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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 398158, 6 pages
A Class of Weingarten Surfaces in Euclidean 3-Space
1School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China
2College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, China
Received 21 April 2013; Revised 31 July 2013; Accepted 19 August 2013
Academic Editor: Ondřej Došlý
Copyright © 2013 Yu Fu and Lan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The class of biconservative surfaces in Euclidean 3-space are defined in (Caddeo et al., 2012) by the equation for the mean curvature function and the Weingarten operator . In this paper, we consider the more general case that surfaces in satisfying for some constant are called generalized bi-conservative surfaces. We show that this class of surfaces are linear Weingarten surfaces. We also give a complete classification of generalized bi-conservative surfaces in .
Let be an isometric immersion of a submanifoldinto Euclidean (pseudo-Euclidean) space. We denote by,, andthe position vector, the mean curvature vector field, and the Laplace operator of, respectively, with respect to the induced metric. The submanifoldinis said to biharmonic if it satisfies the equation. According to the well-known Betrami’s formula, the biharmonic condition in Euclidean spaceis also known as the equation.
There is a well-known conjecture of Chen .
Chen’s Conjecture. The only biharmonic submanifolds of Euclidean spaces are the minimal ones.
This conjecture has been proved by some geometers for some special cases. For example, Chen proved that every biharmonic surface in the Euclidean 3-spaceis minimal. Hasanis and Vlachos  proved that every biharmonic hypersurface inis minimal, also see . For the general case, the conjecture is still open so far.
Very recently, Caddeo et al. introduced the notion of biconservative submanifolds in , which is a natural generalization of biharmonic submanifolds. It is interesting that biconservative submanifolds form a much bigger family of submanifolds including biharmonic submanifolds.
Recall a well-known result; see, for instance, .
Theorem A. Letbe a hypersurface with mean curvature vector. Then,is biharmonic if and only if the following equations hold: whereis the shape operator of the hypersurface with respect to the unit normal vector.
Following the definition of Caddeo et. al. in , a hypersurfacein an -dimensional Euclidean spaceis called biconservative if In general, a submanifold is biconservative if the divergence of the stress bienergy tensor vanishes.
In 1995, Hasanis and Vlachos, in , firstly studied biconservative hypersurfaces, which are also called -hypersurfaces. The authors gave a classification of biconservative hypersurfaces in Euclidean 3-spaces and 4-spaces. Recently, Caddeo et al.  investigated biconservative surfaces in the three-dimensional Riemannian space forms. Moreover, they proved that a biconservative surface in Euclidean 3-space is either a CMC (constant mean curvature) surface or a surface of revolution. This class of surfaces carry some interesting geometry. It was proved in  that the mean curvature functionof a non-CMC biconservative surface in a three-dimensional space formsatisfies the following relation: whereanddenote the Gaussian curvature and mean curvature of the surfaces, respectively. Clearly, the forementioned relation implies that all the biconservative surfaces in the Euclidean 3-space are linear Weingarten surfaces.
A surface is called a Weingarten surface if there exists the Jacobi equationbetween the Gaussian curvatureand the mean curvatureon the surface. Weingarten surfaces were introduced by Weingarten in 1861 in the context of the problem of finding all surfaces isometric to a given surface of revolution. Along the history, they have been of interest for geometers. There is a great amount of literature on Weingarten surfaces, beginning with works of Chern, Hartman, Winter, and Hopf in the fifties of the last century. For a long time, many geometers tried to look for examples of linear Weingarten surfaces, for example, see .
For surfaces in, the biconservative condition is equivalent to the equation Observe from  that the forementioned equation corresponds to a class of linear Weingarten surfaces, which include CMC surfaces and a family of surfaces of revolution. From the view of equation, a natural idea is to extend this class of surfaces in order to search for more examples of Weingarten surfaces of revolution. Hence, from the view of geometry, we propose to study the surfaces insatisfying a more general equation: We would like to call this new class of surfaces is generalized biconservative surfaces.
In this note, we focus on the equation and study this class of surfaces in. Precisely, we will prove that any generalized biconservative surface in Euclidean 3-space is a linear Weingarten surface satisfying a more general relationfor some constant. A local classification of generalized biconservative surfaces inis also obtained. Note that our method is slightly different from the method developed by Caddeo et al. in .
Let be an isometric immersion of a surfaceinto. Denote the Levi-Civita connections of andbyand, respectively. Letanddenote vector fields tangent to, and letbe a normal vector field. The Gauss and Weingarten formulas are given, respectively, by (cf. [8, 15]) where,are the second fundamental form and the shape operator. It is well known that the second fundamental formand the shape operatorare related by The Gauss and Codazzi equations are given respectively by whereis the curvature tensor of the Levi-Civita connection on. The mean curvature vector fieldand the Gauss curvature ofare given respectively, by As known from the Introduction, a surfaceinis biconservative if the mean curvature functionsatisfies Motivated by the above equation for biconservative surfaces in, we propose to the notion of generalized biconservative surfaces in.
Definition 1. A surface in Euclidean 3-spaceis called generalized biconservative surface if the mean curvature functionand the Weingarten operator satisfy a equation for some.
