Table Of ContentLIGHTLIKE SURFACES WITH PLANAR NORMAL SECTIONS
IN MINKOWSKI 3− SPACE
3
1 FEYZAESRAERDOG˘AN,BAYRAMS¸AHI˙N∗,ANDRIFATGU¨NES¸
0
2
Abstract. In this paper we study lightlike surfaces of Minkowski 3− space
n suchthattheyhavedegenerateornon-degenerateplanarnormalsections. We
a first show that every lightlike surface of Minkowski 3− space has degenerate
J
planarnormalsections. Thenwestudylightlikesurfaceswithnon-degenerate
4 planarnormalsectionsandobtainacharacterizationforsuchlightlikesurfaces
2
]
G
1. Introduction
D
. Surfaces with planar normal sections in Euclidean spaces were first studied by
h
Bang-YenChen[6]. Latersuchsurfacesorsubmanifoldshavebeenstudiedbymany
t
a authors[6],[5],[7],[8],[9]. In[7],Y.H.Kiminitiatedthestudyofsemi-Riemannian
m
settingofsuchsurfaces. Butasfarasweknow,lightlikesurfaceswithplanarnormal
[ sections have not been studied so far. Therefore, in this paper we study lightlike
surfaces with planar normal sections of R3.
1 1
v
2 We firstdefine thenotionofsurfaceswithplanarnormalsectionsasfollows. Let
7 M be a lightlike surface of R3. For a point p in M and a lightlike vector ξ which
1
7
spans the radical distribution of a lightlike surface, the vector ξ and transversal
5
spacetr(TM)toM atpdeterminea2-dimensionalsubspaceE(p,ξ)inR3 through
. 1
1 p. The intersection of M and E(p,ξ) gives a lightlike curve γ in a neighborhood of
0
p, which is called the normal section of M at the point p in the direction of ξ.
3
1
: Fornon-degenerateplanarnormalsections,we presentthe followingnotion. Let
v
wbeaspacelikevectortangenttoM atpwhichspansthechosenscreendistribution
i
X of M. Then the vector w and transversal space tr(TM) to M at p determine a 2-
r dimensional subspace E(p,w) in R3 through p. The intersectionof M and E(p,w)
1
a
gives a spacelike curve γ in a neighborhood of p which is called the normal section
of M at p in the direction of w. According to both identifications above, M is
said to have degenerate pointwise and spacelike pointwise planar normal sections,
respectively if each normalsection γ at p satisfies γ′∧γ′′∧γ′′′ =0 at for each p in
M.
For a lightlike surface with degenerate planar normal sections, in fact, we show
that every lightlike surface of Minkowski 3− space has degenerate planar normal
sections. Then for a lightlike surface with non-degenerate planar normal sections,
we obtain two characterizations.
* Corresponding author
1
2 FEYZAESRAERDOG˘AN,BAYRAMS¸AHI˙N∗,ANDRIFATGU¨NES¸
We first show that a lightlike surface M in R3 is a lightlike surface with non-
1
degenerate planar sections if and only if M is either screen conformal and totally
umbilical or M is totally geodesic. We also obtain a characterization for non-
umbilicalscreenconformallightlikesurfacewithnon-degenerateplanarnormalsec-
tions.
2. Preliminaries
Let (M¯,g¯) be an (m+2)-dimensional semi-Riemannian manifold with the in-
definite metric g¯ of index q ∈ {1,...,m+1} and M be a hypersurface of M¯. We
denote the tangent space at x∈M by T M. Then
x
T M⊥ ={V ∈T M¯|g¯ (V ,W )=0,∀W ∈T M}
x x x x x x x x
and
RadT M =T M ∩T M⊥.
x x x
Then, M is called a lightlike hypersurface of M¯ if RadT M 6={0} for any x∈M.
x
Thus TM⊥ = T M⊥ becomes a one- dimensional distribution RadTM on
x∈M x
M. Then thereTexists a vector field ξ 6=0 on M such that
g(ξ,X)=0, ∀X ∈Γ(TM),
where g is the induced degenerate metric tensor on M. We denote F(M) the alge-
bra of differentialfunctions on M andby Γ(E) the F(M)- module of differentiable
sections of a vector bundle E over M.
