tóm tắt tiếng anh một số tính chất địa phương và toàn cục của mặt đối chiều hai trong không gian lorentz-minkowski

MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY DANG VAN CUONG SOME LOCAL AND GLOBAL PROPERTIES OF THE SURFACES OF CO-DIMENSION TWO IN LORENTZ-MINKOWSKI SPACE Speciality: Geometry and Topology Code: 62 46 10 01 A SUMMARY OF MATHEMATICS DOCTORAL THESIS NGHE AN – 2013 Work completed at: Vinh University Advisor: 1. Assoc. Prof. Dr. Doan The Hieu 2. Dr. Nguyen Duy Binh Reviewer 1: Reviewer 2: Reviewer 3: Thesis will be presented and protected at school - level thesis evaluating Council at: at……… ….h…………, date………mouth……….year Thesis can be found at: LIST OF POSTGRADUATE'S WORKS RELATED TO THE THESIS [1] Binh Ng. D, Cuong. D. V , Hieu. D. Th (2013), “Hyperplanarity of surfaces in four dimensional spaces”, pre-print. [2] Cuong. D. V (2008), “The flatness of spacelike surfaces of codimension two in 1  n  '', Vinh university Journal of science.,37 (2A), 11-20. [3] Cuong. D. V (2009), “The umbilicity of spacelike surfaces of codimension two in 1  n  '', Vinh university Journal of science., 38 (3A), 5-14. [4] Cuong. D. V (2010), “On general Gauss maps of surfaces”, East- West J. of Mathematics., 12 (2), 153-162. [5] Cuong. D. V (2012), “ r LS -valued Gauss maps and pacelike surfaces of revolution in 4 1  '', App. Math. Sci., 6 (77), 3845 - 3860. [6] Cuong. D. V and Hieu. D. Th (2012), “ r HS -valued Gauss maps and umbilic spacelike sufaces of codimension two”, submitted. [7] Cuong. D. V (2013), “Surfaces of Revolution with constant Gaussian curvature in four-Space”, Asian-Eur. J. Math., DOI 10.1142/S1793557113500216. [8] Cuong. D. V (2012), “The bi-normal fields on spacelike surfaces in 4 1  ”, submitted. 1 INTRODUCTION 1. Rationale 1.1 The study of the local and global properties of surfaces is one of basic problems of the differential geometry. The local properties are de- pendent on the choose the parametrization of surface while global prop- erties are not. It is well known, in the classical differential geometry, that the Gauss map gives us a useful method in order to study the surfaces of co- dimension one. The following notions are followed by the Gauss map: Gauss curvature; mean curvature; principal curvature,. . . . The Gauss map plays an important role in the study of the behaviour or geometric invari- ants of surfaces of co-dimension one. For example, using the property of principle curvature of surfaces we have: “ a regular surface in R 3 is umbilic if and only if it is either (a part of) sphere or (a part of) plan". For the global properties of surfaces, the Jacobi field along a geodesic plays an important role in the study the connection between the local and global properties. Using this method some global properties was showed. For example, “ a regular surface in R 3 is developable surface if and only if its Gauss curvature is zero". In this thesis, we would like to give some properties of the space-like surfaces of co-dimension two in Lorentz-Minkowski space that is similar the properties of surfaces in R 3 . 1.2 The Geometry of surfaces in R 4 has studied by some mathematical, for example: Romero Fuster, Izumiya, Pei, Little, Ganchev, Milousheva, Weiner, . . . . We can list some main results of this fields. In 1969, Little introduced some geometric invariants on the surfaces in R 4 , for instance ellipse curvature, in order to study the singularities on the manifolds of two dimensions. Authors, in this paper, showed that a surfaces whose all normal fields are bi-normal if and only if it is developable surface. In 1995, Mochida and et.al. showed that a surface admitting two bi-normal fields if and only if it is strictly locally convex. These results was ex- panded to surfaces of codimension two in R n+2 by them in 1999. These methods are used later by M.C. Romero-Fuster and F. S´anchez-Brigas (2002) to study the umbilicity of surfaces. In this paper they gave the connection between the following surfaces: ν-umbilical surfaces; surfaces admitting two orthogonal asymptotic directions anywhere; semi-umbilical 2 surfaces and surfaces with normal curvature identify zero. In 2010, Nu  no- Ballesteros and Romero-Fuster introduced the notion curvature locus, it is expansion of ellipse curvature for the surfaces of co-dimension two in R n+2 , to study the properties of the surfaces of co-dimension two. In this paper authors modify the results of surfaces in R 4 to the manifolds of co-dimension two in R n+2 . In this thesis we would like to extend the properties of both surfaces in R 4 and manifolds of co-dimension two in R n+2 to the spacelike sur- faces of co-dimension two in Lorrentz-Minkowski space. 