a r X i v :0708.3480v 1 [m a t h .D G ] 26 A u g 2007
ON THE THEORY OF SURF ACES IN THE FOUR-DIMENSIONAL
EUCLIDEAN SPACE
GEORGI GANCHEV AND VELICHKA MILOUSHEVA
Abstract.For a two-dimensional surface M 2in the four-dimensional Euclidean space E 4we introduce an invariant linear map of Weingarten type in the tangent space of the surface,which generates two invariants k and κ.
The condition k =κ=0characterizes the surfaces consisting of ?at points.The minimal surfaces are characterized by the equality κ2?k =0.The class of the surfaces with ?at normal connection is characterized by the condition κ=0.For the surfaces of general type we obtain a geometrically determined orthonormal frame ?eld at each point and derive Frenet-type derivative formulas.
We apply our theory to the class of the rotational surfaces in E 4,which prove to be surfaces with ?at normal connection,and describe the rotational surfaces with constant invariants.
1.Introduction
In [3]T.ˉOtsuki introduced curvatures λ1,λ2,...,λn (λ1≥λ2≥···≥λn )for a surface
M 2
in a (2+n )-dimensional Euclidean space E 2+n ,de?ning a quadratic form in the normal space of the surface.In a suitable local frame of the normal space this quadratic form can be written in a diagonal form and the functions λα,α=1,...,n are the coe?cients in the diagonalized form (λαis called the α-th curvature of M 2).These curvatures are closely related to the Gauss curvature K of M 2:
K =λ1+λ2+···+λn .
The local cross-section,which diagonalizes the quadratic form is called a Frenet cross-section (Frenet-frame )of the surface.
For a surface M 2in the four-dimensional Euclidean space E 4the curvatures λ1and λ2are the maximum and minimum,respectively of the Lipschitz-Killing curvature of the surface [4].The function λ1is called the principal curvature and the function λ2-the secondary curvature of M 2in E 4.
Using the idea of the Frenet-frames,Shiohama [5]proved that a complete connected orientable surface M 2in E 4with curvatures λ1=λ2=0is a cylinder.The same result is proved in [6]for a surface in a higher dimensional space E 2+n .
Our aim is to ?nd invariants of a surface M 2in E 4,considering a geometrically determined linear map (of Weingarten type)in the tangent space of the surface,as well as to obtain a geometric Frenet-type frame ?eld of M 2.
In Section 2we de?ne a geometrical linear map in the tangent space of a surface M 2in E 4and determine a second fundamental form II of the surface.We ?nd invariants k and κof M 2(which are analogous to the Gauss curvature and the mean curvature of a surface in
2GEORGI GANCHEV AND VELICHKA MILOUSHEVA
E3).These invariants divide the points of M2into four types:?at,elliptic,parabolic and hyperbolic.
In Section3we give a local geometric description of the surfaces consisting of?at points, proving that they are either planar surfaces(Proposition3.1)or developable ruled surfaces
(Proposition3.2).
In Section4we characterize the minimal surfaces in E4in terms of the invariants k and κ(Proposition4.1).
For the surfaces of general type(which are not minimal and which have no?at points)in
Section5we obtain a geometrically determined orthonormal frame?eld{x,y,b,l}at each point of the surface and derive Frenet-type derivative formulas.The tangent frame?eld {x,y}is determined by the de?ned second fundamental form II,while the normal frame ?eld{b,l}is determined by the mean curvature vector?eld of the surface.
We also characterize the surfaces with?at normal connection in terms of the invariantκ(Theorem5.1).
In the last section we apply our theory to the class of the rotational surfaces in E4,which prove to be surfaces with?at normal connection,and describe the rotational surfaces with k=const.
2.The Weingarten map
We denote by g the standard metric in the four-dimensional Euclidean space E4and by ?′its?at Levi-Civita connection.All considerations in the present paper are local and all functions,curves,surfaces,tensor?elds etc.are assumed to be of the class C∞.
Let M2:z=z(u,v),(u,v)∈D(D?R2)be a2-dimensional surface in E4.The tangent space to M2at an arbitrary point p=z(u,v)of M2is span{z u,z v}.
For an arbitrary orthonormal normal frame?eld{e1,e2}of M2we have the standard derivative formulas:
(2.1)?′z u z u=z uu=Γ111z u+Γ211z v+c111e1+c211e2;?′z u z v=z uv=Γ112z u+Γ212z v+c112e1+c212e2;?′z v z v=z vv=Γ122z u+Γ222z v+c122e1+c222e2,
whereΓk ij are the Christo?el’s symbols and c k ij,i,j,k=1,2are functions on M2.
