Blaschke Type Normalization on Light-Like Hypersurfaces
In this paper we construct and study a Blaschke type normalization on the lightlike hypersurface immersions with 1-degenerate second fundamental form. We discuss basic examples and establish fundamental equations for this canonical transversal vector bundle. As an application, we characterize the Ri...
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irk-123456789-1066502016-10-02T03:02:32Z Blaschke Type Normalization on Light-Like Hypersurfaces Atindogbé, C. In this paper we construct and study a Blaschke type normalization on the lightlike hypersurface immersions with 1-degenerate second fundamental form. We discuss basic examples and establish fundamental equations for this canonical transversal vector bundle. As an application, we characterize the Ricci at 1-degenerate Blaschke immersions. 2010 Article Blaschke Type Normalization on Light-Like Hypersurfaces / C. Atindogbé // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 4. — С. 362-382. — Бібліогр.: 20 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106650 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In this paper we construct and study a Blaschke type normalization on the lightlike hypersurface immersions with 1-degenerate second fundamental form. We discuss basic examples and establish fundamental equations for this canonical transversal vector bundle. As an application, we characterize the Ricci at 1-degenerate Blaschke immersions. |
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Article |
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Atindogbé, C. |
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Atindogbé, C. Blaschke Type Normalization on Light-Like Hypersurfaces Журнал математической физики, анализа, геометрии |
| author_facet |
Atindogbé, C. |
| author_sort |
Atindogbé, C. |
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Blaschke Type Normalization on Light-Like Hypersurfaces |
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Blaschke Type Normalization on Light-Like Hypersurfaces |
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Blaschke Type Normalization on Light-Like Hypersurfaces |
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Blaschke Type Normalization on Light-Like Hypersurfaces |
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Blaschke Type Normalization on Light-Like Hypersurfaces |
| title_sort |
blaschke type normalization on light-like hypersurfaces |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Blaschke Type Normalization on Light-Like Hypersurfaces / C. Atindogbé // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 4. — С. 362-382. — Бібліогр.: 20 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT atindogbec blaschketypenormalizationonlightlikehypersurfaces |
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2025-07-07T18:49:13Z |
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2025-07-07T18:49:13Z |
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1837015135998181376 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 4, pp. 362�382
Blaschke Type Normalization on Light-Like
Hypersurfaces
Cyriaque Atindogbé
Université d'Abomey-Calavi
Institut de Mathématiques et de Sciences Physiques (IMSP)
01 BP 613 Porto-Novo, Bénin
E-mail:atincyr@gmail, atincyr@imsp-uac.org
Received February 27, 2009
In this paper we construct and study a Blaschke type normalization on
the lightlike hypersurface immersions with 1-degenerate second fundamental
form. We discuss basic examples and establish fundamental equations for
this canonical transversal vector bundle. As an application, we characterize
the Ricci �at 1-degenerate Blaschke immersions.
Key words: 1-degenerate lightlike hypersurface, Blaschke normalization
(structure), null transversal vector �eld, characteristic vector �eld.
Mathematics Subject Classi�cation 2000: 53C50, 53A15.
1. Introduction
Before we indicate the aim of this article, let us recall some facts about the
geometry of hypersurfaces in the Lorentzian spaces Rn+2
1 . In such spaces, due to
the causal character of three categories (space-like, time-like and null) of the vector
�elds, there are three types of hypersurfaces M , namely, the Riemannian, the
Lorentzian and the lightlike (null) ones. The induced metric g is a non-degenerate
metric tensor �eld or a degenerate symmetric tensor �eld on M depending on
whether M is of the �rst two types or the third one. As space-like and time-like
hypersurfaces have a non-degenerate (semi)-Riemannian metric, one can consider
all the fundamental intrinsic and extrinsic geometric notions. In particular, a
well de�ned (up to sign) notion of the unit orthogonal vector �eld is known to
lead to a canonical decomposition of the ambient tangent space Rn+2
1 into two
factors: tangent (to M ) and orthogonal. Therefore, by respective projections,
one has fundamental equations such as the Gauss, the Codazzi, the Weingarten
equations,... along with the second fundamental form, sharp operator, induced
connection, etc.
c© Cyriaque Atindogbé, 2010
Blaschke Type Normalization on Light-Like Hypersurfaces
As for lightlike hypersurfaces, the normal bundle is a subbundle of the tan-
gent one, the basic nuisance in studying their extrinsic geometry arises from the
normalization problem.
Several authors considered this problem in various ways (Akivis�Goldberg
[1, 2], Penrose [16], Katsuno [10], Dautcourt [7, 8], Rosca [18, 19], Carter [6],
Taub [20], Larsen [14, 15], Pinl [17], ...). For the most part, these studies are
speci�c for a given problem and a general theory is still desirable.
Following are two important attempts. In [11, 12, 13], Kupeli developed an ap-
proach using the factor vector bundle TM? = TM/TM⊥, where TM⊥ is the cha-
racteristic null line bundle, and used the canonical projection π : TM −→ TM?
in studying the intrinsic geometry of the degenerate semi-Riemannian manifolds.
This approach switches the null geometry of the submanifold for a non-degenerate
one. In 1991, Duggal and Bejancu [9] introduced a general geometric technique to
deal with the above anomaly. Their approach is basically extrinsic in contrast to
the intrinsic one developed by Kupeli, that is very close and consistent with the
known theory of non-degenerate submanifolds. This approach introduces a non-
degenerate screen distribution (or equivalently a null transversal vector bundle)
so as to get three factors splitting the ambient tangent space and derive the main
induced geometric objects such as second fundamental forms, sharp operators,
induced connections, curvature, etc. Unfortunately, the screen distribution is not
unique and there is no preferred one in general. As a consequence, it is a system-
atic task in this approach to study a dependence of the discussed structures and
the induced geometric objects with respect not only to the screen distribution but
also to the choice of the normalizing pair of null vectors.