1.3 In the recent years, some results of the study the spacelike surfaces of co-dimension two in Lorentz-Minkowski has published. We can list some main results of this field. Using the curvatures associated with a normal vector field, in 2004, Izumiya and et.al. showed that if a space- like surface of co-dimension two contained a pseudo-sphere then it is ν-umbilic, where ν is position field. For the reverse direction, by adding the condition of parallel of ν they showed that if the surface is ν-umbilic then it is contained in a pseudo-sphere. In this paper the authors also modified the notion ellipse curvature for spacelike surfaces of two di- mension in Lorrentz-Minkowski and showed the connection between the ν-umbilical surfaces and the semi-umbilical surfaces (the surfaces with ellipse curvature degenerating in to a segment). Since the normal plane of the spacelike surfaces of co-dimension two is timelike 2-plane, it is easy to show that it admits a orthonormal basic where one timelike and the other spacelike vector. Using sum and difference of two vector of this basic, in 2007 Izumiya and et.al. introduced the notion lightcone Gauss map and studied the flatness of the spacelike surfaces of co-dimension two. In this thesis, we would like to define a normal field on a spacelike surfaces of co-dimension two, as the Gauss map, it is usful to study the properties of surfaces. 1.4 Characterization of planarity, i.e. lying on a plane, or sphericity, i.e. lying on a sphere of space curves is one of the most natural problems in classical differential geometry. The planarity of a space curve is charac- terized by the torsion only. It is well-known that a curve is planar, i.e. containing in a plane, if and only if its torsion is zero, i. e. the bi-normal field is constant. More slight assumptions that imply the planarity of a curve in term of osculating planes was defined. In this thesis, we would like to define some sufficient conditions in order to a spacelike surface of co-dimension contained in a hyperplane. 3 1.5 The study of the special class of surfaces in the space, for example ruled surfaces, surfaces of revolution . . . , are also interested by Geome- tricians. Giving a method to study of properties of surfaces is useful if it can classify some these special surfaces. We would like to give some theorems classifying some special surfaces in Lorentz-Minkowski, for ex- ample maximal ruled surface, maximal surfaces of revolution, umbilical surfaces of revolution,. . . . For the above reasons, we have named the doctoral thesis: “ Some lo- cal and global properties of surfaces of co-dimension two in Lorentz- Minkowski space". 2. Aims In this thesis, we study the properties of surfaces of co-dimension two in Lorentz-Minkowski with the following purposes. (1) Introducing an effective tool to study the properties of spacelike surfaces of co-dimension two. (2) Studying the notions umbilic on the spacelike surfaces of co-dimension two, giving some classified results of ν-umbilical and umbilical spacelike surfaces of co-dimension two. (3) Studying the relationship between ν-umbilical and ν-planar space- like surfaces of co-dimension two. (4) Studying the conditions of hyper-planarity, i.e. contained a hyper- plane, of the surfaces in R 4 then extending to the spacelike surfaces in R 4 1 . (5) Applying the above theoretical results to some special surfaces in Lorentz-Minkowski R 4 1 , including ruled surfaces and surfaces of revolution. 