Geometry of lightlike hypersurfaces of a statistical manifold
OGUZHAN BAHADIR,
Department of Mathematics
Kahramanmaras Sutçu Imam University,
Faculty of Sciences, 46040, Kahramanmaras,
TURKEY
MUKUT MANI TRIPATHI,
Department of Mathematics
Banaras Hindu University
Institute of Sciences, Varanasi 221005
INDIA
Abstract: In this paper, we introduced the lightlike hypersurfaces of a statistical manifold. It is shown that a
lightlike hypersurface of a statistical manifold is not a statistical manifold with respect to the induced connections,
but the screen distribution has a canonical statistical structure. Some relations between induced geometric objects
with respect to dual connections in a lightlike hypersurface of a statistical manifold are obtained. An example is
presented. Induced Ricci tensors for lightlike hypersurface of a statistical manifold are computed.
Key-Words: Lightlike hypersurface, Statistical manifolds, Dual connections.
Received: November 9, 2022. Revised: May 7, 2023. Accepted: May 29, 2023. Published: June 16, 2023.
1 Introduction
A statistical manifold, the Riemannian connection
used to model the information, the fields of informa-
tion geometry, as such a generalization of the Rieman-
nian manifold equipped with a relatively new math-
ematics branch, uses the differential geometry tool
to examine the statistical inference, information loss
and prediction, [6]. In 1975, the role of differential
geometry in statistics was first emphasized by [12].
Later, Amari used differential geometric tools to de-
velop this idea, [1], [2].
A Riemannian manifold (f
M, e
g)with a Rieman-
nian metric e
gand the Levi-Civita connection e
D0is
called a statistical manifold if there exists a pair of
torsion-free connection (e
D, e
D∗)such that the fol-
lowing relation satisfies for any tangent vector fields
X, Y and Zon f
M
e
g(X, e
D∗
ZY) = Ze
g(X, Y )−e
g(e
DZX, Y ),(1)
where
e
D0=1
2(e
D+e
D∗).(2)
In 1989, [28], initiated the study of geometry
of submanifolds of statistical manifolds. He ob-
tained Gauss-Weingarten formulas, Gauss and Co-
dazzi equations, etc.. Later, in 2009, [14], studied
hypersurfaces of a statistical manifold. Also, studied
submanifolds of statistical manifolds of constant cur-
vature, [3]. In addition to, many authors have studied
on different types of statistical manifolds, [15], [26],
[27].
On the other hand, lightlike geometry is one of the
important research areas in differential geometry and
has many applications in physics and mathematics.
The geometry of lightlike submanifolds of a semi-
Riemannian manifold was presented by [9], (see also,
[10], [11]). Lightlike hypersurfaces in various spaces
have been studied by many authors including those of
[4], [5], [7] [8], [10], [13], [17], [18], [19], [21], [22],
[23], [24], [25].
Motivated by these circumstances, in this paper,
we initiate the study of lightlike geometry of statisti-
cal manifolds. In section 2, we present basic defini-
tions and results about statistical manifolds and light-
like hypersurfaces. In Section 3, we show that in-
duced connections on a lightlike hypersurface of a sta-
tistical manifold are not dual connections and a light-
like hypersurface is not statistical manifold. More-
over, we show that the second fundamental forms
are not degenerate. Later, we characterize the par-
allelness and integrability of the screen distribution.
Equivalent conditions are also obtained between the
induced objects. This section concludes with an ex-
ample. In section 4, we obtain formula for curvature
tensors of a lightlike hypersurface of a statistical man-
ifold. In general, in lightlike geometry, Ricci tensor is
not symmetric, so we also obtain new conditions for
Ricci tensor to be symmetric.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
466
Volume 22, 2023
2 Preliminaries
Let (¯
M, ¯g)be an (m+ 2)-dimensional semi-
Riemannian manifold with index(¯g) = q≥1. Let
(M, g)be a hypersurface of (¯
M, ¯g)with g= ¯g|M.