Obviously, this situation is very close to the classical a�ne di�erential ge-
ometry in which the fundamental fact is the existence of the Blaschke structure.
It is our aim in this article to introduce and study a natural analogue of the
Blaschke structure for the class of lightlike hypersurfaces in the Lorentz spaces
Rn+2
1 , (n ≥ 1). More precisely, for a 1-degenerate lightlike hypersurface immer-
sion, we will �rst show the existence of a unique (up to sign) normalized null
transversal vector �eld that is equia�ne and satis�es some apolarity condition.
Thereafter we make a systematic study of the geometry of this structure.
The paper is organized as follows. In Section 2 we make a general set up
on the lightlike hypersurfaces and establish some technical results. In Section 3
we introduce an admissible invariant metric volume form used in Section 4 to
construct the Blaschke structure. Section 5 is devoted to some basic examples.
In Section 6 we study the Blaschke fundamental equations and characterize the
Ricci �at 1-degenerate Blaschke immersions.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 363
Cyriaque Atindogbé
2. Basic Facts on Lightlike Hypersurfaces
Consider a hypersurface M of an (n + 2)-dimensional semi-Riemannian man-
ifold (M, g) of constant index 0 < ν < n + 2. In the classical theory of non-
degenerate hypersurfaces, the normal bundle has trivial intersection {0} with
the tangent bundle and plays an important role in the introduction of the main
induced geometric objects on M . In case of lightlike (degenerate, null) hypersur-
faces, the situation is totally di�erent. The normal bundle TM⊥ is a rank-one
distribution on M : TM⊥ ⊂ TM and then coincides with the so-called radical
distribution RadTM = TM ∩ TM⊥. Hence, the induced metric tensor �eld g is
degenerate on M and it has a constant rank n.
A complementary bundle of TM⊥ in TM is a rank n non-degenerate distri-
bution on M . It is called a screen distribution on M and is often denoted by
S(TM). A lightlike hypersurface with a speci�c screen distribution is denoted
by the triple (M, g, S(TM)). As TM⊥ lies in the tangent bundle, the following
result is important in studying the extrinsic geometry of a lightlike hypersurface.
Theorem 2.1. [9] Let (M, g, S(TM)) be a lightlike hypersurface of (M, g)
with a given screen distribution S(TM). Then there exists a unique rank 1 vector
subbundle tr(TM) of TM |M such that for any non-zero section ξ of TM⊥ on a
coordinate neighbourhood U ⊂ M there exists a unique section N of tr(TM) on
U satisfying
g(N, ξ) = 1, g(N,N) = g(N,W ) = 0, ∀ W ∈ Γ(ST (M)|U ). (2.1)
Throughout the paper, all manifolds will be assumed to be smooth, connected
and paracompact. We denote by Γ(E) the F(M)-module of the smooth sections
of a vector bundle E over M , F(M) being the algebra of smooth functions on M .
Also, by ⊕Orth and ⊕ we denote the orthogonal and non-orthogonal direct sum
of two vector bundles. By Theorem 2.1, we may write down the following decom-
positions:
TM = S(TM)⊕Orth TM⊥ (2.2)
and
TM |M = S(TM)⊕Orth (TM⊥ ⊕ tr(TM)) = TM ⊕ tr(TM). (2.3)
As it is well known, we have the following:
De�nition 2.1. Let (M, g, S(TM)) be a lightlike hypersurface of (M, g) with
a given screen distribution S(TM). The induced connection, say ∇, on M is
de�ned by
∇XY = Q(∇XY ), (2.4)
where ∇XY denotes the Levi-Civita connection on (M, g) and Q is the projection
onto TM with respect to the decomposition (2.3).
364 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
Remark 2.1. Notice that the induced connection ∇ on M depends on both g
and the speci�c given screen distribution S(TM) on M . Also, a choice of the
null line bundle tr(TM) is equivalent to the choice of S(TM).
The projections Q and I − Q give rise to the Gauss and the Weingarten
formulas in the form
∇XY = ∇XY + h(X, Y ), ∀X,Y ∈ Γ(TM), (2.5)
∇XV = −AV X +∇t
XV, ∀X,∈ Γ(TM) ∀V ∈ Γ(tr(TM)). (2.6)
Here, ∇XY and AV X belong to Γ(TM). Hence, h is a Γ(tr(TM))- valued sym-
metric F (M)-bilinear form on Γ(TM), AV is an F (M)-linear operator on Γ(TM),
and ∇t is a linear connection on the lightlike transversal vector bundle tr(TM).
Let P denote the projection morphism of Γ(TM) onto Γ(S(TM)) with respect
to the decomposition (2.2). We have
∇XPY =
?
∇X PY + h?(X,PY ) ∀X, Y ∈ Γ(TM), (2.7)
∇XU = − ?
AU X +∇?t
XU, ∀X,∈ Γ(TM) ∀U ∈ Γ(TM⊥). (2.8)
Here
?
∇X PY and
?
AU X belong to Γ(S(TM)),
?
∇ and ∇?t are the linear con-
nections on S(TM) and TM⊥ , respectively. Then, h? is a TM⊥ )-valued F (M)-
bilinear form on Γ(TM)×Γ(S(TM)), and
?