We use the standard denotations E(u,v)=g(z u,z u),F(u,v)=g(z u,z v),G(u,v)= g(z v,z v)for the coe?cients of the?rst fundamental form and set W=
√
W ,M(u,v)=
?2
W
.
ON THE THEORY OF SURF ACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE3 If
(2.2)u=u(ˉu,ˉv);
v=v(ˉu,ˉv),
(ˉu,ˉv)∈ˉD,ˉD?R2
is a smooth change of the parameters{u,v}on M2with J=uˉu vˉv?uˉv vˉu=0,then
zˉu=z u uˉu+z v vˉu,
zˉv=z u uˉv+z v vˉv.
Let
σ(zˉu,zˉu)=ˉc111e1+ˉc211e2,
σ(zˉu,zˉv)=ˉc112e1+ˉc212e2,
σ(zˉv,zˉv)=ˉc122e1+ˉc222e2.
Di?erentiating(2.2)and taking into account(2.1)we?nd
(2.3)ˉc k11=u2ˉu c k11+2uˉu vˉu c k12+v2ˉu c k22,
ˉc k12=uˉu uˉv c k11+(uˉu vˉv+uˉv vˉu)c k12+vˉu vˉv c k22,
ˉc k22=u2ˉv c k11+2uˉv vˉv c k12+v2ˉv c k22.
(k=1,2)
Using(2.3),we obtain
(2.4)?2=J(2uˉu uˉv?1+(uˉu vˉv+uˉv vˉu)?2+2vˉu vˉv?3);
4GEORGI GANCHEV AND VELICHKA MILOUSHEVA Further we denote
(2.8)γ11=
F M?GL
EG?F2
,γ12=
F N?GM
EG?F2
and consider the linear map
γ:T p M2→T p M2 determined by the conditions
(2.9)γ(z u)=γ11z u+γ21z v,
γ(z v)=γ12z u+γ22z v,
γ= γ11γ21γ12γ22 .
Then a tangent vector X=λz u+μz v is transformed into the vector X′=γ(X)=λ′z u+μ′z v so that λ′μ′ =γt λμ .
We have
Lemma2.1.The linear mapγgiven by(2.9)is geometrically determined.
Proof:Let the change of the parameters be given by(2.2).Then we have
zˉu zˉv =T z u z v ,T= uˉu vˉu uˉv vˉv .
If we denote
g= E F F G ,h= L M M N ,
then the de?ning conditions(2.8)implyγ=?hg?1.
With respect to the new coordinates(ˉu,ˉv)the linear mapˉγis determined by the equality ˉγ=?ˉhˉg?1.
On the other hand,the equalities(2.5)and(2.7)express that
ˉg=T gT t,ˉh=εT hT t.
Thus we obtainˉγ=?ˉhˉg?1=εTγT?1,which implies thatˉγ=εγ.
Further,let{ e1, e2}be another orthonormal normal frame?eld of M2.Then
e1=cosθ e1+ε′sinθ e2;
e2=?sinθ e1+ε′cosθ e2;
θ=∠( e1,e1),
andε′=1(ε′=?1)if the normal frame?elds{e1,e2}and{ e1, e2}have the same(opposite) orientation.The relation between the corresponding functions c k ij and c k ij,i,j,k=1,2is given by the equalities
c1ij=cosθc1ij?sinθc2ij;
c2ij=ε′(sinθc1ij+cosθc2ij);i,j=1,2. Thus, ?i=ε′?i,i=1,2,3,and L=ε′L, M=ε′M, N=ε′N,which imply that γ=ε′γ.
ON THE THEORY OF SURF ACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE5
The linear mapγ:T p M2→T p M2is said to be the Weingarten map at the point p∈M2. The following statement follows immediately from Lemma2.1.
Lemma2.2.The functions
(2.10)k:=detγ=LN?M2
2
trγ=
EN+GL?2F M
E
γ21 2+4EG?F2
6GEORGI GANCHEV AND VELICHKA MILOUSHEVA
3.Surfaces consisting of flat points
In this section we consider surfaces M 2:z =z (u,v ),(u,v )∈D consisting of ?at points,i.e.surfaces satisfying the conditions (3.1)
k (u,v )=0,
κ(u,v )=0,
(u,v )∈D .