If the induced metric gon Mis degenerate, then M
is called a lightlike (null or degenerate) hypersurface
([9], [10], [11]). In this case, there exists a null vector
field ξ= 0 on Msuch that
g(ξ, X) = 0,∀X∈Γ (T M).(3)
The radical or the null space of TxM, at each point
x∈M, is a subspace Rad TxMdefined by
Rad TxM={ξ∈TxM:gx(ξ,X)=0, X∈Γ(T M )}.(4)
The dimension of Rad TxMis called the nullity de-
gree of g. We recall that the nullity degree of gfor a
lightlike hypersurface of (¯
M, ¯g)is 1. Since gis de-
generate and any null vector being orthogonal to it-
self, TxM⊥is also null and
Rad TxM=TxM∩TxM⊥.(5)
Since dim TxM⊥= 1 and dim Rad TxM= 1,we
have Rad TxM=TxM⊥. We call Rad T M a rad-
ical distribution and it is spanned by the null vector
field ξ. The complementary vector bundle S(T M)of
Rad T M in T M is called the screen bundle of M. We
note that any screen bundle is non-degenerate. This
means that
T M =Rad T M ⊥S(T M ),(6)
with ⊥denoting the orthogonal-direct sum. The com-
plementary vector bundle S(T M)⊥of S(T M)in
T¯
Mis called screen transversal bundle and it has
rank 2. Since Rad T M is a lightlike subbundle of
S(T M)⊥there exists a unique local section Nof
S(T M)⊥such that
¯g(N, N) = 0,¯g(ξ, N) = 1.(7)
Note that Nis transversal to Mand {ξ, N }is a local
frame field of S(T M)⊥and there exists a line sub-
bundle ltr(T M )of T¯
M, and it is called the lightlike
transversal bundle, locally spanned by N. Hence we
have the following decomposition:
T¯
M=T M ⊕ltr(T M)
=S(T M)⊥Rad T M ⊕ltr(T M ),(8)
where ⊕is the direct sum but not orthogonal ([9],
[10]). From the above decomposition of a semi-
Riemannian manifold ¯
Malong a lightlike hypersur-
face M, we can consider the local quasi-orthonormal
field of frames of ¯
Malong Mgiven by
{E1, . . . , Em, ξ, N },
where {E1, . . . , Em}is an orthonormal basis of
Γ(S(T M)). Let ¯
∇is the Levi-Civita connection of
(¯
M, ¯g). In view of the splitting (8), we have the fol-
lowing Gauss and Weingarten formulas, respectively,
¯
∇XY=∇XY+h(X, Y ),(9)
¯
∇XN=−ANX+∇t
XN(10)
for any X, Y ∈Γ(T M), where ∇XY, ANX∈
Γ(T M)and h(X, Y ),∇t
XN∈Γ(ltr(T M )). If we
set
B(X, Y ) = ¯g(h(X, Y ), ξ), τ (X) = ¯g(∇t
XN, ξ),
then (9) and (10) become
∇XY=∇XY+B(X, Y )N, (11)
∇XN=−ANX+τ(X)N, (12)
respectively. Here, Band Aare called the second
fundamental form and the shape operator of the light-
like hypersurface M, respectively, [9]. Let Pbe the
projection of T(M)on S(T(M)). Then, for any
X∈Γ(T M), we can write
X=P X +η(X)ξ, (13)
where ηis a 1-form given by
η(X) = ¯g(X, N ).(14)
From (11), we have
(∇Xg)(Y, Z) = B(X, Y )η(Z) + B(X, Z)η(Y),
(15)
for all X, Y, Z ∈Γ(T M), where the induced connec-
tion ∇is a non-metric connection on M. From (6),
we have
∇XW=∇∗
XW+h∗(X, W ) = ∇∗
XW+C(X, W )ξ,
(16)
∇Xξ=−A∗
ξX−τ(X)ξ(17)
for all X∈Γ(T M),W∈Γ(S(T M)), where ∇∗
XW
and A∗
ξXbelong to Γ(S(T M)). Here C,A∗
ξand
∇∗are called the local second fundamental form, the
local shape operator and the induced connection on
S(T M), respectively. We also have
g(A∗
ξX, W ) = B(X, W ), g(A∗
ξX, N ) = 0
B(X, ξ) = 0, g(ANX, N) = 0.(18)
Moreover, from the first and third equations of (18),
we have
A∗
ξξ= 0.(19)
The mean curvature Hof Mwith respect to an
{Ei}, i = 1, . . . m, orthonormal basis of Γ(S(T M))
is defined by
H=1
m
m
X
i=1
εiB(Ei, Ei), εi=g(Ei, Ei).(20)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
467
Volume 22, 2023
3 Lightlike hypersurfaces of a
statistical manifold
Let (f
M,e
g)be a semi-Riemannain manifold. If there
exists a torsion free connection e
Dsubject to the fol-
lowing identity
(e
DXe
g)(Y, Z) = ( e
DYe
g)(X, Z)(21)
for all X, Y, Z ∈Γ(Tf
M)then f
Mis called statistical,
[14]. For a statistical manifold (f
M, e
g), the e
g−dual of
e
D, denoted by e
D∗, is defined by the following iden-
tity:
e
g(X, e
D∗
ZY) = Ze
g(X, Y )−e
g(e
DZX, Y ).(22)
It is easy to check that e
D∗is torsion free. If e
D0is the
Levi-Civita connection of e
g, then we can write
e
D0=1
2(e
D+e
D∗).(23)
Note that a statistical manifold is represented by
(f
M, e
g, e
D, e
D∗).
