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

We would like to give sometheorems classifying some special surfaces in Lorentz-Minkowski, for ex-ample maximal ruled surface, maximal surfaces of revolution, umbilicalsurfaces of revolu

DANG VAN CUONG

SOME LOCAL AND GLOBAL PROPERTIES OF THE SURFACES OF CO-DIMENSION TWO IN

Work completed at: Vinh University

Advisor:

1 Assoc Prof Dr Doan The Hieu

2 Dr Nguyen Duy Binh

at……… ….h…………, date………mouth……….year

Thesis can be found at :

[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  '', Vinh university Journal of science.,37 n1

[5] Cuong D V (2012), “ LS -valued Gauss maps and pacelike r

surfaces of revolution in 41'', App Math Sci., 6 (77), 3845 -

3860

[6] Cuong D V and Hieu D Th (2012), “ HS -valued Gauss maps r

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 41”, submitted

1 Rationale

1.1 The study of the local and global properties of surfaces is one ofbasic 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 Gaussmap 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 mapplays 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 R3 isumbilic 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 geodesicplays an important role in the study the connection between the local andglobal properties Using this method some global properties was showed.For example, “ a regular surface in R3 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-likesurfaces of co-dimension two in Lorentz-Minkowski space that is similarthe properties of surfaces in R3

1.2 The Geometry of surfaces in R4 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, Littleintroduced some geometric invariants on the surfaces in R4, for instanceellipse curvature, in order to study the singularities on the manifolds oftwo dimensions Authors, in this paper, showed that a surfaces whose allnormal 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-normalfields if and only if it is strictly locally convex These results was ex-panded to surfaces of codimension two in Rn+2 by them in 1999 Thesemethods 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 theconnection between the following surfaces: ν-umbilical surfaces; surfacesadmitting two orthogonal asymptotic directions anywhere; semi-umbilical

surfaces and surfaces with normal curvature identify zero In 2010, NueBallesteros and Romero-Fuster introduced the notion curvature locus, it

no-is expansion of ellipse curvature for the surfaces of co-dimension two in

Rn+2, to study the properties of the surfaces of co-dimension two Inthis paper authors modify the results of surfaces in R4 to the manifolds

of co-dimension two in Rn+2

In this thesis we would like to extend the properties of both surfaces

in R4 and manifolds of co-dimension two in Rn+2 to the spacelike faces of co-dimension two in Lorrentz-Minkowski space

sur-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 listsome main results of this field Using the curvatures associated with anormal 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 isν-umbilic, where ν is position field For the reverse direction, by addingthe condition of parallel of ν they showed that if the surface is ν-umbilicthen it is contained in a pseudo-sphere In this paper the authors alsomodified the notion ellipse curvature for spacelike surfaces of two di-mension in Lorrentz-Minkowski and showed the connection between theν-umbilical surfaces and the semi-umbilical surfaces (the surfaces withellipse curvature degenerating in to a segment) Since the normal plane

of the spacelike surfaces of co-dimension two is timelike 2-plane, it iseasy to show that it admits a orthonormal basic where one timelike andthe other spacelike vector Using sum and difference of two vector of thisbasic, in 2007 Izumiya and et.al introduced the notion lightcone Gaussmap and studied the flatness of the spacelike surfaces of co-dimensiontwo

In this thesis, we would like to define a normal field on a spacelikesurfaces of co-dimension two, as the Gauss map, it is usful to study theproperties 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 inclassical 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-normalfield is constant More slight assumptions that imply the planarity of acurve in term of osculating planes was defined

In this thesis, we would like to define some sufficient conditions inorder to a spacelike surface of co-dimension contained in a hyperplane

1.5 The study of the special class of surfaces in the space, for exampleruled 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 sometheorems classifying some special surfaces in Lorentz-Minkowski, for ex-ample maximal ruled surface, maximal surfaces of revolution, umbilicalsurfaces 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

(3) Studying the relationship between ν-umbilical and ν-planar like surfaces of co-dimension two

space-(4) Studying the conditions of planarity, i.e contained a plane, of the surfaces in R4then extending to the spacelike surfaces

hyper-in R41

(5) Applying the above theoretical results to some special surfaces inLorentz-Minkowski R41, including ruled surfaces and surfaces ofrevolution

3 Subject of the research

The spacelike surfaces of co-dimension two; the tools for study like surfaces of co-dimension two; the properties of spacelike surfaces ofco-dimension two in Lorentz-Minkowski space

space-4 Scope of the research

In this thesis, we study the local and global properties of the spacelikesurfaces 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 like surfaces of co-dimension two in Lorentz-Minkowski space:

space-(1) Giving two methods to define a differential normal vector field onthe 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 theflatness on the surfaces and give some theorems expressing theproperties of ν-flat surfaces

