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,
8
Theorem 4.2.12 the parametrization of maximal spacelike surfaces of rev-
olution (hyperbolic type and elliptic type). Proposition 4.2.8 and 4.2.14
show that the surfaces of revolution (hyperbolic type and elliptic type)
admit exactly two binormal fields and there exists only one normal field
ν such that it is ν-umbilic. Theorem 4.2.16 shows that the constant prop-
erty of the Gaussian curvature of surfaces of revolution of hyperbolic type
and elliptic type are coincident, moreover it depends only on the radius
of rotation. We also show that the number of bi-normal fields on the
rotational spacelike surface whose meridians lie in two-dimension space
are depended on the properties of its meridian and give the corespondent
examples.

7.2. The organization of the research
The thesis is carried out in four chapters. Besides, the thesis has
the statement of authorship, the acknowledgements, the introduction, the
conclusion 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. Sec-
tion 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 umbilic (ν-umbilic) on the spacelike
surfaces 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 the
ν-umbilical surfaces; Section 2.2 introduces the notion l
±
r
-Gauss maps
and its applications to the study the ν-umbilical surfaces; Section 2.3
classifies 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 hyper-
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-planar
surfaces; Section 3.2 presens the study of the sufficient conditions of the
hyperplanarity of surfaces in R
4
and R
4
1
; Section 3.3 gives some examples
about the ν-planar surfaces and same examples related to the results of
chapter. The results in Section 3.1 are local, and the results in Section
9
3.2 are global.
Chapter 4 presents the results about properties of the ruled surfaces
and surfaces of revolution in R
4
1
, it includes the following: Section 4.1
studies the properties of the ruled surfaces in R
4
1
; Section 4.2 studies the
properties of the surfaces of revolution (of hyperbolic type and ellipse
type) and surface whose meridians 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 R
n+1
= {(x
1
, . . . , x
n+1
) : x
i
∈ R; i = 1, . . . n+
1} with the pseudo scalar product given by
x, y =
n

i=1
x
i
y
i
− x
n+1
y
n+1
where x = (x

1
, . . . , x
n+1
); y = (y
1
, . . . , y
n+1
) ∈ R
n+1
.
Definition 1.1.2. A vector x ∈ R
n+1
1
is called
1. spacelike if x, x > 0 or x = 0,
2. imelike if x, x < 0,
3. lightlike if x, x = 0 vµ x = 0.
In this thesis, a spacelike surface of codimension two M is mean that
a cnnected, oriented (n −1)-dimensional manifold imbedding in to R
n+1
1
such that for each p ∈ M the tangent space T
p
M is spacelike. Locally
M is given by an immersion X : U → R
n+1
1
, where U is a connected
open domain in R
n−1

and (u
1
, u
2
, . . . , u
n−1
) is the local coordinates.
1.2 The curvatures of surfaces of co-dimension two in R
n+1
1
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-umbilical
surface; ν-planar surface; umbilical surface; totally planar surface; bi-
normal field; asymptotic field; osculating hyperplanes;. . . . These are the
objects studied in this thesis.
b) Ellipse curvature
The notion of ellipse curvature of surface in R
4
was introduced by
Little and followed by Izumiya for spacelike surfaces in R
4
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
R

4
1
. These results will used to study the properties of the surfaces in the
following chapters.
Chapter 2
The HS
r
-, LS
r
-valued Gauss maps and the
properties of ν-umbilical surfaces
2.1 The HS
r
-valued Gauss maps and the n
±
r
-umbilical sur-
faces
a) The n
±
r
-Gauss maps
Lemma 2.1.1 ([5],[9]). Let Π be a timelike 2-plane passing through
the origin. Then, for a given r > 0, {x = (x
1
, x
2
, . . . , x
n+1
) ∈ Π ∩