Let (M, g)be a lightlike hypersurface of a statisti-
cal manifold (f
M, e
g, e
D, e
D∗). Then, Gauss and Wein-
garten formulas with respect to dual connections are
given by [14]
e
DXY=DXY+B(X, Y )N, (24)
e
DXN=−A∗
NX+τ∗(X)N, (25)
e
D∗
XY=D∗
XY+B∗(X, Y )N, (26)
e
D∗
XN=−ANX+τ(X)N(27)
for all X, Y ∈Γ(T M ), N ∈Γ(ltrT M), where
DXY,D∗
XY,ANX,A∗
NX∈Γ(T M)and
B(X, Y ) = e
g(e
DXY, ξ), τ∗(X) = e
g(e
DXN, ξ),
B∗(X, Y ) = e
g(e
D∗
XY, ξ), τ(X) = e
g(e
D∗
XN, ξ).
Here, D,D∗,B,B∗,ANand A∗
Nare called the
induced connections on M, the second fundamental
forms and the Weingarten mappings with respect to
e
Dand e
D∗, respectively. Using Gauss formulas, we
obtain
Xg(Y,Z)=g(e
DXY,Z)+g(Y, e
D∗
XZ),
=g(DXY,Z)+g(Y,D∗
XZ)
+B(X,Y )η(Z)+B∗(X,Z)η(Y).(28)
From the equation (28), we have the following re-
sult.
Theorem. Let (M, g)be a lightlike hypersurface of a
statistical manifold (f
M, e
g, e
D, e
D∗). Then:
(i) Induced connections Dand D∗are not dual con-
nections.
(ii) A lightlike hypersurface of a statistical manifold
need not to be a statistical manifold with respect
to the dual connections.
Using Gauss and Weingarten formulas in (28), we
get
(DXg)(Y, Z)+(D∗
Xg)(Y, Z) = B(X, Y )η(Z)
+B(X, Z)η(Y) + B∗(X, Y )η(Z).
+B∗(X, Z)η(Y)(29)
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then the fol-
lowing assertions are true:
(i) Induced connections Dand D∗are symmetric
connections.
(ii) The second fundamental forms Band B∗are
symmetric.
Proof. We know that Te
D= 0. Moreover,
TeD(X,Y )=e
DXY−e
DYX−[X,Y ]
=DXY−DYX−[X,Y ]
+B(X,Y )N−B(Y,X)N=0.(30)
Comparing the tangent and transversal components of
(30), we obtain
B(X, Y ) = B(Y, X), T D= 0,
where TDis the torsion tensor field of D. Thus, sec-
ond fundamental form Bis symmetric and induced
connection Dis symmetric connection.
Similarly, it can be shown that the second funda-
mental form B∗is symmetric and the induced con-
nection D∗is a symmetric connection.
Let Pdenote the projection morphism of Γ(T M)
on Γ(S(T M)) with respect to the decomposition (6).
Then, we have
DXP Y =∇XP Y +h(X, P Y ),(31)
DXξ=−AξX+∇t
Xξ(32)
for all X, Y ∈Γ(T M )and ξ∈Γ(RadT M), where
∇XP Y and AξXbelong to Γ(S(T M)),∇and ∇tare
linear connections on Γ(S(T M)) and Γ(RadT M )
respectively. Here, hand Aare called screen sec-
ond fundamental form and screen shape operator of
S(T M), respectively. If we define
C(X, P Y ) = g(h(X, P Y ), N),(33)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
468
Volume 22, 2023
ε(X) = g(∇t
Xξ, N ),∀X, Y ∈Γ(T M).(34)
One can show that
ε(X) = −τ(X).
Therefore, we have
DXP Y =∇XP Y +C(X, P Y )ξ, (35)
DXξ=−AξX−τ(X)ξ, ∀X, Y ∈Γ(T M ).(36)
Here C(X, P Y )is called the local screen fundamen-
tal form of S(T M).
Similarly, the relations of induced dual objects on
S(T M)are given by
D∗
XP Y =∇∗
XP Y +C∗(X, P Y )ξ, (37)
D∗
Xξ=−A∗
ξX−τ∗(X)ξ, ∀X, Y ∈Γ(T M ).(38)
Using (28), (35), (37) and Gauss-Weingarten formu-
las, the relationship between induced geometric ob-
jects are given by
B(X, ξ)+B∗(X, ξ) = 0, g(ANX+A∗
NX, N ) = 0,
(39)
C(X, P Y ) = g(ANX, P Y ),
C∗(X, P Y ) = g(A∗
NX, P Y ).(40)
Now, using the equation (39) we can state the fol-
lowing result.
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then second
fundamental forms Band B∗are not degenerate.
Additionally, due to e
Dand e
D∗are dual connec-
tions we obtain
B(X, Y ) = g(A∗
ξX, Y ) + B∗(X, ξ),(41)
B∗(X, Y ) = g(AξX, Y ) + B(X, ξ).(42)
Using (41) and (42) we get
A∗
ξξ+Aξξ= 0.
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then the
screen distribution (S(T M), g, ∇,∇∗)has a statisti-
cal structure.
Proof. From (28), for any X, Y ∈Γ(S(T M)) we
obtain
Xg(Y, Z) = g(DXY, Z) + g(Y, D∗
XZ).