(3) Giving some theorems expressing the classification for the ν-umbilicalsurfaces contained in a pseudo-sphere and the umbilical surfaces.(4) Giving a standard to check if a normal field is binormal Definingthe relationship between the ν-umbilical surfaces and the ν-planarsurfaces

(5) Giving some sufficient conditions in order to a surface in four-space(R4 and R41) is contained in a hyperplane

(6) Giving some theorems expressing the properties of some specialsurfaces in R41 : maximal ruled surface; maximal surfaces of rev-olution (hyperbolic type and elliptic type); umbilical surfaces ofrevolution (hyperbolic type and elliptic type) Defining the number

of binormal fields on ruled surfaces, surfaces of revolution bolic type and elliptic type) Giving the equivalent conditions of

(hyper-meridians for defining the number binormal fields on the Generalrotational surface whose meridians lie in two-dimensional plane.Defining the normal field ν on the ruled surface and surfaces ofrevolution 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 givetwo 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 islightlike, 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 theν-umbilical anf ν-planar surfaces, defines the number binormal fields onthe ν-umbilical surface In the Chapter 3, we also study the conditions inorder to a surface in four-space, R4 and R41, is contained in a hyperplane

In Chapter 4, we study the properties of some special surfaces in R41, theyare ruled surfaces and surfaces of revolution

surfaces of co-dimension two, for example Izumiya, Pei, Romero-Fuster, .They supposed that there exists a normal field ν (spacelike, timelike orlightlike), introduced the curvatures associated with ν, then showed someproperties of the ν-umbilical surfaces However they can not show theexistence 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 acter (spacelike, timelike and lightlike) In Chapter 2 of this thesis, wegive 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

char-is spacelike and the other char-is lightlike Thchar-is char-is useful to practice on anyspecific parametric surface An overview of this process is as follows:for each p ∈ M, the normal plane NpM of M at p is a 2-timelike plane,the intersection of this plane and the the hyperbolic n-space with center

v = (0, 0, , 0, −1) and radius R = 1 (corresponding, lightcone) is ahyperbola (corresponding, two rays) For each r > 0, the hyperplane

xn+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 theaxis xn+1 (Theorem 2.1.5) We also give some necessary and sufficientconditions 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 ture is not constant Theorem 2.1.14 gives some properties of a surface

curva-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 lel is equivalent to surface is contained in the intersection of a hyperbolicand the hyperplanexn+1 = c (Theorem 2.1.15) We also give a condi-

paral-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 detailedcomputations in the section 2.1 (c) We obtain the similar results whenuse normal field l∗r to study the l∗r-umbilical surface This is showed inTheorem 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 studythe notions umbilic Connecting the properties of ν-umbilical surface andexistence of parallel frame on a flat connection we give the properties ofthe umbilical surfaces in Theorem 2.3.2

bi-normal, define the relationship between the ν-umbilical surfaces and theν-planar surfaces, study the sufficient conditions of the hyperplanarity ofsurfaces in R4 and R41

In the first section of Chapter 3, using the vector product of threevectors, 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 andν-planar surfaces, Theorem 3.1.3 shows that a ν-umbilical surface (notν-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-planarsurfaces 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-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 conditionfor 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 R3 to the surfaces in four-space in general is not true Using erties of tangent plane, Proposition 3.2.5 gives the sufficient conditionsfor a surface in R4 to be ν-flat Developing this results to the properties

prop-of ν-hyperplanes, Proposition 3.2.6 gives the sufficient conditions for asurface in R4 to be ν-planar However, these conditions is not enough to

a surface be contained a hyperplane Adding the hypothesis, Proposition3.2.7 gives four sufficient condition for a surface in R4 to be contained

in a hyperplane However, these results hold also for spacelike surfaces

in R41 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 and3.2.7 for spacelike surfaces in R4 The hyperplanarity of the spacelikesurfaces coincide to the surfaces in R4 when the nornal field is eitherspacelike or timelike Perhaps, the most interesting case is the one wherethe normal field ν is lightlike Often the appearance of lightlike vectorscauses some interesting differences Proposition 3.2.13 and 3.2.15 givesome sufficient conditions for a spacelike surface to be contained a hy-perplane, but it is only true for the lightlike normal fields We also givesome interesting examples in order to unravel the results in this section

