized by x(u; v) = (u cos v; u sin v; av); a 6= 0 has the Gauss map and its Laplacian respectively given by 1 G = p (a sin v; −a cos v; u) a2 + u2 and 2a2 ∆G = G: (a2 + u2)2 The right (or circular) cone Ca with parametrization x(u; v) = (u cos v; u sin v; au); a ≥ 0 has the Gauss map 1 G = p (a cos v; a sin v; −1) 1 + a2 which satisfies 1 1 ∆G = (G + (0; 0; p ) u2 1 + a2 (cf. [4, 8]). The Gauss maps above are similar to 1-type, but it is not of 1-type Gauss map in the usual sense. Based upon such cases, B.-Y. Chen and the present authors defined the notion of pointwise 1-type Gauss map ([4]). Definition 1.1. A submanifold M in Em is said to have pointwise 1-type Gauss map if the Gauss map G of M satisfies ∆G = f(G + C) for some non-zero smooth function f and a constant vector C: In particular, if C is zero, then the Gauss map is said to be of pointwise 1-type of the first kind. Otherwise, it is said to be of pointewise 1-type of the second kind. Let p be a point of E3 and β = β(s) a unit speed curve such that p does not lie on β. A surface parametrized by x(s; t) = p + tβ(s) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 September 2018 doi:10.20944/preprints201809.0407.v1 Peer-reviewed version available at Symmetry 2018, 10, 514; doi:10.3390/sym10100514 EXTENSION OF EIGENVALUE PROBLEMS 3 is called a conical surface. A typical conical surface is a right cone and a plane. Let us consider a following example of a conical surface. Example 1.2. ([15]) Let M be a surface in E3 parameterized by x(s; t) = (s cos2 t; s sin t cos t; s sin t): Then, the Gauss map G can be obtained by 1 G = p (− sin3 t; (2 − cos2 t) cos t; − cos2 t): 1 + cos2 t After a considerably long computation, its Laplacian turns out to be ∆G = fG + gC for some non-zero smooth functions f; g and a constant vector C: The surface M is a kind of conical surfaces generated by a spherical curve β(t) = (cos2 t; sin t cos t; sin t) on the unit sphere S2(1) centered at the origin. Inspired by such an example, we would like to generalize the notion of pointwise 1-type Gauss map as follows: Definition 1.3. ([15]) The Gauss map G of a submanifold M in Em is of generalized 1-type if the Gauss map G of M satisfies ∆G = fG + gC (1.1) for some non-zero smooth functions f; g and a constant vector C: Especially we define a conical surface with generalized 1-type Gauss map. Definition 1.4. A conical surface with generalized 1-type Gauss map is called a conical surface of G-type. Remark 1.5. ([15]) A conical surface of G-type is constructed by the functions f, g and the constant vector C by solving the differential equations generated by (1.1). In the present paper, we classify a ruled surface with generalized 1-type Gauss map in E3 as a plane, a circular cylinder, a cylinder over a base curve of an infinite type generated by the given function f; g and the constant vector C, a helicoid, a right cone and a conical surface of G-type. 2. Preliminaries Let M be a surface of the 3-dimensional Euclidean space E3. The map G : M ! 2 3 2 S (1) ⊂ E which maps each point p of M to a point Gp of S (1) by identifying the unit normal vector Np to M at the point with Gp is called the Gauss map of the surface M; where S2(1) is the unit sphere in E3 centered at the origin. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 September 2018 doi:10.20944/preprints201809.0407.v1 Peer-reviewed version available at Symmetry 2018, 10, 514; doi:10.3390/sym10100514 4 MIEKYUNG CHOI AND YOUNG HO KIM* For the matrixg ~ = (~gij) consisting of the components of the metric on M, we denote byg ~−1 = (~gij) (resp. G ) the inverse matrix (resp. the determinant) of the matrix (~gij): Then the Laplacian ∆ on M is in turn given by 1 X @ p @ ∆ = −p G g~ij : (2.1) @xi @xj G i;j Now, we define a ruled surface M in the 3-dimensional Euclidean space E3: Let α = α(s) be a regular curve in E3 defined on an open interval I and β = β(s) a transversal vector field to α0(s) along α: Then the ruled surface M can be parameterized by x(s; t) = α(s) + tβ(s); s 2 I; t 2 R satisfying hα0; βi = 0 and hβ; βi = 1; where 0 denotes d=ds. The curve α is called the base curve and β the director vector field or ruling. In particular, M is said to be cylindrical if β is constant, or, non-cylindrical otherwise. Throughout this paper, we assume that all the geometric objects are smooth and all surfaces are connected unless otherwise stated. 3. Cylindrical ruled surfaces in E3 with generalized 1-type Gauss map In this section, we study the cylindrical ruled surfaces with generalized 1-type Gauss map in E3: Let M be a cylindrical ruled surface in E3. Without loss of generality, we assume that M is parameterized by x(s; t) = α(s) + tβ; where α(s) = (α1(s); α2(s); 0) is a plane curve parameterized by the arc-length s and β a constant vector, namely β = (0; 0; 1).
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