3. Subject of the research The spacelike surfaces of co-dimension two; the tools for study space- like surfaces of co-dimension two; the properties of spacelike surfaces of co-dimension two in Lorentz-Minkowski space. 4 4. Scope of the research In this thesis, we study the local and global properties of the spacelike surfaces of co-dimension two , and some special surfaces in Lorentz- Minkowski space. 5. Methodology of the research We use theoretical methods. 6. Expected contributions to the knowledge of the research 6.1 The thesis has a contribution to the following problems for the space- like surfaces of co-dimension two in Lorentz-Minkowski space: (1) Giving two methods to define a differential normal vector field on the normal bundle of the spacelike surfaces of co-dimension two, one of them is spacelike and the other is lightlike. (2) Using the normal vector field ν (defined as above) to study the flatness on the surfaces and give some theorems expressing the properties of ν-flat surfaces. (3) Giving some theorems expressing the classification for the ν-umbilical surfaces contained in a pseudo-sphere and the umbilical surfaces. (4) Giving a standard to check if a normal field is binormal. Defining the relationship between the ν-umbilical surfaces and the ν-planar surfaces. (5) Giving some sufficient conditions in order to a surface in four-space (R 4 and R 4 1 ) is contained in a hyperplane. (6) Giving some theorems expressing the properties of some special surfaces in R 4 1 : maximal ruled surface; maximal surfaces of rev- olution (hyperbolic type and elliptic type); umbilical surfaces of revolution (hyperbolic type and elliptic type). Defining the number of binormal fields on ruled surfaces, surfaces of revolution (hyper- bolic type and elliptic type). Giving the equivalent conditions of 5 meridians for defining the number binormal fields on the General rotational surface whose meridians lie in two-dimensional plane. Defining the normal field ν on the ruled surface and surfaces of revolution such that they are ν-umbilic. 6.2 The thesis has a contribution for the students' references, students of master's degree standard and postgraduates in this field of the research. 7. Organization of the research 7.1. Overview of the research The basis knowledge is presented in the Chapter 1. This knowledge is useful for presenting the content of thesis. In the Chapter 2, we give two methods to define a couple normal vector field on the normal bun- dles of the spacelike surfaces, one of them is spacelike and the other is lightlike, then we use these couple normal vector fields to study the prop- erties of ν-umbilical and umbilical surfaces. Chapter 3 gives a standard to check if a normal field is binormal, studies the connection between the ν-umbilical anf ν-planar surfaces, defines the number binormal fields on the ν-umbilical surface. In the Chapter 3, we also study the conditions in order to a surface in four-space, R 4 and R 4 1 , is contained in a hyperplane. In Chapter 4, we study the properties of some special surfaces in R 4 1 , they are ruled surfaces and surfaces of revolution. 7.1.1 In the recent years, some Geometricians have studied the ν-umbilical surfaces of co-dimension two, for example Izumiya, Pei, Romero-Fuster,. . . . They supposed that there exists a normal field ν (spacelike, timelike or lightlike), introduced the curvatures associated with ν, then showed some properties of the ν-umbilical surfaces. However they can not show the existence of the normal field ν. This makes sense in theory but it is dif- ficult to the calculations on a specific surface. For a parametric surface, we now can not both define a normal field and control its causal char- acter (spacelike, timelike and lightlike). In Chapter 2 of this thesis, we give two methods to define a differential normal vector field on the nor- mal bundle of the spacelike surfaces of co-dimension two, one of them is spacelike and the other is lightlike. This is useful to practice on any specific parametric surface. An overview of this process is as follows: for each p ∈ M, the normal plane N p M of M at p is a 2-timelike