Using (35) and (37) in the last equation, we get
Xg(Y, Z) = g(∇XY, Z) + g(Y, ∇∗
XZ).
Thus ∇and ∇∗are dual connections. Moreover, the
torsion tensor of S(T M)with respect to ∇is given
T∇(X, Y ) = ∇XY− ∇YX−[X, Y ].
Using (35) in the last equation we obtain T∇= 0.
Similarly, the torsion tensor of S(T M )with respect
to ∇∗is equal to zero. Also, using (35) we have
(∇Xg)(Y, Z) = (∇Yg)(X, Z).
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then the fol-
lowing assertions are equivalent:
(i) The screen distribution S(T M)is parallel.
(ii) C(X, Y ) = 0 for all X, Y ∈Γ(S(T M)).
(iii) C∗(X, Y ) = 0 for all X, Y ∈Γ(S(T M)).
Proof. For any X, Y ∈Γ(S(T M)), from Gauss-
Weingarten formulas and (40), we obtain
g(D∗
XY, N) = C∗(X, Y ),(43)
g(DXY, N) = C(X, Y ),(44)
Then, the proof is completed.
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then the fol-
lowing assertions are equivalent:
(i) The screen distribution S(T M)is integrable.
(ii) C(Y, X) = C(X, Y )for all X, Y ∈Γ(S(T M )).
(iii) C∗(X, Y ) = C∗(Y, X)for all X, Y ∈
Γ(S(T M)).
Proof. For any X, Y ∈Γ(S(T M)), from Gauss-
Weingarten formulas and (40), we obtain
g([X, Y ], N) = C(X, Y )−C(Y, X).(45)
g([X, Y ], N) = C∗(X, Y )−C∗(Y, X).(46)
These equations prove our assertions.
Considering ([11], [16], [20]), we can give the fol-
lowing definition
Definition. Let (M, g)be a hypersurface of a statis-
tical manifold (f
M, e
g, e
D, e
D∗).
(i) Mis called totally geodesic with respect to e
Dif
B= 0.
(ii) Mis called totally geodesic with respect to e
D∗if
B∗= 0.
(iii) Mis called totally tangentially umbilical with
respect to e
Dif B(X, Y ) = kg(X, Y )for all
X, Y ∈Γ(T M), where kis smooth function.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
469
Volume 22, 2023
(iv) Mis called totally tangentially umbilical with re-
spect to e
D∗if B∗(X, Y ) = k∗g(X, Y ), for any
X, Y ∈Γ(T M), where k∗is smooth function.
(v) Mis called totally normally umbilical with re-
spect to e
Dif A∗
NX=kX for any X, Y ∈
Γ(T M), where kis smooth function.
(vi) Mis called totally normally umbilical with re-
spect to e
D∗if ANX=k∗Xfor all X, Y ∈
Γ(T M), where k∗is smooth function.
In view of (36), (38), (41) and (42), we have the
following proposition.
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then the fol-
lowing assertions are equivalent:
(i) Mis totally geodesic with respect to e
D(resp. M
is totally geodesic with respect to e
D∗).
(ii) A∗
ξvanishes on M(resp. Aξvanishes on M).
(iii) RadT M is a parallel distribution with respect to
e
D(resp. RadT M is a parallel distribution with
respect to e
D∗).
(iv) B∗(X, Y ) = g(AξX, Y )(resp. B(X, Y ) =
g(A∗
ξX, Y )), for all X, Y ∈Γ(T M).
Next, we have the following
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then the fol-
lowing assertions are equivalent:
(i) Mis totally geodesic with respect to e
Dand e
D∗.
(ii) AξX=A∗
ξX= 0 for all X∈Γ(T M).
(iii) DXg+D∗
Xg= 0 for all X∈Γ(T M).
(iv) DXξ+D∗
Xξ∈Γ(RadT M )for all X∈Γ(T M).
Proof. From (39), (41) and (42) we get the equiv-
alence of (i) and (ii). The equation (29) implies the
equivalence of (i) and (iii). Next, by using (36) and
(38) we have the equivalence of (ii) and (iv).
Theorem. Let (M, g)be a lightlike hypersurface of
a statistical manifold (f
M, e
g, e
D, e
D∗). Then, Mis to-
tally tangentially umbilical with respect to e
Dand e
D∗
if and only if
A∗
ξX+AξX=ρX, ∀X∈Γ(T M ),
where ρis smooth function.
Proof. Using (41) and (42) we obtain
kg(X, Y ) = g(A∗
ξX, Y ) + B∗(X, ξ),(47)
and
k∗g(X, Y ) = g(AξX, Y ) + B(X, ξ).(48)
If we add the equations (47) and (48) side by side and
using (39) we complete the proof.
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). If Mis totally
normally umbilical with respect to e
Dand e
D∗. Then
C(X, P Y ) + C∗(X, P Y ) = 0,∀X∈Γ(T M).