H
n
+
(v, 1) | x
n+1
= r} contains exactly two vectors.
Using Lemma 2.1.1 we have the notion of n
±
r
-Gauss maps.
Definition 2.1.2 ([5],[9]).Let M be a surface of co-dimension two in
R
n+1
1
, the maps
n
±
r
: M → HS
r
:= H
n
+
(v, 1) ∩{x
n+1
= r}
p → n
±
r
(p).

are n
±
r
-Gauss maps of M.
Theorem 2.1.3 ([5],[9]). n
±
r
-Gauss maps are smooth.
n
±
r
are spacelike M.
From now on, we will use the symbol “∗" to instead of either “ + " or “
- " in n
±
r
.
b) The n
∗
r
- flat surfaces of co-dimension two
Theorem 2.1.5 ([9]). The following statements are equivalent:
1. there exists r > 0 such that M is n
∗
r
-flat;
2. there exists r > 0 such that n
∗
r
is constant;

12
3. there exists a spacelike vector a = (a
1
, a
2
, . . . , a
n
, a
n+1
), a
n+1
=
0 and a real number c such that M ⊂ HP
a
(c).
c) The n
∗
r
-umbilical surfaces of co-dimension two
Theorem 2.1.12 ([9]). Let M be a surface of co-dimension two immersed
in H
n
+
(0, R). The following statements are equivalent:
1. there exists r > 0 such that M is n
∗
r
-umbilic;
2. M is umbilic;
3. M is contained in a hyperplane.

Theorem 2.1.14 ([9]). Let M be a surface of co-dimension two immersed
in H
n
+
(0, R). The following statements are equivalent:
1. M is contained in

x
n+1
= c

;
2. n
∗
r
is parallel, for each r > 0;
3. there exist two linearly independent everywhere and parallel nor-
mal fields n
∗
r
1
, n
∗
r
2
;
4. there exists r > 0 such that A
n
∗
r

= −αid, where α is constant.
Theorem 2.1.15 ([9]). Let M be a surface of co-dimension two. The
following statements are equivalent:
1. M is n
∗
r
-umbilic for some non-constant and parallel normal vector
field n
∗
r
;
2. M is contained in the hyperplane

x
n+1
= c

.
Theorem 2.1.16 ([9]). Suppose that M is n
∗
r
-umbilic for some r > 0, i.e.
A
n
∗
r
= −αid. Then n
∗
r
is constant if and only if

∂
∂u
j


∂
∂u
i
n
∗
r

T

=
∂
∂u
i


∂
∂u
j
n
∗
r

T

(0.1)

for every i, j ∈ {1, 2, . . . , n − 1}.
d) The examples of ν-umbilical surfaces in R
4
1
13
In this section, we give some great examples in order to show that:
there exist the n
∗
r
-umbilical surfaces not umbilical; there exist the umbil-
ical surfaces whose principal curvatures are not constant; there exist the
ν-umbilical surfaces not bot n
+
r
-umbilic and n
−
r
-umbilic, for any r > 0;
there exist surfaces contained in a pseudo-sphere, of course it are X-
umbilic, not umbilic; there exist the ν-flat surfaces not contained in any
hyperplane.
2.2 The LS
r
-valued Gauss maps and the l
±
r
-umbilical sur-
faces
a) The l
±

r
-Gauss maps
Set LS
r
= LC
∗
∩HP (v, 0) where v = (0, 0, . . . , 0, r). The l
±
r
-Gauss
maps will be introduced by using the following Lemma.
Lemma 2.2.1 ([6]). Let Π be a timelike 2-plane passing through the
origin. Then the set
Π ∩ LS
r
contains exactly two vectors.
Since the normal planes of M are timelike 2-plane, we have the following
definition.
Definition 2.2.2 ([6]). The following maps
l
±
r
: M → LS
r
, p → l
±
r
(p)
are called l
±

r
-Gauss maps.
b) The l
∗
r
-umbilical surfaces of co-dimension two
The obstacles when using n
±
r
-Gauss maps to study the properties of
the surfaces on de Sitter n-space S
n
1
(a, R) are n
±
r
and the position direc-
tion may be dependent linear. The l
±
r
-Gauss maps can solve this problem.
Theorem 2.2.7 ([6]). Let M be a surface of co-dimension two contained
in a de Sitter n-space with center a and radius R, S
n
1
(a, R). The fol-
lowing statements are then equivalent:
(1) M is l
∗
r