AU is a Γ(S(TM))-valued F (M)-linear
operator on Γ(TM). They are a second fundamental form and a shape operator
of the screen distribution, respectively.
Equivalently, consider a normalizing pair ξ, N as in Theorem 2.1. Then (2.5)
and (2.6) take the form
∇XY = ∇XY + BN (X,Y )N, ∀X, Y ∈ Γ(TM |U ) (2.9)
and
∇XN = −ANX + τN (X)N, ∀X ∈ Γ(TM |U ), (2.10)
where we put locally on U
BN (X, Y ) = g(h(X, Y ), ξ), ∀X,Y ∈ Γ(TM |U ), (2.11)
τN (X) = g(∇t
XN, ξ), ∀X ∈ Γ(TM |U ). (2.12)
It is important to emphasize that the local second fundamental form BN in (2.11)
does not depend on the choice of the screen distribution.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 365
Cyriaque Atindogbé
We also de�ne (locally) on U the following:
CN (X, PY ) = g(h?(X, PY ), N), ∀X,Y ∈ Γ(TM |U ). (2.13)
Thus, one has for X, Y ∈ Γ(TM |U )
∇XPY =
?
∇X PY + CN (X, PY )ξ (2.14)
and
∇XN = − ?
Aξ X − τN (X)ξ. (2.15)
It is straightforward to verify that for X, Y ∈ Γ(TM)
BN (X, ξ) = 0, BN (X, Y ) = g(
?
Aξ X,Y ),
?
Aξ ξ = 0, CN (X, PY ) = g(ANX,Y )
(2.16)
The induced connection is torsion-free, but not necessarily g-metric, and we have
for all tangent vector �elds X, Y and Z in TM
(∇Xg)(Y, Z) = BN (X,Y )η(Z) + BN (X,Z)η(Y ), (2.17)
where η is a 1-form de�ned by
η(·) = g(N, ·). (2.18)
From (2.17) it follows that ∇ is g-metric if and only if M is totally geodesic
(i.e., BN = 0). On the other hand, the linear connection
?
∇ is a metric connection
on S(TM).
The following lemma accounts for a relationship between the induced geomet-
ric objects described above with respect to the choice of two di�erent normalizing
pairs of the null vector �elds as in Theorem 2.1.
Lemma 2.1. Let {ξ, N} be a normalizing pair as in Theorem 2.1 and make a
change {ξ̃, Ñ} with Ñ = φN + ζ, where ζ ∈ Γ(TM) and φ ∈ C∞(M)?. Then
(a) ξ̃ =
1
φ
ξ,
(b) 2φη(ζ) + ||ζ||2 = 0,
(c) BÑ (X,Y ) =
1
φ
BN (X, Y ),
(d) P̃ = PY − 1
φ
g(ζ, Y )ξ,
366 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
(e) CÑ (X, P̃Y ) = φCN (X, PY )− g(∇Xζ, PY )
+ [τN (X) +
X · φ
φ
+
1
φ
BN (ζ,X)]g(ζ, Y ),
(f) ∇̃XY = ∇XY − 1
φ
BN (X,Y )ζ,
(g) τ Ñ = τN + d ln |φ|+ 1
φ
BN (ζ, ·),
(h) A
Ñ
= φAN −∇.ζ + [τN + d ln |φ|+ 1
φ
BN (ζ, ·)]ζ,
(i)
?
Aξ̃
=
1
φ
?
Aξ − 1
φ2
BN (ζ, ·)ξ,
for all tangent vector �elds X and Y .
P r o o f. The �rst two relations in items (a) and (b) are immediate con-
sequences of the relations g(Ñ , ξ̃) = 1, g(Ñ , Ñ) = 0 and dim(Rad(TM)) = 1.
Writing the Gauss, respectively the Weingarten, formulas for both pairs {ξ, N}
and {ξ̃, Ñ}, we obtain by identi�cation the relations in items (c) and (f) (respec-
tively, (g) and (h) ). Now let Y ∈ Γ(TM), we have
Y = P̃ Y + η̃(Y )ξ̃
= P̃ Y + η̃(Y )(
1
φ
ξ)
= P̃ Y +
1
φ
η̃(Y )ξ.
Then
P̃ Y = Y − 1
φ
η̃(Y )ξ
= Y − 1
φ
g(Ñ , Y )ξ
= Y − 1
φ
g(φN + ζ, Y )ξ
= Y − 1
φ
[φη(Y ) + g(ζ, Y )]ξ
= Y − η(Y )ξ − 1
φ
g(ζ, Y )ξ
= PY − 1
φ
g(ζ, Y )ξ
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 367
Cyriaque Atindogbé
and the item (d) is derived. By using the de�nition of CÑ , we have
CÑ (X, P̃Y ) = g(A
Ñ
X, P̃Y )
(h)
= g(φANX −∇Xζ + [τN (X)
+X · (ln |φ|) +
1
φ
BN (ζ, X)]ζ, P̃ Y )
(d)
= g(φANX −∇Xζ + [τN (X)
+X · (ln |φ|) +
1
φ
BN (ζ, X)]ζ, PY − 1
φ
g(ζ, Y )ξ)
(d)
= g(φANX −∇Xζ + [τN (X)
+X · (ln |φ|) +
1
φ
BN (ζ, X)]ζ, PY )
= CÑ (X, PY ) = φCN (X, PY )− g(∇Xζ, PY )
+[τN (X) +
X · φ
φ
+
1
φ
BN (ζ, X)]g(ζ, Y )
which establishes relation (e). Finally, we have
∇̃X ξ̃ = − ?