Proof. Let kand k∗be smooth functions and let
A∗
NX=kX and ANX=k∗X, then using (39) we
get k+k∗= 0. Thus, from (40) proof is completed.
It is known that Mis screen locally conformal
lightlike hypersurface of a statistical manifold f
Mif
AN=φA∗
ξ, A∗
N=φ∗Aξ,(49)
where φand φ∗are non-vanishing smooth functions
on M. Using (40) and (49) we get the following
proposition.
Proposition. Let (M, g)be a lightlike hypersurface
of a statistical manifold (f
M, e
g, e
D, e
D∗). Then, Mis
screen locally conformal if and only if
C(X, Y ) + C∗(X, Y ) = σ(B(X, Y ) + B∗(X, Y )),
,for all X, Y ∈Γ(S(T M)) ,where σis non-vanishing
smooth functions on M.
Now, we give an example.
Example. Let (R4
2,e
g)be a 4-dimensional semi-
Euclidean space with signature (−,−,+,+) of the
canonical basis (∂0, . . . , ∂3). Consider a hypersurface
Mof R4
2given by
x0=x1+√2qx2
2+x2
3.
For simplicity, we set f=qx2
2+x2
3. It is easy to
check that Mis a lightlike hypersurface whose radical
distribution RadT M is spanned by
ξ=f(∂0−∂1) + √2(x2∂2+x3∂3).
Then the lightlike transversal vector bundle is given
by
ltr(T M ) = Span{N=1
4f2{f(−∂0+∂1)
+√2(x2∂2+x3∂3)}}.
It follows that the corresponding screen distribution
S(T M)is spanned by
{W1=∂0+∂1, W2=−x3∂2+x2∂3}.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
470
Volume 22, 2023
Then, by direct calculations we obtain
e
∇XW1=e
∇W1X= 0,
e
∇W2W2=−x2∂2−x3∂3,
e
∇ξξ=√2ξ, e
∇W2ξ=e
∇ξW2=√2W2,
for any X∈Γ(T M), [11].
We define an affine connection e
Das follows
e
DXW1=e
DW1X= 0,e
DW2W2=−2x2∂2
e
Dξξ=√2ξ−√2N, (50)
e
DW2ξ=e
DξW2=√2W2−√2W1.
Then using (23) we obtain
e
D∗
XW1=e
D∗
W1X= 0,e
D∗
W2W2=−2x3∂3
e
D∗
ξξ=√2ξ+√2N, (51)
e
D∗
W2ξ=e
D∗
ξW2=√2W2+√2W1.
Then (R4
2,e
g, e
D, e
D∗)is a statistical manifold. Thus,
by using Gauss formulas (24) and (26) we obtain
B(X, W1) = B(W1, X) = 0,
B(W2, W2) = −2√2x2
2, B(ξ, ξ) = −√2
B(X, W2) = B(W2, X) = 0,(52)
and
B∗(X, W1) = B∗(W1, X) = 0,
B∗(W2, W2) = −2√2x2
3, B∗(ξ, ξ) = √2
B∗(X, W2) = B∗(W2, X) = 0.(53)
The equations (50), (51), (52) and (53) imply that in-
duced connections Dand D∗are symmetric connec-
tions and the second fundamental forms Band B∗are
symmetric. This verifies Proposition 3. Moreover,
the equations B(ξ, ξ) = −√2and B∗(ξ, ξ) = √2
show the accuracy of the Proposition 3.
Using (50), (51), (52) and (53) we get
DXW1=DW1X= 0, Dξξ=√2ξ,
DW2W2=√2x2
2
2f(−∂0+∂1)
+1
4f2{(4x3
2−2x2)∂2+ 4x3x2
2∂3)},
DW2ξ=DξW2=√2W2−√2W1,(54)
and
D∗
XW1=D∗
W1X= 0, D∗
ξξ=√2ξ,
D∗
W2W2=√2x2
3
2f(−∂0+∂1)
+1
4f2{4x2
3x2∂2+ (4x3
3−2x3)∂3)},
D∗
W2ξ=D∗
ξW2=√2W2+√2W1.
(55)
If we choose X=W2,Y=W2and Z=ξ, (54) and
(55) indicate that induced connections D∗and Dare
not dual connections. This verifies Theorem 3.
From (35) and (37), we have
C(X, W1) = C(W1, X) = 0,
C(W2, W2) = −√2
2(x2
f)2,
C(ξ, W2) = 0 (56)
and
C∗(X, W1) = C∗(W1, X) = 0,
C∗(W2, W2) = −√2
2(x3
f)2,
C∗(ξ, W2) = 0.(57)
From (56) and (57), we say that Cand C∗are sym-
metric. Thus we have Proposition 3.