We give a local geometric description of these surfaces.
For the sake of simplicity,we shall assume that the parametrization of M 2is orthogonal,
i.e.F =0.Denote the unit vector ?elds x =z u
E ,y =z v G
.Then we write (2.1)in the form
(3.2)
?′
x x =γ1y +c 111E
e 2,
?′
x y =?γ1x +c 112EG e 1+c 212EG
e 2,
?′
y x =?γ2y +
c 112EG e 1+c 212EG e 2,?′
y y =γ2y +c 122G e 2.
Obviously,the surface M 2lies in a 2-plane if and only if M 2is totally geodesic,i.e.c k ij =0,i,j,k =1,2.
Now,let at least one of the coe?cients c k ij not be zero.Then
rank c 111c 112c 1
22
c 211c 212c 2
22
=1and the vectors σ(x,x ),σ(x,y ),σ(y,y )are collinear.Let {b,l }be a normal frame ?eld of M 2,consisting of orthonormal vector ?elds,such that b is collinear with σ(x,x ),σ(x,y ),and σ(y,y ).It is clear that the normal frame ?eld {b,l }is invariant.Then the derivative formulas of M 2can be written as follows:
(3.3)
?′x x =γ1y +ν1b,?′x b =?ν1x ?λy +β1l,?′x y =?γ1x +λb,
?′y b =?λx ?ν2y +β2l,
?′y x =?γ2y +λb,
?′x l =?β1b,?′y y =γ2x
+ν2b,
?′y l =
?β2b,
for some functions ν1,ν2,λ,β1,β2,γ1,γ2on M 2.
The Gauss curvature K of M 2is expressed by (3.4)
K =ν1ν2?λ2.
Further we denote β=β21+β2
2.It follows immediately that βdoes not depend on the change (2.2)of the parameters.
Since the curvature tensor R ′of the connection ?′is zero,then the equalities R ′(x,y,b )=0and R ′(x,y,l )=0together with (3.3)imply that either K =0or β=0.
A surface M 2is said to be planar if there exists a hyperplane E 3?E 4containing M 2.First we shall characterize the planar surfaces.
ON THE THEORY OF SURF ACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE7
Proposition3.1.A surface M2is planar if and only if
k=0,κ=0,β=0.
Proof:I.Let M2?E3and b be the usual normal to M2in E3.Choosing l to be the normal to the hyperplane E3,from(3.3)we get L=M=N=0andβ=0.
II.Under the conditions k=κ=β=0,from(3.3)it follows that l=const and M2lies in a hyperplane E3orthogonal to l.
A ruled surface M2is a one-parameter system{g(v)},v∈J of straight lines g(v),de?ned in an interval J?R.The straight lines g(v)are called generators of M2.A ruled surface M2={g(v)},v∈J is said to be developable,if the tangent space T p M2at all regular points p of an arbitrary?xed generator g(v)is one and the same.
Each ruled surface M2can be parameterized as follows:
(3.5)z(u,v)=x(v)+u e(v),u∈R,v∈J,
where x(v)and e(v)are vector-valued functions,de?ned in J,such that the vectors e(v)and x′(v)+u e′(v)are linearly independent for all v∈J.The tangent space of M2is spanned by the vectors
z u=e(v);
z v=x′(v)+u e′(v).
The ruled surface M2determined by(3.5)is developable if and only if the vectors e(v), e′(v)and x′(v)are linearly dependent.
We shall characterize the developable ruled surfaces in terms of the invariants k,κand the Gauss curvature K.
Proposition3.2.A surface M2is locally a developable ruled surface if and only if
k=0,κ=0,K=0.
Proof:I.Let M2be a developable ruled surface,de?ned by the equality(3.5),where e(v),e′(v)and x′(v)are linearly dependent.Without loss of generality we assume that e2(v)=1.Then,the vector?elds e(v)and e′(v)are orthogonal and the tangent space of M2 is span{e(v),e′(v)}.Since x′(v)∈span{e(v),e′(v)},then x′(v)is decomposed in the form x′(v)=p(v)e(v)+q(v)e′(v)for some functions p(v)and q(v).Hence,the tangent space of M2is spanned by
z u=e;
z v=p e+(u+q)e′.
Considering only the regular points of M2(where u=?q),we choose an orthonormal tangent frame?eld{x,y}of M2in the following way:
(3.6)x=e=z u;
y=
e′
(e′)2
=?
p
(e′)2
z u+
1
(e′)2
z v.