Aξ̃
X − τ Ñ (X)ξ̃.
But using (f), we get
∇̃X ξ̃ = ∇X ξ̃ − 1
φ
BN (X, ξ̃)ζ
= ∇X ξ̃
(a)
= ∇X(
1
φ
ξ)
= −X · (φ)
φ2
ξ +
1
φ
(− ?
Aξ X − τN (X)ξ)
= −[
X · (φ)
φ2
+
1
φ
τN (X)]ξ − 1
φ
?
Aξ X.
Identifying the above two expressions of ∇̃X ξ̃, we get
− ?
Aξ̃
X =
1
φ
?
Aξ X + [
X · (φ)
φ2
+
1
φ
τN (X)]ξ − 1
φ
τ Ñ (X)ξ
=
1
φ
?
Aξ X + [
X · (φ)
φ2
+
1
φ
τN (X)]ξ − 1
φ
[τN (X) +
X · (φ)
φ
+
1
φ
BN (ζ,X)]ξ
=
1
φ
?
Aξ X − 1
φ2
BN (ζ, X)ξ,
and we obtain the relation in (i), which completes the proof.
368 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
3. An Invariant Metric Volume Form
Consider a lightlike hypersurface immersion f : M −→ Rn+2
1 , and let g denote
the metric tensor �eld induced on f(M). We have
g(X,Y )|x = 〈f?X, f?Y 〉|f(x)
for any X, Y tangent to M , where 〈, 〉 := g denotes the Lorentz metric on Rn+2
1 ,
and f? denotes the tangent map. In the sequel, we identify M and f(M) and
write x and M instead of f(x) and f(M). Also, throughout the text, we consider
on Rn+2
1 a parallel volume form given by the standard determinant det.
Let N be a null transversal vector �eld on M . As BN is degenerate, it is not
possible to de�ne a volume form ωBN relative to BN in a usual way. By item
(c) in Lemma 2.1, it is remarkable that the rank of second fundamental form BN
is invariant under the change of transversal null vector �eld N . We de�ne this
invariant as a rank of the lightlike hypersurface immersion. Now consider the case
when BN has the maximal rank n (or equivalently has nullity degree 1). In this
case we say that the lightlike hypersurface immersion is 1-degenerate, which we
assume from now on.
The following range of indices will be in the order: i, j, ··· = 0, . . . , n; a, b, ··· =
1, . . . , n and α, β, · · · = 0, . . . , n+1. Let Q = TM/TM⊥ denote the factor bundle
by the characteristic line bundle, and for X, Y ∈ Γ(Q) set
BN (X, Y ) = BN (X,Y ). (3.1)
As B(ξ , ) = 0, BN is well de�ned. Furthermore, it is non-degenerate on Q. Let
F denote the frame bundle of M . Then each point of F is (x, e0, . . . , en), where
(e0, . . . , en) is a basis of TxM with (without loss of generality) e0 generating the
characteristic null space Rad(TxM). Consider some patch with coordinates (xi)
so that the coordinate vector �elds ∂i form a local basis of TM , with Rad(TM) =
span∂0 = ξ. These coordinate systems are called F-admissible coordinate systems
and a passage of a coordinate system (xi) in a coordinate system (yi) with ∂yc
∂x0
=
0, and y0 = εx0 + λ (ε = ±1, λ ∈ R) is called an admissible coordinate change.
At each point in the domain of such an admissible coordinate system, the matrix
of BN with respect to (∂i) has the form
(BN
ab) =
0 . . . . . . 0
...
... BN
ab
0
, (3.2)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 369
Cyriaque Atindogbé
where BN
ab = BN (∂a, ∂b) = BN (∂a, ∂b) = BN
ab are the entries of the invertible rank
n matrix of BN with respect to the (local) frame (∂a) of Γ(Q).
Let us de�ne a metric volume form on M relative to BN by
ωBN =
√
|det(BN
ab)|dx0 ∧ dx1 ∧ · · · dxn. (3.3)
The (n + 1)-form ωBN is indeed invariant under admissible coordinate changes.
Let (yj) be the admissible coordinates of another chart intersecting the one of
(xi)'s . In terms of the y-coordinates, the volume form is
|det(
∂xa
∂yc
∂xb
∂yd
BN
ab)|
1
2 dy0 ∧ dy1 ∧ · · · dyn,
and noting that ∂yc
∂x0
= 0, dy0 = εdx0 with ε = ±1 and dyc =
∂yc
∂xa
dxa, this
becomes
|det(
∂xa
∂yc
)|
√
|det(BN
ab)|εdx0 ∧ det(
∂yc
∂xa
)dx1 ∧ · · · dxn,
that is equal to
±ε
√
|det(BN
ab)|dx0 ∧ dx1 ∧ · · · dxn.
Hence, by appropriate choice of orientation, we get ±ε = 1, and ωBN is invariant
under admissible coordinate changes.
Remark 3.1. Starting with a null transversal vector �eld N , make a change
N = φN + ζ. Then, by item (c) in Lemma 2.1, we have
det(BÑ
ab) = φ−ndet(BN
ab),
hence we have
ω
BÑ = φ−
n
2 ωBN . (3.4)
4. The Blaschke Structure on the 1-Degenerate M
Let θ be an arbitrary volume form in the neighbourhood U of a point x.
