Total Mean Curvature and Submanifolds of Finite Type
... Then the fundamental equations for M are (cf. [16]) ...
... Using Equations (16) and (22) in Equation (21), we have ...
... Proof. Let M be a compact and connected minimal hypersurface M of S n+1 with We have on using Equation (16) ...
Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2<n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n>2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.
... Hence the Gauss map n satisfies, (see [19]) ...
... On account of (19), (20), (21), (22) and (4), we conclude that (−1) k 3 k sin 2k+2 ψ (2r) k+1 (δ cos ψ) 2k+1 h + 1 δ 2k+1 cos 2k ψ F k+1 (cos ψ, sin ψ) + (−1) k−1 ρ 1 3 k−1 sin 2k ψ (2r) k (δ cos ψ) 2k−1 h + ρ 1 1 δ 2k−1 cos 2k−2 ψ F k (cos ψ, sin ψ) + · · · − ρ k−1 3 sin 4 ψ 4r 2 δ 3 cos 3 ψ h + ρ k−1 1 δ 3 cos 2 ψ F 2 (cos ψ, sin ψ) + ρ k sin 2 ψ 2rδ cos ψ h + ρ k 1 δ F 1 (cos ψ, sin ψ) = 0 . (23) For simplicity, equation (23) can be written as follows 3 k sin 2k+2 ψ (2r) k+1 cos ψ h = Q 1 (cos ψ, sin ψ)h + Q 2 (cos ψ, sin ψ)b, ...
In this study, we continue the classification of finite type Gauss map surfaces in the 3-dimensional Euclidean space E 3. To do this, we investigate an important family of surfaces, namely, tubular surfaces in E 3. We show that the Gauss map of a tubular surface is of an infinite type regarding the second fundamental form.
... Chen in the early 1970s and since then, it has become a source of interest for many researchers in this field. The reader can refer to [10] for more details. In the framework of this kind of study the first-named author with S. Stamatakis have given in [18] a new generalization to this area of study by giving a similar definition of surfaces of finite type. ...
... The second differential parameter of Beltrami with respect to the fundamental form J = I, II, III of S is defined by [10] ...
In this paper, we consider surfaces of revolution in the 3-dimensional Euclidean space E3 with nonvanishing Gauss curvature. We introduce the finite Chen type surfaces with respect to the third fundamental form of the surface. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.
... Later in [22] O. Garay generalized Chen's condition studied surfaces for which surfaces in E 3 satisfy the condition ∆ I r = Ar ( ‡) where A ∈ R 3×3 or ∆ I N = AN where N is the Gauss map of S. It was shown that a surface S in E 3 satisfies ( ‡) if and only if it is an open part of a minimal surface, a sphere, or a circular cylinder. Surfaces satisfying ( ‡) are said to be of coordinate finite type, meanwhile Surfaces satisfying ∆ I N = AN are said to be of coordinate finite type Gauss map. ...
In this paper, we consider tubes in the Euclidean 3-space whose Gauss map N is of coordinate finite II-type, i.e., the position vector N satisfies the relation , where is the Laplace operator with respect to the second fundamental form I of the surface and is a square matrix of order 3. We show that circular cylinders are the only class of surfaces mentioned above of coordinate finite I-type Gauss map.
... There have been extensive studies on biharmonic maps. We refer to [5,11,13,15,20,22] for an introduction to this topic. We observe that, obviously, any harmonic map is trivially biharmonic and an absolute minimum for the bienergy. ...
The flat torus admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere given by , where is the minimal Clifford torus and is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion . After, we shall study in the detail the kernel of the generalised Jacobi operator . We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper we shall analyse the specific contribution of to the biharmonic index and nullity of . In this context, we shall study a more general composition , where , , , is a minimal immersion and is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of is nonnegatively defined on . Then we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the p-harmonic index and nullity of . In the final section we compare our general results with those which can be deduced from the study of the equivariant second variation.
The purpose of this talk is to present six research topics in differential geometry in which the position vector field play important roles.
We investigate biharmonic Ricci soliton hypersurfaces (Mn, g, 𝜉, ⋋) whose potential field 𝜉 satisfies certain conditions. We obtain a result based on the average scalar curvature of the compact Ricci soliton hypersurface Mn, where 𝜉 is a general vector field. Then we prove that there are no proper biharmonic Ricci soliton hypersurfaces in the Euclidean space En+1 provided that the potential field 𝜉 is either a principal vector in grad H⊥ or ξ=gradHgradH.
Minimal surfaces with constant Gauss curvature in real space forms are studied.
The main purpose of this note is to give a survey of some recent developments for submanifolds with parallel mean curvature and for total mean curvature.