-umbilic;
(2) M is umbilic;
(3) M is contained in a hyperplane.
Theorem 2.2.8 ([6]). Let M be a surface of co-dimension two contained
in S
n
1
(a, R). The following statements are then equivalent:
14
1. M is contained in

x
n+1
= c

;
2. l
∗
r
is parallel, for every r > 0;
3. there exist two parallel normal fields l
∗
r
1
, l
∗
r
2
;
4. there exists r > 0 such that A

l
∗
r
= −λid, where λ is constant.
Theorem 2.2.9 ([6]). Let M be a surface of co-dimension two in R
n+1
1
.
The following statements are equivalent:
1. there exists r > 0 such that l
∗
r
is parallel but not constant, and M
is l
∗
r
-umbilic;
2. M is contained in a hypersphere S
n
1
(a, R) ∩

x
n+1
= c

.
2.3 The umbilical surfaces
HÖ qu¶ 2.3.1. If M is totally umbilic, then for any p ∈ M there exist a
local neighbourhood U

p
⊂ M of p and a parallel normal frame u, v on
U
p
.
Theorem 2.3.2 ([6]). Let M be a surface of co-dimension two. We then
have:
1. if either u is spacelike or both u and v are lightlike on U
p
, then U
p
is contained in the intersection of a hyperbolic and a hyperplane;
2. if u is timelike on U
p
, then U
p
is contained in the intersection of a
de Sitter and a hyperplane.
Conclusions of Chapter 2
In this chapter, we get the following results:
(1) Giving a method to define a couple spacelike, differential normal
fields, n
±
r
, on a spacelike surface of co-dimension two, then by us-
ing these normal field, we give some properties of the nu-umbilical
surfaces specially surfaces are contained in a hyperbolic.
(2) Giving a method to define a couple timelike, differential normal
fields, l
∗

r
, on a spacelike surface of co-dimension two, then by
using these normal field, we give some properties of the ν-umbilical
surfaces specially surfaces are contained in a de Sitter.
(3) Giving the characters of the umbilical surfaces.
The results of Chapter 2 was published on [5],[6] vµ [9].
15
Chapter 3
The properties of the ν-planar surfaces in R
4
1
3.1 The relationship between the ν-umbilical surfaces and
the ν-planar surfaces
Proposition 3.1.2 ([8]). Let ν be a normal field on M. Then ν is binor-
mal if and only if either ν ∧ ν
u
∧ ν
v
= 0 or 0 = ν ∧ ν
u
∧ ν
v
is parallel
ti T
p
M.
There exist ν-planar surfaces but not ν-umbilic, this will showed in the
en of this chap trer. However, the following theorem shows that the class
of ν-umbilical surfaces is contained in the class of ν-planar surfaces.
Theorem 3.1.3 ([2]). If M is ν-umbilic (not ν-flat) then M admits at

least one and at most two binormal fields. Then M admits only one
binormal field if and only if M is umbilic.
Proposition 3.1.10 ([2]). M is planar if and only if there exist two nor-
mal fields ν
1
, ν
2
on M such that M is both ν
1
-flat and ν
2
-planar.
3.2 Hyperplanarity of surfaces in 4-dimensional spaces
The study of the sufficient conditions for a curve to be contained
in a hyperplane is a natural problem of the classical differential geometry.
A space curve is planar (lying on a plane) if and only if its torsion is zero
or equivalently, the bi-normal vector field of the curve is constant. Some
well-known, but weaker, conditions in terms of the osculating planes for
a space curve to be planar are:
(1) the osculating planes of the curve are parallel to a fixed direction
but the tangent lines of the curve are not;
(2) the osculating planes of the curve contain a fixed point but the
tangent lines of the curve do not.
The purpose of this section is to find similar results for two dimen-
sional surfaces in R
4
as well as for two dimensional spacelike surfaces
in R
4
1