A complementary vector bundle S(TM) of TM⊥ =RadTM in TM i,e.,
(2.1) TM =RadTM ⊕ S(TM)
orth
is called a screen distribution on M. It follows from the equation above that
S(TM) is a non-degenerate distribution. Moreover, since we assume that M is
paracompact, there always exists a screen S(TM). Thus, along M we have the
decomposition
(2.2) TM¯ =S(TM)⊕ S(TM)⊥, S(TM)∩S(TM)⊥ 6={0},
|M orth
that is, S(TM)⊥ is the orthogonal complement to S(TM) in TM¯ | . Note that
M
S(TM)⊥ is also a non-degenerate vector bundle of rank 2. However, it includes
TM⊥ =RadTM as its sub-bundle.
Let (M,g,S(TM)) be a lightlike hypersurface of a semi-Riemannian manifold
M¯,g¯ . Then there exists a unique vector bundle tr(TM) of rank 1 over M, such
(cid:0)that f(cid:1)or any non-zero section ξ of TM⊥ on a coordinate neighborhood U ⊂ M,
there exists a unique section N of tr(TM) on U satisfying: TM⊥ in S(TM)⊥ and
take V ∈ Γ(F | ),V 6= 0. Then g¯(ξ,V) 6= 0 on U, otherwise S(TM)⊥ would be
U
degenerate at a point of U [4]. Define a vector field
1 g¯(V,V)
N = V − ξ
g¯(V,ξ)(cid:26) 2g¯(V,ξ) (cid:27)
on U where V ∈ Γ(F | ) such that g¯(ξ,V)6=0. Then we have
U
(2.3) g¯(N,ξ)=1, g¯(N,N)=0, g¯(N,W)=0, ∀W ∈Γ(S(TM)| )
U
LIGHTLIKE SURFACES 3
Moreover,from (2.1) and (2.2) we have the following decompositions:
(2.4) TM¯ | =S(TM)⊕ TM⊥⊕tr(TM) =TM ⊕tr(TM)
M orth
(cid:0) (cid:1)
Locally, suppose {ξ,N} is a pair of sections on U ⊂ M satisfying (2.3). Define a
symmetric ̥(U)-bilinear from B and a 1-form τ on U . Hence on U, for X,Y ∈
Γ(TM | )
U
(2.5) ∇¯ Y = ∇ Y +B(X,Y)N
X X
(2.6) ∇¯ N = −A X +τ(X)N,
X N
equations (2.5) and (2.6) are local Gauss and Weingarten formulae. Since ∇¯ is a
metric connection on M¯, it is easy to see that
(2.7) B(X,ξ)=0,∀X ∈Γ(TM | ).
U
Consequently, the second fundamental form of M is degenerate [4]. Define a local
1-from η by
(2.8) η(X)=g¯(X,N),∀∈Γ(TM | ).
U
Let P denote the projection morphism of Γ(TM) on Γ(S(TM)) with respect to
the decomposition (2.1). We obtain
(2.9) ∇ PY = ∇∗ PY +C(X,PY)ξ
X X
∇ ξ = −A∗X +ε(X)ξ
X ξ
(2.10) = −A∗X −τ(X)ξ
ξ
where ∇∗ Y and A∗X belong to Γ(S(TM)),∇ and ∇∗t are linear connections
X ξ
onΓ(S(TM))andTM⊥ respectively, h∗is a Γ TM⊥ -valued̥(M)-bilinearform
on Γ(TM)× Γ(S(TM)) and A∗ is Γ(S(TM)(cid:0))-value(cid:1)d ̥(M)-linear operator on
ξ
Γ(TM). We called them the screen fundamental form and screen shape operator
of S(TM), respectively. Define
(2.11) C(X,PY) = g¯(h∗(X,PY),N)
(2.12) ε(X) = g¯ ∇∗tξ,N ,∀X,Y ∈Γ(TM),
X
(cid:0) (cid:1)
one can show that ε(X) = −τ(X). Here C(X,PY) is called the local screen
fundamental formof S(TM). Precisely,the two localsecondfundamental forms of
M and S(TM) are related to their shape operators by
(2.13) B(X,Y) = g¯ Y,A∗X ,
ξ
(2.14) A∗ξ = 0(cid:0), (cid:1)
ξ
(2.15) g¯ A∗PY,N = 0 ,
ξ
(2.16) (cid:0)C(X,PY(cid:1)) = g¯(PY,A X),
N
(2.17) g¯(N,A X) = 0.
N
A lightlike hypersurface (M,g,S(TM)) of a semi-Riemannian manifold is called
totally umbilical[4] if there is a smooth function ̺, such that
(2.18) B(X,Y)=̺g(X,Y),∀X,Y ∈Γ(TM)
where ̺ is non-vanishing smooth function on a neighborhood U in M.