Using (54) and (55) in (35) and (37) we obtain
∇XW1=∇W1X= 0,
∇W2W2=1
f2{(2x3
2−x2
2)∂2+ 2x3x2
2∂3},
∇ξW2=√2W2−√2W1,(58)
and
∇∗
XW1=∇∗
W1X= 0,
∇∗
W2W2=1
f2{2x2
3x2∂2+ (2x3
3−x3
2)∂3},
∇∗
ξW2=√2W2+√2W1.(59)
From (58) and (59), the torsion tensors vanish with
respect to ∇and ∇∗. Furthermore, ∇and ∇∗are dual
connections. This situation verifies Proposition 3.
4 Curvature tensors of a lightlike
hypersurface of a statistical
manifold
We denote by e
Rand e
R∗the curvature tensor of e
Dand
e
D∗, respectively. The curvature tensors satisfy
eg(e
R∗(X,Y )Z,W )=−eg(e
R(X,Y )W,Z).(60)
Using Gauss-Weingarten formulas, the curvature ten-
sors e
Rand e
R∗of the connection e
Dand e
D∗are given
by
e
R(X, Y )Z=R(X, Y )Z−B(Y, Z)A∗
NX
+ (B(Y, Z)τ∗(X)−B(X, Z)τ∗(Y))N
+ ((DXB)(Y, Z)−(DYB)(X, Z))N,
+B(X, Z)A∗
NY(61)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
471
Volume 22, 2023
and
e
R∗(X,Y )Z=R∗(X,Y )Z−B∗(Y,Z)ANX
+(B∗(Y,Z)τ(X)−B∗(X,Z)τ(Y))N
+((D∗
XB∗)(Y,Z)−(D∗
YB∗)(X,Z))N
+B∗(X,Z)ANY, (62)
where Rand R∗are the curvature tensor with respect
to Dand D∗, respectively. Consider curvature tensors
e
Rand e
R∗of type (0,4). From the above equation and
the Gauss-Weingarten equations for Mand S(T M )
we obtain
g(e
R(X,Y )Z,P W )=g(R(X,Y )Z,P W )
−B(Y,Z)C∗(X,P W )
+B(X,Z)C∗(Y,P W ),(63)
g(e
R∗(X,Y )Z,P W )=g(R∗(X,Y )Z,P W )
−B∗(Y,Z)C(X,P W )
+B∗(X,Z)C(Y,P W ),(64)
g(e
R(X,Y )Z,ξ)=B(Y,Z)τ∗(X)
+(DXB)(Y,Z)−(DYB)(X,Z)
−B(X,Z)τ∗(Y)(65)
g(e
R∗(X,Y )Z,ξ)=B∗(Y,Z)τ(X)
−B∗(X,Z)τ(Y)
+(D∗
XB∗)(Y,Z)
−(D∗
YB∗)(X,Z),(66)
g(e
R(X,Y )Z,N )=g(R(X,Y )Z,N )
−B(Y,Z)g(A∗
NX,N )
+B(X,Z)g(A∗
NY,N ),(67)
g(e
R∗(X,Y )Z,N )=g(R∗(X,Y )Z,N )
−B∗(Y,Z)g(ANX,N)
+B∗(X,Z)g(ANY,N),(68)
g(e
R(X,Y )ξ,N )=g(R(X,Y )ξ,N)
−B(Y,ξ)g(A∗
NX,N )
+B(X,ξ)g(A∗
NY,N ),(69)
g(e
R∗(X,Y )ξ,N )=g(R∗(X,Y )ξ,N)
−B∗(Y,ξ)g(ANX,N)
+B∗(X,ξ)g(ANY,N ),(70)
where
g(R(X, Y )ξ, N ) = C(Y, AξX)−C(X, AξY)
−2dτ(X, Y ),
g(R∗(X, Y )ξ, N ) = C∗(Y, A∗
ξX)
−C∗(X, A∗
ξY)−2dτ(X, Y ).
Now, let Mbe a lightlike hypersurface of a (m+
2)-dimensional statistical manifold f
M. We consider
the local quasi-orthonormal basis {Ei, ξ, N }, i =
1, . . . m, of f
Malong M, where {E1, . . . , Em}is an
orthonormal basis of Γ(S(T M)). Then, we obtain
RD(0,2)(X, Y ) =
m
X
i=1
εig(R(X, Ei)Y, Ei)
+e
g(R(X, ξ)Y, N ),(71)
where εidenotes the causal character (∓1) of respec-
tive vector field Ei. Using Gauss-Weingarten equa-
tions we have
g(R(X, Ei)Y, Ei) = g(e
R(X, Ei)Y, Ei)
+B(Ei, Y )C∗(X, Ei)
−B(X, Y )C∗(Ei, Ei)(72)
Substituting this in (71), using (40) and (41) we obtain
RD(0,2)(X, Y ) = g
Ric(X, Y )−B(X, Y )trA∗
N
+g(A∗
NX, A∗
ξY)
+g(R(X, ξ)Y, N )(73)
where g
Ric(X, Y )is the Ricci tensor of f
Mwith re-
spect to e
D. Similarly, dual tensor of Mwith respect
to D∗as follows:
RD∗(0,2)(X, Y ) = g
Ric∗(X, Y )−B∗(X, Y )trAN
+g(ANX, AξY)
+g(R∗(X, ξ)Y, N )(74)
From First Bianchi identities and (73) we get
RD(0,2)(X, Y )−RD(0,2)(Y, X)
=
m
X
i=1
εi((B(Ei, Y )C∗(X, Ei)
−B(Ei, X)C∗(Y, Ei) + g(e
R(X, Y )Ei, Ei))
+g(e
R(X, Y )ξ, N ).(75)
Therefore, RD(0,2) is not symmetric.