plane, the intersection of this plane and the the hyperbolic n-space with center 6 v = (0, 0, . . . , 0, −1) and radius R = 1 (corresponding, lightcone) is a hyperbola (corresponding, two rays). For each r > 0, the hyperplane  x n+1 = r  intersects this hyperbola (corresponding, two rays) exactly two vector, denoted by n ± r (corresponding l ± r ). We can show that the nor- mal fields n ± r (corresponding, l ± r ) are spacelike (corresponding, lightlike) and smooth (Theorem 3.1.3), therefore we can define the curvatures asso- ciated with them in order to study the n ∗ r -umbilical and the l ∗ r -umbilical surfaces. Although n ∗ r is not parallel but if a surface is n ∗ r -flat then n ∗ r is constant, i.e. surface is contained in a hyperplane not contain the axis x n+1 (Theorem 2.1.5). We also give some necessary and sufficient conditions for a surface immersed in a hyperbolic to be (a part of) a hyper-sphere or a right hyper-sphere (Theorem 2.1.12). Since n ∗ r is not parallel, if M is n ∗ r -umbilic then in the general the n ∗ r -principal curva- ture is not constant. Theorem 2.1.14 gives some properties of a surface contained in a hyperbolic, n ∗ r -umbilic such that n ∗ r -principal curvature is constant. For a general surface, the condition of n ∗ r -umbilic and n ∗ r paral- lel is equivalent to surface is contained in the intersection of a hyperbolic and the hyperplane  x n+1 = c  (Theorem 2.1.15). We also give a condi- tion that is equivalent to n ∗ r is parallel (Theorem 2.1.16). As applications of n ∗ r -Gauss maps, we introduce some concrete examples with detailed computations in the section 2.1 (c). We obtain the similar results when use normal field l ∗ r to study the l ∗ r -umbilical surface. This is showed in Theorem 2.2.7, 2.2.8 and 2.2.9. Note that l ∗ r is useful for studying the surfaces contained in a de Sitter, where n ∗ r may be not favorable to study the notions umbilic. Connecting the properties of ν-umbilical surface and existence of parallel frame on a flat connection we give the properties of the umbilical surfaces in Theorem 2.3.2. 7.1.2. In Chapter 3, we give a standard to check if a normal field is bi- normal, define the relationship between the ν-umbilical surfaces and the ν-planar surfaces, study the sufficient conditions of the hyperplanarity of surfaces in R 4 and R 4 1 . In the first section of Chapter 3, using the vector product of three vectors, we give a standard to check if a normal field is binormal (Propo- sition 3.1.2). For the relationship between the ν-umbilical surfaces and ν-planar surfaces, Theorem 3.1.3 shows that a ν-umbilical surface (not ν-flat) admits at least one and at most two binormal fields, i.e. it is ν- planar. Moreover, we give the examples to show that there exist ν-planar surfaces are not ν-umbilical. It is mean that class of ν-umbilical surfaces is contained class of ν-planar surfaces and the reverse is not true. Propo- 7 sition 3.1.10 gives a necessary and sufficient condition for a surface to be totally planar. In the second section of Chapter 3, we study the sufficient condition for a surface in four-dimensional space be contained a hyperplane. Ex- ample 3.2.1 and 3.2.2 show that the improvement the planarity of curves in R 3 to the surfaces in four-space in general is not true. Using prop- erties of tangent plane, Proposition 3.2.5 gives the sufficient conditions for a surface in R 4 to be ν-flat. Developing this results to the properties of ν-hyperplanes, Proposition 3.2.6 gives the sufficient conditions for a surface in R 4 to be ν-planar. However, these conditions is not enough to a surface be contained a hyperplane. Adding the hypothesis, Proposition 3.2.7 gives four sufficient condition for a surface in R 4 to be contained in a hyperplane. However, these results hold also for spacelike surfaces in R 4 1 as well, no matter what the causality of the normal vector feld is. With similar proofs, we obtain the modified Propositions 3.2.5, 3.2.6 and 3.2.7 for spacelike surfaces in R 4 . The hyperplanarity of the spacelike surfaces coincide to the surfaces in R 4 when the nornal field is either spacelike or timelike. Perhaps, the most interesting case is the one where the normal field ν is lightlike. Often the appearance of lightlike vectors causes some interesting differences. Proposition 3.2.13 and 3.2.15 give some sufficient conditions for a spacelike surface to be contained a hy- perplane, but it is only true for the lightlike normal fields. We also give some interesting examples in order to unravel the results in this section. In the end of Chapter 3, we give some great examples in order to illuminate the results in the this chapter. 