An admissible basis (∂0, . . . , ∂n) of TxM is said to be unimodular for θ if
θ(∂0, . . . , ∂n) = 1. For a lightlike hypersurface immersion f : M −→ Rn+2
1 , let N
be a null transversal vector �eld and consider the parallel volume form on Rn+2
1
given by the standard determinant det. In addition to the induced geometric
objects discussed in Section , we set
θN (X0, . . . , Xn) = det[f?(X0), . . . , f?(Xn), N ]. (4.1)
370 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
Then θN is a volume form on M called the induced volume form (with respect to
N). Now, for an admissible basis (∂0, . . . , ∂n) of TxM , consider the matrix (BN
ab)
of BN . We have
det(BN
ab) =
∣∣∣∣∣∣∣∣∣∣∣∣
1 0 . . . . . . 0
0
...
... BN
ab
0
∣∣∣∣∣∣∣∣∣∣∣∣
=
∣∣∣∣∣∣∣∣∣∣∣∣
1 0 . . . . . . 0
0
...
... BN
ab
0
∣∣∣∣∣∣∣∣∣∣∣∣
as BN
ab = BN
ab using (3.1). Let
EN
a =
0
BN
1a
...
...
BN
na
and EN
0 =
1
0
...
...
0
.
Then,
det(BN
ab) = det(EN
0 , EN
1 , . . . , EN
n ) = ψ(BN )det(∂0, . . . , ∂n).
As there exists a non-vanishing function ρ (independent of (∂0, . . . , ∂n)) such that
det(∂0, . . . , ∂n) = ρθN (∂0, . . . , ∂n), it follows that
det(BN
ab) = ψ(BN )ρθN (∂0, . . . , ∂n).
Hence, if we restrict on a unimodular admissible basis for θN , then the determinant
of the matrix (BN
ab) is independent of the choice of unimodular admissible basis
(∂0, . . . , ∂n) for θN . We denote this number ψ(BN )ρ by detθN BN and call it the
determinant of BN relative to θN .
Remark 4.1. It follows that for an arbitrary admissible basis (∂0, . . . , ∂n) we
have
det(BN
ab) = detθN BNθN (∂0, . . . , ∂n). (4.2)
The following lemma shows what is the behaviour of detθN BN with respect
to a change in null transversal vector �eld.
Lemma 4.1. In the lightlike hypersurface immersion f : M −→ Rn+2
1 , suppose
we change a null transversal vector �eld N to Ñ = φN + ζ. Then,
det
θÑ BÑ = φ−(n+2)(detθN BN ). (4.3)
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Cyriaque Atindogbé
P r o o f. Using (4.1) for Ñ we �nd θÑ = φθN Then, (∂0, . . . , ∂n) be-
ing a unimodular admissible basis for θN , it follows that (∂0, φ
−1∂1, . . . , ∂n) is a
unimodular admisible basis for unimodular admissible basis for θÑ . Hence, we
obtain
det
θÑ BN = φ−2(detθN BN ). (4.4)
On the other hand, by item (c) in Lemma 2.1, we have BÑ =
1
φ
BN and
detθN BÑ = φ−n(detθN BN ). (4.5)
Finally, we get
det
θÑ BÑ (4.4)
= φ−2(detθN BÑ )
(4.5)
= φ−2φ−n(detθN BN )
= φ−(n+2)(detθN BN ).
Now, it is our aim to achieve, by an appropriate choice of the null transversal
vector �eld N , the following two goals:
(B1) (∇N , θN ) is an equia�ne structure, i.e ∇NθN = 0,
(B2) θN coincides with the volume element ωBN relative to the 1-degenerate
second fundamental form BN .
We prove the following result.
Theorem 4.1. Let f : M −→ Rn+2
1 be a 1-degenerate lightlike hypersurface
(isometric) immersion. For each point x0 ∈ M there is a null transversal vector
�eld de�ned in a neighbourhood of x0 satisfying conditions (B1) and (B2) above,
such a null transversal vector �eld is unique up to sign and gives rise to a nor-
malization of the null characteristic vector �eld.
P r o o f. Start by a tentative null transversal N , and note that by (4.2) in
Remark 4.1 we have
ωBN = |detθN BN | 12 |θN ((∂0, . . . , ∂n))| 12 dx0 ∧ dx1 · · · ∧ dxn.
It follows that θN = ωBN if and only if |detNθ BN | 12 = 1. Now make the change Ñ =
φN+ζ, ζ ∈ Γ(TM). Then, by Lemma 4.1, we have det
θÑ BÑ = φ−(n+2)(detθN BN ).
Hence to realize ω
BÑ = θÑ , it is necessary and su�cient to set
det
θÑ BÑ = 1 = φ−(n+2)(detθN BN ),
372 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
that is
|φ| = |detθN BN )| 1
n+2 . (4.6)
On the other hand, let D denote the �at connection on Rn+2
1 . From the standard
parallel volume form det, we have
0 = (DXdet)(X0, . . . , Xn, N) = (∇N
XθN )(X0,...,Xn)− τN (X)θN (X0, . . . , Xn)
for all basis (X1, . . . , Xn) and X is tangent to M . It follows that the equia�ne
condition is equivalent to τN = 0, that is DXN is tangent to M . Hence, in item
(g) in Lemma 2.1, φ being chosen as in (4.6), we have to choose ζ such that
τ Ñ = 0. But,
τ Ñ = 0 ⇐⇒ BN (ζ, )̇ = −φτN − dφ. (4.7)
BN is 1-degenerate with characteristic direction ξ, then the last equality in (4.7)
determines ζ up to the characteristic component. But from item (b) in Lemma 2.1
we have 2φη(ζ) + ||ζ||2 = 0. So, we only need a non-characteristic component
from (4.7), and we use the above relation to complete ζ.