.
a) Hyperplanarity of a surface in R
4
The classical fact: “the osculating planes of a bi-regular curve in R
3
16
are parallel to a fixed direction but its tangent lines are not, then the curve
is planar" is well-known.
The natural generalization of this fact for a surface immersed in R4
admitting a binormal field ν is no longer to be true. The condition (P):
“osculating ν-hyperplanes of the surface are parallel to a fixed plane but
its tangent planes are not" does not imply the hyperplanarity of the sur-
face and the followings are counterexamples.
Example 3.2.1. Consider a parametrization of the Clifford torus in R
4
X(u, v) = (cos u, sin u, cos v, sin v) , 0 < u, v < 2π.
Example 3.2.2. This example shows that, even though the osculating ν-
hyperplanes contain a fixed plane, M may be not hyperplanar. Consider
the parametric surface M given by


















X(u, v) = (1, u, v, 0), u ∈ (−π, 0), v ∈ R;
X(u, v) = (cos u, sin v, v, 0), u ∈ [0,
π
2
), v ∈ R;
X(u, v) = (u −
π
2
, 1, v, 0), u ∈ [
π
2
, π), v ∈ R;
X(u, v) = (
π
2
+ sin u, 1, v, 1 + cos u), u ∈ [π,
3π
2
)v ∈ R.
From some the binding conditions of the tangent planes we get some
interesting properties of surfaces.
Proposition 3.2.5 ([2]) If one of the followings is satisfied, then M is ν-
at for some normal vector field ν.
1. Tangent planes of M are parallel to a fixed direction;

2. Tangent planes of M contain a fixed point A.
Proposition 3.2.6 ([2]). If one of the followings is satisfied, then M is
ν-planar, i. e. ν is bi-normal.
1. ν-hyperplanes of M are parallel to a fixed plane Q;
2. ν-hyperplanes of M contain two fixed points A, B such that A, B /∈
M.
17
The following Proposition gives some sufficient conditions for a sur-
face in R
4
to be contained in a hyperplane.
Proposition 3.2.7 ([2]). Each one of the followings is a sufficient condi-
tion for M to be hyperplanar.
1. M is ν-umbilic and ν-hyperplanes are parallel to a fixed plane Q,
which is not parallel to any tangent plane of M.
2. M is ν-umbilic and ν-hyperplanes contain two fixed points A, B,
which is not contained in any tangent plane of M.
3. M is ν-flat and ν-hyperplanes are parallel to a fixed line d, which
is not parallel to any tangent plane of M.
4. M is ν-flat and ν-hyperplanes contain a fixed points A which is
not contained in any tangent plane of M.
b) Hyperplanarity of a spacelike surface in R
4
1
Let M be a spacelike surface immersed in R
4
1
. As in the case of sur-
faces in R
4

, a spacelike surface in R
4
1
admitting a bi-normal vector field
ν, that satisfies Condition (P) may be not contained in any hyperplane.
The surface
X(u, v) = (cos u, sin u, sinh v, cosh v) , 0 < u < 2π, v ∈ R
is an example similar to the Example 3.2.1 for the surfaces in R
4
1
.
When study a surface with a normal vector field, it is often to assume
that the normal vector field is unitary or of constant length or parallel.
These assumptions play an important role in the study. Note that a par-
allel 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 vector field ν is invariant, locally
we can assume that its length to be 1 provided ν is spacelike or timelike
or we can assume the last coordinate to be 1 if ν is lightlike.
However, the results in the previous section hold also for spacelike
surfaces in R
4
1
as well, no matter what the causality of the normal vector
18
field is. With similar proofs, we obtain the modified Propositions 3.2.5,
3.2.6 and 3.2.7 for spacelike surfaces. The proofs of the modified Propo-