7.1.3. The study of the properties of special surfaces, for example ruled surfaces or surfaces of revolution, is always interested to the geometri- cians. As application the results in the Chapter 2 and 3, Chapter 4 studies the properties of ruled spacelike surfaces and spacelike surfaces of revo- lution in R 4 1 . Propositon 4.1.3 defines the number binormal direction at each poit on a ruled surface. Proposition 4.1.5 shows that the necessary and sufficient condition for a ruled spacelike surface to be maximal is it is contained a timelike hyperplane and maximal, a ruled spacelike surface is ν-umbilic iff it is umbilic. For the surfaces of revolution in R 4 1 , we consider two type of surfaces that are the orbit of a curve by rotating it around a plane and the obit of a plane curve rotated around both two planes. Theorem 4.2.4 and Theorem 4.2.10, by using l ± r -Gauss maps, give the parametrization of umbilical spacelike surfaces of revolution (hyper- bolic type and elliptic type). Applying l ± r -Gauss maps, Theorem 4.2.6, [...]... sections Section 1.1 presents the basic notions about Lorentz-Minkowski Section 1.2 introduces the tools used in the thesis, it has two following subsection: Subsection a) presents the notions curvatures associated with a normal vector field and the notions of surfaces; Subsection b) presents the notion of ellipse curvature for spacelike surfaces in Lorentz-Minkowski space Chapter 2 studies the notions... of the surfaces of revolution (of hyperbolic type and ellipse type) and surface whose meridians lie in two-dimension space in R4 1 10 Chapter 1 Basis knowledge 1.1 The Lorentz-Minkowski space n+1 is the (n + 1)Definition 1.1.1 The Lorentz-Minkowski space R1 n+1 = {(x , , x dimensional vector space R 1 n+1 ) : xi ∈ R; i = 1, n+ 1} with the pseudo scalar product given by n xi yi − xn+1 yn+1 x,... order to define the shapes of them 2 Introducing the tools to study th timelike and lightlike surfaces of co-dimension two in Lorentz-Minkowski space 3 Introducing the notion of th helicoid surface in R4 and studying 1 the properties of it 4 Study the manifolds with density in Lorentz-Minkowski space ... Note that a parallel normal vector field has constant length In Euclidean spaces, if the surface is oriented, locally the normal vector field can be assumed to be unitary without loss of generality In Lorentz-Minkowski spaces, this can not always be the case, because the causal character of a normal vector field ν may vary from point to point Let see Example 3.3.2 If the causal character of a normal... The notion of ellipse curvature of surface in R4 was introduced by Little and followed by Izumiya for spacelike surfaces in R4 1 11 Conclusions of Chapter 1 In this chapter, we briefly introduced the Lorentz-Minkowski space, represented the notions of curvatures associated with a normal field of the surfaces of co-dimension two and the ellipse curvature of surface in R4 These results will used to . lie in two-dimension space in R 4 1 . 10 Chapter 1 Basis knowledge 1.1 The Lorentz-Minkowski space Definition 1.1.1. The Lorentz-Minkowski space R n+1 1 is the (n + 1)- dimensional vector space. thesis, we would like to give some properties of the space-like surfaces of co-dimension two in Lorentz-Minkowski space that is similar the properties of surfaces in R 3 . 1.2 The Geometry of. space. 1.3 In the recent years, some results of the study the spacelike surfaces of co-dimension two in Lorentz-Minkowski has published. We can list some main results of this field. Using the curvatures