The statistical manifold (f
M, e
g)is called of con-
stant curvature cif
e
R(X, Y )Z=c(Y, Z)X−g(X, Z)Y. (76)
Moreover, if (e
D, e
g)is a statistical structure of con-
stant c, then using (60) we can easily see that (e
D∗,e
g)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
472
Volume 22, 2023
is also a statistical structure of constant c . Then, us-
ing (40), (41), (69) and (76) in (75) we have
RD(0,2)(X, Y )−RD(0,2)(Y, X)
=C∗(X, A∗
ξY)−C∗(Y, A∗
ξX),
and similarly
RD∗(0,2)(X, Y )−RD∗(0,2)(Y, X)
=C(X, AξY)−C(Y, AξX).
Then we have the following theorem
Theorem. Let (M, g)be a lightlike hypersurface of
a statistical manifold (f
Mn+2(c),e
g)of constant sec-
tional curvature c. Then the following assertions are
true:
(i) The tensor RD(0,2)(X, Y )is symmetric if and
only if
C∗(X, A∗
ξY) = C∗(Y, A∗
ξX).
(ii) The tensor RD∗(0,2)(X, Y )is symmetric if and
only if
C(X, AξY) = C(Y, AξX).
Thus, in view of Propositon 3, we have the follow-
ing:
Corollary. Let (M, g)be a lightlike hypersurface of
a statistical manifold (f
Mn+2(c),e
g)of constant sec-
tional curvature c. If S(T M)is parallel then the ten-
sors RD(0,2) and RD∗(0,2) are symmetric with respect
to connections Dand D∗, respectively.
5 Conclusion
Neural networks are useful for solving many com-
plex optimization problems in electromagnetic the-
ory. In 2019, the Event Horizon Telescope (EHT) col-
laboration released the first image of a black hole’s
shadow with the help of deep learning algorithms.
This image provides direct evidence for the existence
of black holes and the general theory of relativity, and
indirectly for the existence of lightlike geometry in
the universe. A statistical manifold is the emerging
branch of mathematics that generalizes the Rieman-
nian manifold and is used to model information; and
also uses differential geometry tools to study statisti-
cal inference, loss of information, and prediction. It
can be applied to many fields such as statistical man-
ifolds, neural networks, machine learning, and artifi-
cial intelligence. On the other hand, the study of light-
like manifolds is one of the most important research
areas in differential geometry, with many applications
in physics and mathematics, such as general relativ-
ity, electromagnetism, and black hole theory.
In this paper, we introduced a new structure on sta-
tistical manifolds. This is called lightlike hypersur-
face of a statistical manifold. We have characterized
some tensors of lightlike hypersurfaces on statistical
manifolds.
This study, which is made with a new perspective,
will open the way for scientists working in the field
of differential geometry and physics. Differential ge-
ometers and physicists can produce many studies by
applying the different types of this structure we de-
veloped on any kind of complex manifold.
References:
[1] S. Amari, Differential Geometry of curved ex-
ponential families - curvature and information
loss. Ann. Statist. 10 (1982), no. 2, 357–385.
MR0653513.
[2] S. Amari, Differential geometrical methods
in statistics. Lecture Notes in Statistics, 28.
Springer-Verlag, New York, 1985. MR0788689.
[3] M. E. Aydin, A. Mihai, I. Mihai, Some inequal-
ities on submanifolds in statistical manifolds of
constant curvature. Filomat 29 (2015), no. 3, 465–
476. MR3359440.
[4] C. Atindogbe, J. P. Ezin, J. Tossa, Lightlike Ein-
stein hypersurfaces in Lorentzian manifolds with
constant curvature. Kodai Math. J. 29 (2006), no.
1, 58–71. MR2222167.
[5] J.K. Beem, P.E. Ehrlich, K. L. Easley, Global
Lorentzian geometry. Second edition. Mono-
graphs and Textbooks in Pure and Applied Math-
ematics, 202. Marcel Dekker, Inc., New York,
1996. MR1384756 (97f:53100).
[6] O. Calin, C. Udriste, Geometric modeling
in probability and statistics. Springer, 2014.
MR3308142.
[7] K. L. Duggal, On scalar curvature in lightlike ge-
ometry. J. Geom. Phys. 57 (2007), no. 2, 473–
481. MR2271200.
[8] K. L. Duggal, Foliations of lightlike hyper-
surfaces and their physical interpretation. Cent.
Eur. J. Math. 10 (2012), no. 5, 1789–1800.