Note that this class of surfaces include all the biconservative surfaces as a subclass when.
Clearly, all of the CMC surfaces inare trivially generalized biconservative surfaces. This is also the case of biconservative surfaces. We are interested in the case of non-CMC surfaces in.
3. The Characterizations of Generalized Biconservative Surfaces
In this section, let us focus on the situation of non-CMC generalized biconservative surfaces in.
Suppose thaton any point. It follows from (12) thatis a principal direction andis the corresponding principal curvature. We can choose a local orthonormal frame fieldsuch thatis parallel to. Therefore, we have. Since (12) gives, it follows that. According to the Gauss equation, the Gaussian curvatureis given by which implies the following.
Theorem 2. The generalized biconservative surfaces inare linear Weingarten surfaces.
If we put, then. Using the remark above, the Codazzi equation reduces to Sinceis nonconstant, fromone has. So, the second equation of (14) yields. Moreover, the first equation of (14) implies that. Without the loss of generality, one assumes that .
According to the second equation of (14), one divides it into the following two cases.
Case A (). In this case, the surface is flat. Then, the second equation of (14) yieldsas well. Choose the local coordinates onasand. By applying the Gauss and Weingarten formulas (6) and (7) respectively, the immersion satisfies that
By solving the second and third equations of (15), we obtain that for a constant vectorand a curvein. Substitute (17) into the first equation of (15) and the first equation of (16), respectively. Combining these equations, we obtain a three-order differential equation as follows: In order to solve the above equation, we introduce two functions,(vector-valued function) andby puttingand. Note thatis the nonzero principal curvature andis not constant.
Denote by “” the derivative with respect to the new variable. With these symbols, (18) becomes whose solution is given by whereandare constant vectors in. Consequently, by a suitable translation, the immersionis given by Considering the metric of surfaces, we may choose that Hence, the surface can be expressed as Remark that the surface (23) is a cylinder, but not a circular cylinder, since the curvature of the curveis not constant.
Case B (). Let. Since, it follows from the second equation of (14) that. Therefore, there exist local coordinatesonsuch thatand. Then, the metric tensor ofis given by
Since, we have thatis a function depending only on the variable. Consequently, the Levi-Civita connectionis given by the expressions and the second fundamental form is given by Moreover, it follows from (25) and second fund form that the Gauss and Weingarten formulas (6) and (7) yield, respectively By (28), the compatibility condition of PDE system (27) is given by Integrating on (29), we obtain for some integral constant. Clearly, we havefor nonconstant function. Solving the second equation of (27), the immersionis given by for two vector-valued functionsandin.
Case B.1 (k = 0). In this case, (27) and (28) become It follows from (30) that for some constant, and. Substituting (31) into the first equation of (32), after a suitable translation, we obtain the immersion whereis another curve in. Substituting (35) into the third equation of (32) and applying (33), we have the following three-order differential equation: By (30), the solution of (36) is given by for constant vector,in. Hence, the immersion becomes In view of the metric (24), one can obtain that,,are mutual orthonormal and After choosing,, andas the immersion can be expressed as Note that this surface is a cone.
Case B.2 (k ≠ 0,1,2). Substituting (31) into the first and third equations of (27), we have which is equivalent to Substituting (29) and (30) into the previous equation in succession, we have In view of (44), the two sides of the equation have different variables, respectively. Hence, we have for some constant vectorin. Solving (45) gives for two constant vectorsandin. Looking at (31), we may assume that . In fact, the immersion can be rewritten as whereand. Hence, the immersion becomes Solving (46) gives for a constant vector.
One can compute from (49) that It follows from the above expressions and the metric (24) that Combining the previous expression (52) with (30) gives After a change of the variable, we can assume. Hence, the three vectors,,incan be chosen as Now, let us consider (50), which can be rewritten as whereis the derivative ofwith respect to. By applying (30), (55) becomes By solving (56) for some constant vectorin.
In summary, we have the following classification result.
Theorem 3. Letbe a nondegenerate generalized biconservative surface immersed in the 3-dimensional Euclidean space. Then, the immersionis either a CMC surface or locally given by one of the following three surfaces: (1) a cylinder given by where the function satisfies ; (2) a cone given by where ; (3) a surface of revolution given by where is defined as for .
4. Some Examples of Generalized Biconservative Surfaces
In this section, we give some examples of generalized biconservative surfaces (3) in Theorem 3, depending on different values for.
Example 1. In the case, the functioncan be integrated as for some integral constant. Hence, by a suitable translation, the non-CMC biconservative surface in(see also ) is given by whereis defined as See Figure 1.
Example 2. For, the functioncan be integrated as for some integral constantand. We have a non-CMC generalized biconservative surface (after a suitable translation) in, given by
The authors would like to thank the referee for giving very valuable suggestions and comments to improve the present paper. Lan Li author was supported by the National Natural Science Foundation of China (11201309), the Tianyuan Youth Fund of Mathematics in China (11126070), and the Natural Science Foundation of SZU (Grant no. 201111).
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