Since the tangent space of M2does not depend on the parameter u,then the normal space of M2is spanned by vector?elds b1(v),b2(v).With respect to the basis{e(v),e′(v), b1(v),b2(v)}the derivatives of b1(v)and b2(v)are decomposed in the form
(3.7)b′1=?c1e′+c0b2, b′2=?c2e′?c0b1,
8GEORGI GANCHEV AND VELICHKA MILOUSHEVA where c0,c1,c2are functions of v.
Then the equalities(3.6)and(3.7)imply
?′x b1=0,
?′y b1=?c1(u+q)
u+q y?
c0
(e′)2
b1.
Consequently,L=M=N=0and K=0.
II.Let M2be a surface for which L=M=N=0and K=0.We consider an orthonormal frame?eld{x,y,b,l}of M2,satisfying the equalities(3.3).Since K=0,thenν1ν2?λ2=0. Ifν1=ν2=0,then M2lies in a plane E2.So we assume that there exists a neighborhood D?D such thatν2|e D=0(orν1|e D=0)and we consider the surface M2=M2|e D. Let{y}be the orthonormal tangent frame?eld of M2,de?ned by
y=?sin?x+cos?y,
where tan?=?λ
x,x,
x
γ1x b=
x
γ1y b=?y+
y
γ2x l=?
y
γ2ν2b,?′β2b,
where
x,x,
γ1=0,
x x
b=0,
?′y=0,?′
u0,u0,u)=z(v0)be the integral curve of the vector?eld x
x
b=0and?′
x,
ON THE THEORY OF SURF ACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE9
4.Minimal surfaces
We recall that a surface M2is said to be minimal if the mean curvature vector
1
√√
E e1+
c211
√√√√
G e1+
c222 G
=
?1
E =
N
E
.Hence trσ=0.
5.Surfaces of general type
From now on we consider surfaces,satisfying the condition
κ2?k=0
and call them surfaces of general type.
As in the classical di?erential geometry of surfaces in E3the second fundamental form determines conjugate tangents at a point p of M2.A tangent g:X=λz u+μz v is said to
10GEORGI GANCHEV AND VELICHKA MILOUSHEVA
be principal if it is perpendicular to its conjugate.The equation for the principal tangents at a point p ∈M 2is
E F L M
λ2+ E G L N λμ+ F G M N μ2=0.A line c :u =u (q ),v =v (q );q ∈J on M 2is said to be a principal curve if its tangent at
any point is principal.
The surface M 2is parameterized with respect to the principle lines if and only if
F =0,
M =0.
Let M 2be parameterized with respect to the principal lines and denote the unit vector
?elds x =z u E ,y =z v
G
.
Since the mean curvature vector ?eld H =0,we determine the unit normal vector ?eld
b by the equality b =
H
E ),γ2=?x (ln
√
κ2?k
ON THE THEORY OF SURF ACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE11 Let z=g(z,x)x+g(z,y)y be an arbitrary tangent vector?eld of M2.We de?ne the one-formθby the equality
θ(z)=g(?′z b,l).
Then the formulas(5.1)imply that
θ(z)=g(β1x+β2y,z),
which shows that the one-formθcorresponds to the tangent vector?eldβ1x+β2y and
θ =
12GEORGI GANCHEV AND VELICHKA MILOUSHEVA
Obviously,M2is a surface with?at normal connection if and only ifν1=ν2=:ν.So,the Frenet-type formulas(5.1)of a surface M2with?at normal connection take the form:
(5.5)?′x x=γ1y+νb;?′x b=?νx?λy+β1l;?′x y=?γ1x+λb+μl;?′y b=?λx?νy+β2l;?′y x=?γ2y+λb+μl;?′x l=?μy?β1b;?′y y=γ2x+νb;?′y l=?μx?β2b.
Hence the invariants k and K are expressed by
k=?4ν2μ2,K=ν2?(λ2+μ2).
Remark2.The curvature of the normal connection of a surface M2in E4is the Gauss torsionκG of M2,which is de?ned in terms of the of the ellipse of normal curvature at a point p∈M2[1].
6.Rotational surfaces
Now we shall apply our theory to the class of the rotational surfaces in E4.
We denote by Oe1e2e3a?xed orthonormal base of E3.Let c: z= z(u),u∈J be a smooth curve in E3,parameterized by
z(u)=(x1(u),x2(u),r(u));u∈J.