MR2949651.
[9] K. L. Duggal, A. Bejancu, Lightlike submani-
folds of semi-Riemannian manifolds and appli-
cations. Mathematics and its Applications, 364.
Kluwer Academic Publishers Group, Dordrecht,
1996. MR1383318 (97e:53121).
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
473
Volume 22, 2023
[10] K.L. Duggal, D.H. Jin, Null curves and hyper-
surfaces of semi-Riemannian manifolds. World
Scientific Publishing Co. Pte. Ltd., Hackensack,
NJ, 2007. MR2358723 (2008k:53149).
[11] K. L. Duggal, B. Sahin, Differential geometry of
lightlike submanifolds. Frontiers in Mathematics.
Birkhauser Verlag, Basel, 2010. MR2598375.
[12] B. Efron, Defining the curvature of a statistical
problem (with applications to second order effi-
ciency). With a discussion by C. R. Rao, Don A.
Pierce, D. R. Cox, D. V. Lindley, Lucien LeCam,
J. K. Ghosh, J. Pfanzagl, Niels Keiding, A. P.
Dawid, Jim Reeds and with a reply by the au-
thor. Ann. Statist. 3 (1975), no. 6, 1189–1242.
MR0428531.
[13] M. Gulbahar, E. Kilic, S. Keles, Chen-like
inequalities on lightlike hypersurfaces of a
Lorentzian manifold. J. Inequal. Appl. 2013,
2013:266, 18 pp. MR3112038.
[14] H. Furuhata, Hypersurfaces in statistical mani-
folds. Differential Geom. Appl. 27 (2009), no. 3,
420–429. MR2521902 (2010c:53028).
[15] H. Furuhata, Statistical hypersurfaces in the
space of Hessian curvature zero. Differential
Geom. Appl. 29 (2011), suppl. 1, S86–S90.
MR2832004 (2012g:53076).
[16] H. Furuhata, I. Hasegawa, Submanifold the-
ory in holomorphic statistical manifolds. Geome-
try of Cauchy-Riemann submanifolds, 179–215,
Springer, Singapore, 2016. MR3525856.
[17] S. Izumiya, Lightlike hypersurfaces along
spacelike submanifolds in anti-de Sitter space. J.
Math. Phys. 56 (2015), no. 11, 112502, 29 pp.
MR3427566.
[18] S. Kazan, B. Sahin, Pseudosymmetric lightlike
hypersurfaces in indefinite Sasakian space forms.
J. Appl. Anal. Comput. 6 (2016), no. 3, 699–719.
MR3472439.
[19] E. Kilic, O. Bahadir, Lightlike hypersurfaces
of a semi-Riemannian product manifold and
quarter-symmetric nonmetric connections. Int. J.
Math. Math. Sci. 2012, Art. ID 178390, 17 pp.
MR2964936.
[20] T. Kurose, Conformal-projective geometry
of statistical manifolds. Interdiscip. Inform.
Sci. 8 (2002), no. 1, 89–100. MR1923487
(2003g:53027).
[21] X. Liu, Q. Pan, On horizontal lightlike hyper-
surfaces of Robertson-Walker spacetimes. Com-
mun. Korean Math. Soc. 30 (2015), no. 2, 109–
121. MR3346485.
[22] F. Massamba, Lightlike hypersurfaces of indef-
inite Sasakian manifolds with parallel symmet-
ric bilinear forms. Differ. Geom. Dyn. Syst. 10
(2008), 226–234. MR2390019.
[23] F. Massamba, Killing and geodesic lightlike
hypersurfaces of indefinite Sasakian manifolds.
Turkish J. Math. 32 (2008), no. 3, 325–347.
MR2436262.
[24] F. Massamba, Lightlike hypersurfaces in indef-
inite trans-Sasakian manifolds. Results Math. 63
(2013), no. 1-2, 251–287. MR3009686.
[25] N. O. Poyraz, E. Yasar, Lightlike hypersurfaces
of a golden semi-Riemannian manifold. Mediterr.
J. Math. 14 (2017), no. 5, Art. 204, 20 pp.
MR3703454.
[26] A.N. Siddiqui, Y.J Suh, O. Bahadir, Extrem-
ities for Statistical Submanifolds in Kenmotsu
Statistical Manifolds, Filomat, Vol. 35, No. 2,
591=E2=80=93603 (2021).
[27] A.N. Siddiqui, B.-Y. Chen, O. Bahadir, Statisti-
cal solitons and inequalities for statistical warped
product submanifolds, Mathematics, Vol. 7, no.
9, 797 (2019).
[28] P. W. Vos, Fundamental equations for statisti-
cal submanifolds with applications to the Bartlett
correction. Ann. Inst. Statist. Math. 41 (1989), no.
3, 429–450. MR1032591 (91a:62013).
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.52
Oguzhan Bahadir, Mukut Mani Tripathi
E-ISSN: 2224-2880
474
Volume 22, 2023