4 FEYZAESRAERDOG˘AN,BAYRAMS¸AHI˙N∗,ANDRIFATGU¨NES¸
A lightlike hypersurface (M,g,S(TM)) of a semi-Riemannian manifold is
calledscreenlocallyconformaliftheshapeoperatorsA andA∗ ofM andS(TM),
N ξ
respectively, are related by
(2.19) A =ϕA∗
N ξ
where ϕ is non-vanishing smooth function on a neighborhood U in M. Therefore,
it follows that ∀X,Y ∈Γ(S(TM)), ξ ∈RadTM
(2.20) C(X,ξ)=0,
For details about screen conformal lightlike hypersurfaces, see: [1] and [4] .
3. Planar normal sections of lightlike surfaces in R3
1
Let M be a lightlike surface of R3. Now we investigate lightlike surfaces with
1
degenerateplanar normalsections. If γ is a null curve,for a point p in M, we have
(3.1) γ′(s) = ξ
(3.2) γ′′(s) = ∇¯ ξ =−τ(ξ)ξ
ξ
(3.3) γ′′′(s) = ξ(τ(ξ))+τ2(ξ) ξ
(cid:2) (cid:3)
Then,γ′′′(0)isalinearcombinationofγ′(0)andγ′′(0). Thus(3.1),(3.2)and(3.3)
give γ′′′(0)∧γ′′(0)∧γ′(0)=0. Thus lightlike surfaces always have planar normal
sections.
Corollary 3.1. Every lightlike surface of R3 has degenerate planar normal sec-
1
tions.
In fact Corollary 3.1 tells us that the above situation is not interesting. Now,
we will check lightlike surfaces with non-degenerate planar normal sections. Let
M be a lightlike hypersurface of R3. For a point p in M and a spacelike vector
1
w∈S(TM) tangent to M at p , the vector w and transversalspace tr(TM) to M
atpdeterminea2-dimensionalsubspaceE(p,w)inR3 throughp. Theintersection
1
of M and E(p,w) give a spacelike curve γ in a neighborhood of p, which is called
thenormalsectionofM atpinthedirectionofw. Now,weresearchtheconditions
for a lightlike surface of R3 to have non-degenerate planar normal sections.
1
Let (M,g,S(TM)) be a totally umbilical and screen conformal lightlike surface
of g¯,R3 . In this case S(TM) is integrable[1]. We denote integral submanifold of
1
S(T(cid:0)M) b(cid:1)y M′. Then, using (2.6), (2.10) and (2.19 ) we find
(3.4) C(w,w)ξ+B(w,w)N =g¯(w,w){αξ+βN}={αξ+βN},
where t, α,β ∈R. In this case, we obtain
(3.5) γ′(s) = w
(3.6) γ′′(s) = ∇¯ w =∇∗w+C(w,w)ξ+B(w,w)N
w w
(3.7) γ′′(s) = ∇∗w+αξ+βN
w
(3.8) γ′′′(s) = ∇∗∇∗w+C(w,∇∗w)ξ+w(C(w,w))ξ−C(w,w)A∗w
w w w ξ
+w(B(w,w))N −B(w,w)A w+B(w,∇∗w)N
N w
(3.9) γ′′′(s) = ∇∗∇∗w+t{αξ+βN}−αA∗w−βA w.
w w ξ N
LIGHTLIKE SURFACES 5
Where ∇∗ and ∇ are linear connections on S(TM) and Γ(TM), respectively and
γ′(s) = w. From the definition of planar normal section and that S(TM) =
Sp{w}, we have
(3.10) w∧∇∗w =0
w
and
(3.11) w∧∇∗∇∗w =0.
w w
Then,from(3.4),(3.6),(3.8)and(3.10),(3.11)weobtainγ′′′(s)∧γ′′(s)∧γ′(s)=0.
Thus, M has planar non-degenerate normal sections.
If M is totally geodesic lightlike surface of R3. Then, we have B = 0, A∗ = 0.
1 ξ
Hence (3.5)-(3.8) become
γ′(s) = w
γ′′(s) = ∇∗w+αξ
w
γ′′′(s) = ∇∗∇∗w+tαξ−βA w.
w w N
Since A w∈Γ(TM), we have γ′′′(s)∧γ′′(s)∧γ′(s)=0.