Let ζ = ζ0ξ + ζa∂a. From (4.7), we have
−φτN (ξ)− ξ(φ) = 0,
and
ζaBN
ac = −φτN (∂c)− ∂c(φ).
As BN
ab = BN
ab, (a, b = 1 . . . , n), we have
ζaBN
ab = ζaBN
ac
= −φτN (∂c)− ∂c(φ).
Hence
ζa = −BN
ac
[φτN (∂c) + ∂c(φ)]. (4.8)
Then we have
‖ζ‖2 = gabB
N
ac
BN
be
[φτN (∂c) + ∂c(φ)][φτN (∂e) + ∂e(φ)],
and by item (b) in Lemma2.1, the null component of ζ is given by
η(ζ) = − 1
2φ
(
gabB
N
ac
BN
be
[φτN (∂c) + ∂c(φ)][φτN (∂e) + ∂e(φ)]
)
. (4.9)
Then ζ is determined by (4.8) and (4.9).
Now, we show that the null transversal vector �eld is unique up to sign.
Suppose N and Ñ = φN + ζ satisfy (B1) and (B2). It follows that
|detθN BN | = 1 = |det
θÑ BÑ |,
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 373
Cyriaque Atindogbé
that is, using (4.6), |φ| = 1 or |φ| = −1. But condition (B1) for both N and
Ñ leads to τN = τ Ñ = 0. Since φ = ±1, relation (4.8) leads to ζa = 0 for all
a = 1, . . . , n and ‖ζ‖ = 0. Then, as φ 6= 0, it follows item (b) in Lemma 2.1 that
η(ζ) = 0. Finally, we obtain ζ = 0, φ = ±1 and Ñ = ±N , which completes the
proof.
De�nition 4.1. A null transversal vector �eld satisfying (B1) and (B2) is
called Blaschke null transversal vector �eld. Locally it is uniquely determined up
to sign. For each point x ∈ M , the line through x in the direction of the Blaschke
null transversal vector Nx is independent of the choice of the sign for N and
is called Blaschke null transverse through x. The triplet (∇N , BN , AN ) is called
the Blaschke structure on the 1-degenerate lightlike hypersurface (M, g). The later
with this structure will be denoted (M, g,NBla). The unique null vector �eld ξ with
〈ξ,N〉 = 1 is called the Blaschke normalized null characteristic (radical) vector
�eld.
5. Some Examples
Beyond all physical considerations, the null cone ∧n+2
0 ⊂ Rn+2
1 is one of the
most important manifolds with lightlike metric. In fact, as we know from [5],
the null cone is, up to homogeneous Riemannian factor, the only homogeneous
lightlike manifold on which a Lie group with �nite center acts faithfully, isometri-
cally and non-properly. In this interest, the following example considers the case
of the lightlike cone ∧3
0 in the Minkowski space R4
1. This example can easily be
generalized to ∧n+2
0 ⊂ Rn+2
1 . Our second example is concerned with more general
Monge hypersurfaces.
5.1. Blaschke structure on the lightcone ∧3
0. Let us conider the lightcone
∧3
0 as the immersion
f : M = R3 \ {0} −→ R4
1
(x, y, z) 7−→
[
x, y, z, ε(x2 + y2 + z2)
1
2
]
, ε = ±1.
Locally, ∧3
0 is the graph t = ε(x2 + y2 + z2)
1
2 and it is an obvious fact that this is
a lightlike hypersurface immersion.
Let us take
N = x∂x + y∂y + z∂z − t∂t
as a tentative null transversal vector �eld. The induced volume form θN is thus
given by
θN (u, v, w) = det [f?u, f?v, f?w,N ] .
374 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
Let p0 = (x0, y0, z0) ∈ M . We may assume (without loss of generality) that
x0 6= 0 as p 6= 0. Then there is a neighbourhood U of p0 such that x 6= 0 on U .
Then, let
r =
1
2t2
(x∂x + y∂y + z∂z),
u1 = −1
x
(y∂x − x∂y),
u2 =
1
t
(z∂x − x∂z).
To see that (r, u1, u2) is a unimodular basis for θN , it is easy to check that
f?r =: e0 = ξ =
1
2t2
(x∂x + y∂y + z∂z + t∂t),
f?u1 =: e1 = −1
x
(y∂x − x∂y),
f?u2 =: e2 =
1
t
(z∂x − x∂z),
and then, θN (r, u1, u2)) = det[f?r, f?u1, f?u2, N ] = 1. We have also
〈e0, e0〉 = 〈e0, e1〉 = 〈e0, e2〉 = 〈e1, eN 〉 = 〈e2, N〉 = 0, and 〈e0, N〉 = 1.
Hence (e0, e1, e2) is an admissible basis on f(U). Now, D being the �at Levi�
Civita connection on R4
1, by direct calculation we have
Dre0 = 0,
Dre1 = 0,
Dre2 = 0,
Du1e0 =
1
2t2
e1,
Du1e1 = − y
x3
(y∂x − x∂y)− 1
x2
(x∂x + y∂y),
which shows that the transversal component of Du1e1 is −x2 + y2
2x2t2
N . Also,
Du1e2 =
y
tx
∂z,
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 375
Cyriaque Atindogbé
then its transversal component is yz
2xt3
N . We also obtain
Du2e0 = 0,
Du2e1 =
yz
2xt3
,
Du2e2 = −x2 + z2
2t4
,
It follows that the second fundamental form BN is given with respect to the
unimodular admissible basis (r, u1, u2) on M by
BN =
0 0 0
0 −x2 + y2
2x2t2
yz
2xt3
0
yz
2xt3
−x2 + z2
2t4
, (5.1)
which shows that ∧3
0 is a 1-degenerate lightlike hypersurface in R4
1 with
detθN BN =
(
1√
2t
)4
. (5.2)
Hence, we obtain
|φ| = 1√
2t
. (5.3)
In the sequel, we choose
φ =
1√
2t
. (5.4)
Now we compute τN . By similar calculations as above, we get
DrN =
1
t
N,
Du1N = e1,
Du2N = e2.