In the end of Chapter 3, we give some great examples in order toilluminate the results in the this chapter

surfaces or surfaces of revolution, is always interested to the cians As application the results in the Chapter 2 and 3, Chapter 4 studiesthe properties of ruled spacelike surfaces and spacelike surfaces of revo-lution in R4

geometri-1 Propositon 4.1.3 defines the number binormal direction ateach poit on a ruled surface Proposition 4.1.5 shows that the necessaryand 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 R41, weconsider 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 twoplanes Theorem 4.2.4 and Theorem 4.2.10, by using l±r-Gauss maps, givethe parametrization of umbilical spacelike surfaces of revolution (hyper-bolic type and elliptic type) Applying l±r-Gauss maps, Theorem 4.2.6,

Theorem 4.2.12 the parametrization of maximal spacelike surfaces of olution (hyperbolic type and elliptic type) Proposition 4.2.8 and 4.2.14show that the surfaces of revolution (hyperbolic type and elliptic type)admit exactly two binormal fields and there exists only one normal field

rev-ν such that it is rev-ν-umbilic Theorem 4.2.16 shows that the constant erty of the Gaussian curvature of surfaces of revolution of hyperbolic typeand elliptic type are coincident, moreover it depends only on the radius

prop-of rotation We also show that the number prop-of bi-normal fields on therotational spacelike surface whose meridians lie in two-dimension spaceare depended on the properties of its meridian and give the corespondentexamples

7.2 The organization of the research

The thesis is carried out in four chapters Besides, the thesis hasthe statement of authorship, the acknowledgements, the introduction, theconclusion and recommendations, the list of postgraduate's works related

to the thesis, the bibliography, and the index

Chapter 1 presents the basis knowledge including two sections tion 1.1 presents the basic notions about Lorentz-Minkowski Section 1.2introduces the tools used in the thesis, it has two following subsection:Subsection a) presents the notions curvatures associated with a normalvector field and the notions of surfaces; Subsection b) presents the notion

Sec-of ellipse curvature for spacelike surfaces in Lorentz-Minkowski space.Chapter 2 studies the notions of umbilic (ν-umbilic) on the spacelikesurfaces of co-dimension two, it includes the following: Section 2.1 in-

troduces the notion n±r-Gauss maps and its applications to the study theν-umbilical surfaces; Section 2.2 introduces the notion l±r-Gauss mapsand its applications to the study the ν-umbilical surfaces; Section 2.3classifies the umbilical surfaces Almost results in this chapter are local,

but the properties of n±r-flat and l±r-flat are global

Chapter 3 studies the properties of the ν-planar surfaces and planarity of surfaces in four dimensional space, it includes the following:Section 3.1 we give a standard to check if a normal field is binormal,define the relationship between the ν-umbilical surfaces and the ν-planarsurfaces; Section 3.2 presens the study of the sufficient conditions of thehyperplanarity of surfaces in R4 and R4

hyper-1; Section 3.3 gives some examplesabout the ν-planar surfaces and same examples related to the results ofchapter The results in Section 3.1 are local, and the results in Section

3.2 are global.

Chapter 4 presents the results about properties of the ruled surfacesand surfaces of revolution in R41, it includes the following: Section 4.1studies the properties of the ruled surfaces in R41; Section 4.2 studies theproperties of the surfaces of revolution (of hyperbolic type and ellipsetype) and surface whose meridians lie in two-dimension space in R41

Chapter 1

Basis knowledge1.1 The Lorentz-Minkowski space

1)-dimensional vector space Rn+1= {(x1, , xn+1) : xi∈ R; i = 1, n+1} with the pseudo scalar product given by

In this thesis, a spacelike surface of codimension two M is mean that

a cnnected, oriented (n − 1)-dimensional manifold imbedding in to Rn+11such that for each p ∈ M the tangent space TpM is spacelike Locally

M is given by an immersion X : U → Rn+11 , where U is a connectedopen domain in Rn−1 and (u1, u2, , un−1) is the local coordinates

1.2 The curvatures of surfaces of co-dimension two inRn+11

a) The normal curvatures associated with a normal field

In this section, we introduce the notions of curvatures associated with

a normal field ν, we then give the notions of: ν-flat surface; ν-umbilicalsurface; ν-planar surface; umbilical surface; totally planar surface; bi-normal field; asymptotic field; osculating hyperplanes; These are theobjects studied in this thesis

b) Ellipse curvature

The notion of ellipse curvature of surface in R4 was introduced byLittle and followed by Izumiya for spacelike surfaces in R41