N
Conversely,weassumethatM hasplanarnon-degeneratenormalsections. Then,
from (3.5), (3.6), (3.8) and (3.10), (3.11) we obtain
(C(w,w)ξ+B(w,w)N)∧ C(w,w)A∗w+B(w,w)A w =0,
ξ N
(cid:0) (cid:1)
thus C(w,w)A∗w+B(w,w)A w =0 or C(w,w)ξ+B(w,w)N =0. If
ξ N
(cid:16) (cid:17)
C(w,w)A∗w+B(w,w)A w =0, then, from
ξ N
B(w,w)
A∗w =− A w
ξ C(w,w) N
atp∈M,M isascreenconformallightlikesurfacewithC(w,w)6=0. IfC(w,w)ξ+
B(w,w)N =0, then RadTM is parallel and M is totally geodesic.
Consequently, we have the following,
Theorem 3.1. Let M be a lightlike surface of R3. Then M has non-degenerate
1
planar normal sections if and only if either M is umbilical and screen conformal
or M is totally geodesic.
Theorem 3.2. Let (M,g,S(TM)) be a screen conformal non-umbilical lightlike
surfaceofR3. Then,forT (w,w)=C(w,w)ξ+B(w,w)N thefollowingstatements
1
are equivalent
(1) ∇¯ T (w,w)=0, every spacelike vector w∈S(TM)
w
(2) (cid:0)∇¯T =(cid:1)0
(3) M has non-degenerate planar normal sections and each normal section at
p has one of its vertices at p
By the vertex of curve γ(s) we mean a point p on γ such that its curvature κ
satisfies dκ2(p) =0, κ2 =hγ′′(s),γ′′(s)i.
ds
6 FEYZAESRAERDOG˘AN,BAYRAMS¸AHI˙N∗,ANDRIFATGU¨NES¸
Proof. From (3.5), (3.6) and that a screen conformal M, we have
∇¯ T (w,w)=∇¯ T (w,w)
w w
which shows (a) ⇔ (b). (cid:0)(b) ⇒(cid:1)(c) Assume that ∇¯T = 0 . If ∇¯T = 0 then
M is totally geodesic and Theorem 3.1 implies that M has (pointwise) planar
normal sections. Let the γ(s) be a normal section of M at p in a given direction
w∈S(TM). Then (3.5) shows that the curvature κ(s) of γ(s) satisfies
κ2(s) = hγ′′(s),γ′′(s)i
= 2C(w,w)B(w,w)
(3.12) = hT(w,w),T (w,w)i
where w=γ′(s). Therefore we find
dκ2(p)
(3.13) = ∇¯ T (w,w),T(w,w) = ∇¯ T (w,w),T (w,w)
w w
ds
(cid:10) (cid:11) (cid:10)(cid:0) (cid:1) (cid:11)
Since ∇¯ T (w,w)=0, this implies
w
dκ2(0)
=0
ds
at p = γ(0). Thus p is a vertex of the normal section γ(s). (c) ⇒ (a) : If M has
planar normal sections, then Theorem 3.1 gives
(3.14) T (w,w)∧ ∇¯ T (w,w)=0.
w
(cid:0) (cid:1)
If p is a vertex of γ(s), then we have
dκ2(0)
=0.
ds
Thus, since M has planar normal sections using (3.13) we find
γ′(s)∧γ′′(s)∧γ′′′(s) = w∧(∇∗w+T (w,w))
w
∧ ∇∗∇∗w+tT (w,w)+ ∇¯ T (w,w) =0
w w w
γ′(s)∧γ′′(s)∧γ′′′(s) = T(cid:0)(w,w)∧ ∇¯ T (w,w)=(cid:0)0 (cid:1) (cid:1)
w
(cid:0) (cid:1)
and
(3.15) ∇¯ T (w,w),T (w,w) =0.
w
(cid:10)(cid:0) (cid:1) (cid:11)
Combining (3.14) and (3.15) we obtain ∇¯ T (w,w) = 0 or T (w,w) = 0. Let us
w
defineU ={w ∈S(TM)|T(w,w)=0}.(cid:0)Ifint(cid:1)(U)6=∅,weobtain ∇¯ T (w,w)=
w
0 on int(U). Thus, by continuity we have ∇¯T =0. (cid:0) (cid:1) (cid:3)
Example 3.1. Consider the null cone of R3 given by
1
∧= (x ,x ,x )|−x2+x2+x2 =0,x ,x ,x ∈IR .