Hence, for all X tangent to U ⊂ M , τN (f?X) =
1
2t2
〈X,N〉, i.e.
τN =
1
2t2
η. (5.5)
376 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
It follows (4.8), (5.4) and (5.5), that ζa = 0, a = 1, 2 as η(e1) = η(e2) = 0 and
e1 · φ = e2 · φ = 0. We also get from (4.9), η(ζ) = 0 and then ζ = 0. Finally, we
obtain the Blaschke null transversal vector �eld along ∧3
0
Ñ =
1√
2t
(x∂x + y∂y + z∂z − t∂t). (5.6)
Remark 5.1. This enables a canonical Blaschke normalization of the null
characteristic (radical) vector �eld along ∧3
0 as follows:
ξ̃ =
1√
2t
(x∂x + y∂y + z∂z + t∂t). (5.7)
5.2. Monge surfaces in R3
1. Consider the graph M of the function F , x =
F (y, z) as the immersion f : Ω ⊂ R2 −→ R3
1 given by (y, z) 7−→ (F (y, z), y, z) ∈
R3
1 with F ∈ C∞(Ω). M is lightlike if and only if
(F ′
y)
2 + (F ′
z)
2 = 1. (5.8)
In this case, using ∂y and ∂z for coordinate vector �elds on R2, we have
f?(∂y) = (F ′
y, 1, 0),
f?(∂z) = (F ′
z, 0, 1),
and the null characteristic (radical) distribution is spanned by the null vector �eld
ξ = ∂x + F ′
y∂y + F ′
z∂z. (5.9)
Set
N = −∂x + F ′
y∂y + F ′
z∂z. (5.10)
As 〈ξ, N〉 = 2 and 〈N,N〉 = 0, let us take N as a tentative null transversal vector
�eld along f . The induced volume form (by the standard determinant) is given by
θN (u, v) = det(f?u, f?v, N).
A unimodular frame �eld for θN is then given by (ξ, W ) with
W =
1
2
(
F ′
z∂y − F ′
y∂z
)
;
in particular, (ξ,W,N) is an admissible frame �eld on R3
1 along f(M), according
to decomposition (2.3), and we have
〈ξ, ξ〉 = 〈ξ, W 〉 = 〈W,N〉 = 0 and 〈ξ,N〉 = 2.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 377
Cyriaque Atindogbé
Now, by straightforward calculation, one sees that the matrix of the local second
fundamental form BN with respect to the unimodular admissible basis ξ, W is
given by
BN =
0 0
0 −1
8
(F”yy + F”zz)
, (5.11)
which shows that the lightlike surface M is 1-degenerate provided ∆F = F”yy +
F”zz be everywhere non-zero on M , which we assume from now on. Then, the
determinant of BN relative to θN is given by detθN BN = −1
8
(F”yy + F”zz).
Hence, we obtain
|φ| =
[
|1
8
(F”yy + F”zz)|
] 1
3
=
1
2
(|∆F |) 1
3 .
As ∆F is continuous and nowhere vanishing, we may assume |φ| = 1
2
(∆F )
1
3 and
choose for the sequel
φ =
1
2
(∆F )
1
3 .
Now, by standard calculations and using di�erentiation of relation (5.8), one
�nds τN = 0. Set
L(F ) = F ′
zF
(3)
yyy + F ′
zF
(3)
yzz − F ′
yF
(3)
yyz − F ′
yF
(3)
zzz.
Then, using (4.8), (4.9), the above expression of φ and τN = 0, we obtain the
Blaschke null transversal vector �eld
Ñ =
1
2
(∆F )
1
3 N − 2
3
(∆F )−
5
3 L(F )W − 1
72
(∆F )−
10
3 L(F )2ξ. (5.12)
Remark 5.2. For ∆F > 0, the canonical Blaschke normalization of the null
characteristic (radical) vector �eld along M is as follows:
ξ̃ = [∆F ]−
1
3 (∂x + F ′
y∂y + F ′
z∂z). (5.13)
Also, if L(F ) = 0, then,
Ñ =
1
2
[∆F ]
1
3 (−∂x + F ′
y∂y + F ′
z∂z). (5.14)
378 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
6. Blaschke Fundamental Equations
Consider a Blaschke 1-degenerate (M, g, NBla). The following theorem sum-
marizes and accounts fundamental equations on this normalization.
Theorem 6.1. For the Blaschke structure (M, g,NBla), with Blaschke null
transversal N , we have the following:
g(R(X, Y )Z, PW ) = B(Y, Z)C(X, PW )−B(X,Z)C(Y, PW ), (6.1)
(∇XB) (Y,Z) = (∇Y B) (X, Z), (6.2)
(∇XC) (Y, PZ) = (∇Y C) (X, PZ), (6.3)
η(R(X,Y )Z) = 0, (6.4)
B(
?
Aξ X, Y ) = B(X,
?