We denote by c1the projection of c on the2-dimensional plane Oe1e2.
Without loss of generality we can assume that c is parameterized with respect to the arc-length,i.e.(x′1)2+(x′2)2+(r′)2=1.We assume also that r(u)>0,u∈J.Let us consider the rotational surface M2in E4given by
(6.1)z(u,v)=(x1(u),x2(u),r(u)cos v,r(u)sin v);u∈J,v∈[0;2π).
The tangent space of M2is spanned by the vector?elds
z u=(x′1,x′2,r′cos v,r′sin v);
z v=(0,0,?r sin v,r cos v).
Hence,
E=1;F=0;G=r2(u);W=r(u).
We consider the following orthonormal tangent vector?elds
y=(0,0,?sin v,cos v),
i.e.z u=y.The second partial derivatives of z(u,v)are expressed as follows
z uu=(x′′1,x′′2,r′′cos v,r′′sin v);
z uv=(0,0,?r′sin v,r′cos v);
z vv=(0,0,?r cos v,?r sin v).
ON THE THEORY OF SURF ACES IN THE FOUR-DIMENSIONAL EUCLIDEAN SPACE 13
Let κand τbe the curvature and the torsion of the curve c (considered as a curve in E 3).We consider the normal vector ?elds e 1and e 2,de?ned by
e 1=
1
κ
(x ′2r ′′?x ′′2r ′,x ′′1r ′?x ′1r ′′,(x ′1x ′′2?x ′′1x ′2)cos v,(x ′1x ′′2?x ′′1x ′
2)sin v ).
Now it is easy to calculate that
L =0;
M =?(x ′1x ′′2?x ′′1x ′
2);N =0.
Hence,
k =?
(x ′1x ′′2?x ′′1x ′2)
2
x,
x x e 1=?κx
y e 1
=
r ′′
y ;
?′
x =
r ′
y ;
?′
y
r
κr e 1?
κ1
y e 2=
κ1
y.
So,the Gauss curvature of M 2is:
K =?
r ′′
2x +√2
2
x ?
√2
2
r
;ν=
2κr ;
λ=
κ4r 2?(r ′′)2?(κ1)2
(κ2r ?r ′′)2+(κ1)2;
μ=
κκ1
(κ2r ?r ′′)2+(κ1)
2.
14GEORGI GANCHEV AND VELICHKA MILOUSHEVA Consequently,the invariants k,κand K of the rotational surface M2are:
k=?(κ1)2
r
.
At the end of the section we shall describe all rotational surfaces,for which the invariant k is constant.
1.The invariant k=0if and only ifκ1=0,which means that the projection of the curve
c on the plane Oe
1e2is a straight line.There are two subcases:
1.1.If K=0,i.e.r′′=0,then M2is a developable ruled surface.
1.2.If K=0,i.e.r′′=0,then M2is a planar surface.
2.The invariant k=const(k=0)if and only if r(u)=a(x′1x′′2?x′′1x′2),a=const. Moreover,if r(u)satis?es r′′(u)=c r(u),then the Gauss curvature K is also a constant.
References
[1]Aminov Yu.,The geometry of submanifolds.Gordon and Breach Science Publishers,2002.
[2]Chen B.-Y.,Geometry of submanifolds.Marcel Dekker,Inc.,New York,1973.
[3]ˉOtsuki T.,On the total curvature of surfaces in Euclidean spaces.Japanese J.Math.,35(1966),61-71.
[4]ˉOtsuki T.,Surfaces in the4-dimensional Euclidean space isometric to a sphere.Kˉo dai Math.Sem.
Rep.18(1966),101-115.
[5]Shiohama K.,Surfaces of curvaturesλ=μ=0in E4.Kˉo dai Math.Sem.Rep.19(1967),75-79.
[6]Shiohama K.,Cylinders in Euclidean space E2+N.Kˉo dai Math.Sem.Rep.19(1967),225-228.
Bulgarian Academy of Sciences,Institute of Mathematics and Informatics,Acad.G. Bonchev Str.bl.8,1113Sofia,Bulgaria
E-mail address:ganchev@math.bas.bg
Bulgarian Academy of Sciences,Institute of Mathematics and Informatics,Acad.G. Bonchev Str.bl.8,1113,Sofia,Bulgaria
E-mail address:vmil@math.bas.bg