sitions 3.2.5 and 3.2.6 for spacelike surfaces do not concern the causality
of ν. The only one that concerns the causality of ν is the proof of the
modified Proposition 3.2.7 for spacelike surfaces. But in this case we can
show that the causality of ν must be invariant, if the surface is assumed
to be connected. Indeed, except the case where ν is lightlike, if µ is
spacelike (res. timelike) at some point then it is spacelike (res. timelike)
in a neighborhood at that point. In this neighborhood, the length of the
normal vector field ν/|ν| is 1. A similar proof as in Proposition 3.2.7
yields ν/|ν| is constant in that neighborhood, i.e. the direction of ν at
each point is fixed. Therefore, the existence of another point where ν is
timelike (res. spacelike) or lightlike is impossible because of the contin-
uousness of the normal vector field and the connectedness of the surface.
Perhaps, the most interesting case is the one where ν is lightlike.
Often the appearance of lightlike vectors causes some interesting differ-
ences.
Proposition 3.2.13 ([2]). If ν is lightlike and ν-hyperplanes are parallel
to a fixed plane, then M is hyperplanar.
The following proposition give a sufficient condition for a lightlike
normal field ν to be binormal.
Proposition 3.2.14 ([2]). If ν is lightlike and ν-hyperplanes contain ei-
ther a given point A or parallel to a given vector a, then M is ν-planar.
Because M is ν-flat if only if M is ν-planar and ν-umbilical, we
have the following by combining Proposition 3.2.12 and 3.2.14.
Proposition 3.2.15 ([2]). Suppose that ν is lightlike and M is ν-umbilical.
If ν-hyperplanes contain either a given point A or parallel to a given
vector a, then M is hyperplanar.
3.3 Examples about ν-planar surfaces
In this section, we give some useful examples in order to show that: there
exist the surfaces not admit any binormal field; there exist the surfaces
admitting one binormal field but not ν-umbilic; there exist the surfaces

admitting two binormal fields but not ν-umbilic; the assumptions of M
in the propositions are relaxed.
19
Conclusions of Chapter 3
In this chapter, we get the following results:
(1) Using the vector product of three vectors, we give a standard to
check if a normal field is binormal.
(2) Defining the number of binormal fields on the ν-umbilical sur-
faces, and the relationship between the ν-flat surfaces; ν-umbilical
surfaces and ν-planar surfaces.
(3) Giving some sufficient conditions for a surface in 4-dimensional
space to be contained in a hyperplane. These results are global
properties of the surfaces in R
4
and R
4
1
.
(4) Giving some great examples in order to unravel the results in this
chapter. The results of Chapter 3 was published on [2] and [8].
Chapter 4
Ruled surfaces and surfaces of revolution
in R
4
1
4.1 Ruled surfaces
A surface M in R
4
1
is called ruled if through every point of M there

is a straight line that lies on M . We have a local parameterization of M
X(u, t) = α(u) + tW (u), (0.2)
where |W | = 1, α is a spacelike curve such thta α

, W = 0, α

, α

 =
1.
Proposition 4.1.3 ([8]). Let M be a ruled surface given by (0.2). We
have:
(1) at the point such that {α

, W, W

} is linearly dependent each normal
vector is bi- normal direction;
(2) at the point such that {α

, W, W

} is linearly independent M admits
only one bi- normal direction.
20
Proposition 4.1.5 ([8]). Let M be a ruled surface in R
4
1
, then we have:
(a) M is maximal if and only if it is contained in a timelike hyperplane

and maximal.
(b) M is ν-umbilic (not ν-flat) if and only if it is umbilic.
4.2 Surfaces of revolution
a) Surfaces of revolution of hyperbolic type
A surface of revolution of hyperbolic type [RH] in R
4
1
, is surface given
by the following parametrization
X(u, v) = (f(u), g(u), ρ(u) sinh v, ρ(u) cosh v) , u ∈ I, v ∈ R. (0.3)
Theorem 4.2.4 ([6]). If [RH] is umbilic then it is contained in a timelike
hyperplane and the curve C is given by
f(u) = ±
1
√
C
sinh