Aξ Y ), (6.5)
C(
?
Aξ X, Y ) = C(X,
?
Aξ Y ), (6.6)
θ = ωB, (6.7)
∇ωB = 0 (6.8)
for X, Y , Z tangent to M , where ξ is the (Blaschke) normalized characteristic
(null) vector �eld.
P r o o f. The last two equalities are a part of the Blaschke conditions.
To show B(
?
Aξ X,Y ) = B(X,
?
Aξ Y ), use (2.16) and the symmetry of B. Now
recall the following equations [9] using the local Gauss�Codazzi equations from
the general setting:
〈R(X,Y )Z, ξ〉 = (∇XB) (Y,Z)− (∇Y B) (X,Z)
+τ(X)B(Y, Z)− τ(Y )B(X, Z), (6.9)
〈R(X, Y )Z, PW 〉 = 〈R(X,Y )Z,PW 〉+ B(X,Z)C(Y, PW )
−B(Y, Z)C(X, PW ), (6.10)
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4 379
Cyriaque Atindogbé
〈R(X, Y )ξ, N〉 = 〈R(X, Y )ξ, N〉 = C(
?
Aξ X, Y )− C(
?
Aξ Y, X)
−2dτ(X, Y ), (6.11)
〈R(X,Y )Z,N〉 = 〈R(X, Y )Z, N〉, (6.12)
〈R(X, Y )PZ,N〉 = (∇XC) (Y, PZ)− (∇Y C) (X, PZ)
+τ(Y )C(X,PZ)− τ(X)C(Y, PZ). (6.13)
Also, we see so far that the equia�ne condition is equivalent to τ = 0. Finally,
as the target space in the Blaschke immersion is the �at Rn+2
1 , set in the above
equations, R = 0 and τ = 0 and the proof is complete.
Corollary 6.1. For the Blaschke structure (Mn+1, g, NBla) with the Blaschke
null transversal N , we have the following:
(i) C = 0 if and only if R = 0.
(2) Ric(X, Y ) = B(X, Y )trAN −B(ANX, Y ) and if n > 1, Ric = 0 if and only
if C = 0.
(iii) For totally geodesic (M, g), if the Blaschke screen is totally umbilical in M
with C = λg, then λ = cte with
P r o o f. Let p ∈ M and assume C = 0 at p. Then, by (6.1) in Theorem 6.1,
g(R(X, Y )Z, PW ) = 0 for all tangent vectors X, Y , Z and W ; i.e. R(X,Y )Z ∈
Rad(TM). But η(R(X, Y )Z) = 0 from (6.4). It follows that R = 0. Conversely,
if R = 0, then B(Y, Z)C(X, PW ) = B(X, Z)C(Y, PW ) using (6.1). At p, as B is
symmetric and real valued, consider a quasi-orthonormal basis (ξ, e1, . . . , en) with
respect to B such that (e1, . . . , en) spans S(TpM). Then, for i, j, k with k 6= i,
we have BkkCkj = 0 and Cij = 0 for all i and j that is C = 0, which proves (i).
Now, the following formula of the Ricci tensor is known [4]:
Ric(X, Y ) = Ric(X,Y )− η(R(ξ, Y )X) + B(X, Y )trAN −B(ANX, Y ). (6.14)
Setting R = 0, we obtain the expression in item (ii). Henceforth, it is imme-
diate that if C = 0, then Ric = 0. Suppose Ric = 0. Then, B(X, Y )trAN −
B(ANX, Y ) = 0, i.e. B(trANX − ANX, Y ) = 0 for all Y . As B is 1-degenerate
with null direction 〈ξ〉, we have trANX − ANX ∈ Rad(TpM). Then ANX =
(trAN )PX for all X. Hence, we get trAN = n(trAN ). It follows that if n > 1,
we obtain trAN = 0, that is C = 0, and (ii) is proved. Let C = λg. We have
(∇XC)(Y, PZ) = ∇X(C(Y, PZ))− C(∇XY, PZ)− C(Y,
?
∇X PZ)
= X · [λg(Y, Z)]− λg(∇XY, PZ)− λg(Y,
?
∇X PZ)
= (X · λ)g(Y, PZ) + λ(∇Xg)(Y, PZ).
380 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 4
Blaschke Type Normalization on Light-Like Hypersurfaces
Then, by (6.3), we have
0 = (∇XC)(Y, PZ)− (∇Y C)(X,PZ)
= X · λ)g(Y, PZ)− Y · λ)g(X,PZ)
+λ[(∇Xg)(Y, PZ)−∇Y g)(X, PZ)].
But ∇Xg)(Y, PZ) = ∇Y g)(X,PZ) = 0 as M is totally geodesic, and we get
g((X · λ)Y − (Y · λ)X, PZ) = 0,
which means that
(X · λ)PY − (Y · λ)PX = 0
for all X, Y ∈ Γ(TM |U ), which shows that X · λ = 0 on U ⊂ M and the proof is
complete.
Corollary 6.2. In Blaschke normalization with n > 1, the induced Ricci is
�at if and only if the Blaschke screen is integrable with totally geodesic leaves (in
Mn+1 ) that are parallel along the null characteristic orbits.
P r o o f. This is immediate from item (ii) in Corollary 6.1 as C = 0 is
a rephrasing of the fact that the screen distribution is integrable with totally
geodesic leaves, parallel along the null characteristic orbits. Note that the later
condition reads C(ξ, PY ) = 0 for all Y while the former means that C(PX, PY )
= 0, for all X, Y .
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