1 2 3 1 2 3 1 2 3
(cid:8) (cid:9)
The radical bundle of null cone is
∂ ∂ ∂
ξ =x +x +x
1∂x 2∂x 3∂x
1 2 3
and screen distribution is spanned by
∂ ∂
Z =x +x
1 2∂x 3∂x
1 2
LIGHTLIKE SURFACES 7
Then the lightlike transversal vector bundle is given by
1 ∂ ∂ ∂
Itr(TM)=Span{N = x +x −x }.
2(−x2+x2)(cid:18) 1∂x 2∂x 3∂x (cid:19)
1 2 1 2 3
Itfollows that the corresponding screen distribution S(TM)is spannedby Z . Thus
1
∂ ∂ ∂
∇ ξ = x +x +x
ξ 1∂x 2∂x 3∂x
1 2 3
∂ ∂ ∂
∇¯ ∇ ξ = x +x +x .
ξ ξ 1∂x 2∂x 3∂x
1 2 3
Then, we obtain
γ′′′(s)∧γ′′(s)∧γ′(s)=0
which shows that null cone has degenerate planar normal sections.
Example 3.2. Let R3 be the space IR3 endowed with the semi Euclidean metric
1
g¯(x,y)=−x y +x y +x y ,(x=(x ,x ,x )).
0 0 1 1 2 2 0 1 2
The lightlike cone ∧2 is given by the equation −(x )2+(x )2+(x )2 =0, x6=0. It
0 0 1 2
is known that ∧2 is a lightlike surface of R3 and the radical distribution is spanned
0 1
by a global vector field
∂ ∂ ∂
(3.16) ξ =x +x +x
0∂x 1∂x 2∂x
0 1 2
on ∧2. The unique section N is given by
0
1 ∂ ∂ ∂
(3.17) N = −x +x +x
2(x )2 (cid:18) 0∂x 1∂x 2∂x (cid:19)
0 0 1 2
and is also defined. As ξ is the position vector field we get
(3.18) ∇¯ ξ =∇ ξ =X, ∀X ∈Γ(TM).
X X
Then, A∗X +τ(X)ξ+X =0. As A∗ is Γ(S(TM))-valued we obtain
ξ ξ
(3.19) A∗X =−PX, ∀X ∈Γ(TM)
ξ
Next, any X ∈ Γ S(T∧2) is expressed by X = X ∂ +X ∂ where (X ,X )
0 1∂x1 2∂x2 1 2
satisfy (cid:0) (cid:1)
(3.20) x X +x X =0
1 1 2 2
and then
2 2
∂X ∂
(3.21) ∇ X = ∇¯ X = x a ,
ξ ξ A
∂x ∂x
AX=0Xa=1 A a
2 2 ∂Xa
(3.22) g¯(∇ X,ξ) = x x =−(x X +x X )=0
ξ a A∂x 1 1 2 2
AX=0Xa=1 A
where (3.20) is derived with respect to x ,x ,x . It is known that ∧2 is a screen
0 1 2 0
conformal lightlike surface with conformal function ϕ= 1 . We also know that
2(x0)2
A ξ =0. By direct compute we find
N
1
A X = A∗X.
N 2(x )2 ξ
0
8 FEYZAESRAERDOG˘AN,BAYRAMS¸AHI˙N∗,ANDRIFATGU¨NES¸
Now we evaluate γ′,γ′′ and γ′′′
γ′ = X =(0,−x ,x )
2 1
γ′′ = ∇ X+B(X,X)N
X
1 ∂ 3 ∂ ∂
= x − x +x
2 0∂x 2 1∂x 2∂x
0 1 2
γ′′′ = ∇¯ ∇ X +X(B(X,X))N +B(X,X)∇¯ N
X X X
= ∇ ∇ X +B(X,∇ X)N +X(B(X,X))N −B(X,X)A X
X X X N
using A X in γ′′′ we get
N
1 ∂ 1 ∂
γ′′′ = x − x .
2 2∂x 2 1∂x
1 2
Therefore γ′′′ and γ′are linear dependence at ∀p∈∧2 and we have
0
γ′∧γ′′∧γ′′′ =0.
Namely, ∧2 has non-degenerate planar normal sections.
0
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Faculty of Arts and Science, Department of Mathematics, Adıyaman University,
02040Adıyaman,TURKEY
E-mail address: [email protected]
Departmentof Mathematics,I˙no¨nu¨ University,44280 Malatya,TURKEY
E-mail address: [email protected]
Departmentof Mathematics,I˙no¨nu¨ University,44280 Malatya,TURKEY
E-mail address: [email protected]