√
C(u − C
1
)

+ m,
g(u) =
C
2
√
C
sinh


√
C(u − C
1
)

+ k,
ρ(u) = ±
1
√
C
cosh

√
C(u − C
1
)

,
where C > 0, C
1
, C
2
, k are constant.
Theorem 4.2.6 ([6]). If [RH] is maximal then it is contained in a timelike
hyperplane and the curve C is given by
f(u) =
√
C
3

1 + C
2
2
arcsin

u − C
1
√
C
3

+ m, g(u) = C
2
f(u) + k,
ρ(u) = ±

C
3
− (u − C
1
)
2
,
where C
1
, C
2
, C
3
> 0, k are constant.

Proposition 4.2.8 ([8]). Suppose that f

g

− f

g

= 0, then we have:
1. [RH] admits exactly two bi-normal fields and its asymptotic fields
are orthogonal, and
21
2. there exists exactly one normal field ν satisfying that [RH] is ν-
umbilic.
b) Surfaces of revolution of ellipse type
A surfaces of revolution of hyperbolic type [RE] in R
4
1
is surface given
by
X(u, v) = (ρ(u) cos v, ρ(u) sin v, f (u), g(u)) , v ∈ R. (0.4)
We have the the properties of [RE] that is similar to the properties of
surface [RH].
Denoting by
E = g
11
= X
u
, X
u

, F = g
12
= X
u
, X
v
 = 0, G = g
22
= X
v
, X
v
,
the Gaussian of [RH] and [RE] are defined as follow
K = −
1
eg

ε
1

g
u
e

u
+ ε
2

e

v
g

v

,
where e = |E|
1/2
, g = |G|
1/2
vµ ε
1
, ε
2
are the sign of E, G respectively.
Theorem 4.2.16 ([7]). The Gaussian of [RH] ([RE]) is constant, K = C,
if and only if
1. ρ(u) = C
1
e
λu
+ C
2
e
−λu
, when εC = −2λ
2
< 0,
2. ρ(u) = C
1

sin(λu) + C
2
cos(λu), when εC = 2λ
2
> 0,
3. ρ(u) = C
1
u + C
2
, when C = 0,
where C
1
, C
2
are constant and ε is sign of E.
c) Rotational spacelike surface whose meridians lie in two-
dimension space
In this section, we show that the number of bi-normal fields on the
rotational spacelike surface whose meridians lie in two-dimension space
are depended on the properties of its meridian and give the corespondent
examples.
22
Conclusions of Chapter 4
In this chapter, we get the following results:
(1) Introducing the notions of ruled surface and ruled developable sur-
face; defining the number of the binormal directions on the ruled
surfaces and normal fields ν such that it is ν-umbilic; giving the
character of the maximal ruled surfaces.
(2) Introducing the notions of surfaces of revolution of hyperbolic and
elliptic type; defining the number of binormal fields on the surfaces

of revolution and the normal fields ν such that they are ν-umbilic;
giving the parametriczation of the maximal surfaces of revolution,
the umbilic surfaces of revolution and the surfaces of revolution
with constant Gaussian curvature.
(3) Introducing the notions of the rotational spacelike surface whose
meridians lie in two-dimension space and showing the number of
bi-normal fields on it.
The results of Chapter 4 was published on [6], [7] and [8].
CONCLUSIONS AND RECOMMENDATIONS FOR
FURTHER STUDY
1 Conclusions
The main results of the thesis.
1. Solving a system of equations, we defined a couple of smooth space-
like normal fields on a spacelike surface of co-dimension two, we
then applied this normal fields to study the properties of the ν-
umbilical surfaces (Theorem 2.1.5, 2.1.12, 2.1.14, 2.1.15, 2.1.16,
2.1.17, 2.2.8, 2.2.9). Connecting the results about the properties of
the ν-umbilical surfaces, we gave the properties of the umbilical
surfaces (Theorem 2.3.2). We also gave some example to unravel
these results. These results was published on [5], [6] and [9].