Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold
We introduce slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold. We establish existence theorems for these submersions. We also investigate the necessary and sufficient conditions for the leaves of the distributions to be totally geodesic foliations in...
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| author | Bhatia, S. S. Kumar, R. Sachdeva, R. Бхатія, С. С. Кумар, Р. Сачдєва, Р. |
| author_facet | Bhatia, S. S. Kumar, R. Sachdeva, R. Бхатія, С. С. Кумар, Р. Сачдєва, Р. |
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| description | We introduce slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold. We
establish existence theorems for these submersions. We also investigate the necessary and sufficient conditions for the
leaves of the distributions to be totally geodesic foliations in indefinite almost Hermitian manifolds. |
| first_indexed | 2026-03-24T02:14:45Z |
| format | Article |
| fulltext |
UDC 517.9
R. Sachdeva, R. Kumar (Dept. Basic and Appl. Sci., Punjabi Univ., India),
S. S. Bhatia (School Math., Thapar Univ., India)
SLANT LIGHTLIKE SUBMERSIONS FROM AN INDEFINITE
ALMOST HERMITIAN MANIFOLD ONTO A LIGHTLIKE MANIFOLD
ПОХИЛI СВIТЛОПОДIБНI ЗАНУРЕННЯ З НЕВИЗНАЧЕНОГО
МАЙЖЕ ЕРМIТОВОГО МНОГОВИДУ В СВIТЛОПОДIБНИЙ МНОГОВИД
We introduce slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold. We
establish existence theorems for these submersions. We also investigate the necessary and sufficient conditions for the
leaves of the distributions to be totally geodesic foliations in indefinite almost Hermitian manifolds.
Введено поняття похилих свiтлоподiбних занурень з невизначеного майже ермiтового многовиду в свiтлоподiбний
многовид. Доведено теореми iснування для таких занурень. Вивчено необхiднi та достатнi умови для того, щоб
листки розподiлiв були повнiстю геодезичними розшаруваннями в невизначених майже ермiтових многовидах.
1. Introduction. Riemannian submersions between Riemannian manifolds were studied by O’Neill
[6] and Gray [5]. Later Watson [14] defined almost Hermitian submersions between almost Hermitian
manifolds. Semi-Riemannian submersions were introduced by O’Neill in [7]. It is known that when
M and B are Riemannian manifolds, then the fibers are always Riemannian manifolds. However,
when the manifolds are semi-Riemannian manifolds, then the fibers may not be semi-Riemannian
manifolds. Therefore in [10], Sahin introduced a screen lightlike submersion from a lightlike manifold
onto a semi-Riemannian manifold and in [11], Sahin and Gündüzalp introduced a lightlike submersion
from a semi-Riemannian manifold onto a lightlike manifold. As a generalization of almost Hermitian
submersions, Sahin [12] introduced slant submersions from almost Hermitian manifolds onto Rieman-
nian manifolds. The geometry of lightlike submanifolds has extensive uses in mathematical physics,
particularly in general relativity [3]. Also, it is well-known that semi-Riemannian submersions are
of interest in physics, owing to their applications in the Yang – Mills theory, Kaluza – Klein theory,
supergravity and superstring theories [1, 2, 4, 13]. Moreover, we obtained nonexistence of totally
contact umbilical proper slant lightlike submanifolds of indefinite Sasakian manifolds in [8]. Thus
all these motivated us to club the theory of lightlike submersions with slant submersions. In this
paper, we introduce slant lightlike submersions from an indefinite almost Hermitian manifold onto a
lightlike manifold. We establish existence theorems for such submersions. We also investigate the
necessary and sufficient conditions for the leaves of the distributions to be totally geodesic foliation
in indefinite almost Hermitian manifolds.
2. Lightlike submersions. The used notations and fundamental equations for lightlike submer-
sions are refereed from [11].
Let (M, g) be a real n-dimensional smooth manifold where g is a symmetric tensor field of type
(0, 2). The radical space \mathrm{R}\mathrm{a}\mathrm{d}TxM of TxM is defined by
\mathrm{R}\mathrm{a}\mathrm{d}TxM =
\bigl\{
\xi \in TxM : g(\xi ,X) = 0 \forall X \in TxM
\bigr\}
.
The dimension of \mathrm{R}\mathrm{a}\mathrm{d}TxM is called the nullity degree of g. If the mapping \mathrm{R}\mathrm{a}\mathrm{d}TM : x \in M \rightarrow
\rightarrow \mathrm{R}\mathrm{a}\mathrm{d}TxM defines a smooth distribution on M of rank r > 0, then \mathrm{R}\mathrm{a}\mathrm{d}TM is called the radical
c\bigcirc R. SACHDEVA, R. KUMAR, S. S. BHATIA, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7 963
964 R. SACHDEVA, R. KUMAR, S. S. BHATIA
distribution of M and the manifold M is called an r-lightlike manifold if 0 < r \leq n, for detail
see [3].
Let (M1, g1) be a semi-Riemannian manifold and (M2, g2) an r-lightlike manifold. Consider
a smooth submersion f : M1 \rightarrow M2, then f - 1(p) is a submanifold of M1 of dimension dimM1 -
dimM2 for p \in M2. The kernel of f\ast at the point p is given by
\mathrm{K}\mathrm{e}\mathrm{r} f\ast =
\bigl\{
X \in Tp(M1) : f\ast (X) = 0
\bigr\}
,
and (\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot is given by
(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot =
\bigl\{
Y \in Tp(M1) : g1(Y,X) = 0 \forall X \in \mathrm{K}\mathrm{e}\mathrm{r} f\ast
\bigr\}
.
Since Tp(M1) is a semi-Riemannian vector space therefore (\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot may not be a complementary
to \mathrm{K}\mathrm{e}\mathrm{r} f\ast so assume \bigtriangleup = \mathrm{K}\mathrm{e}\mathrm{r} f\ast \cap (\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot \not = \{ 0\} . Thus we have the following four cases of
submersions.
Case 1: 0 < \mathrm{d}\mathrm{i}\mathrm{m}\bigtriangleup < \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ), \mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot \bigr\} .Then \bigtriangleup is the radical subspace
of Tp(M1). Since \mathrm{K}\mathrm{e}\mathrm{r} f\ast is a real lightlike vector space therefore complementary nondegenerate
subspace \bigtriangleup in \mathrm{K}\mathrm{e}\mathrm{r} f\ast is S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) and we obtain
\mathrm{K}\mathrm{e}\mathrm{r} f\ast = \bigtriangleup \bot S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ).
Similarly
(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot = \bigtriangleup \bot S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot ,
where S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot is a complementary subspace of \bigtriangleup in (\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot . Since S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot is nondegen-
erate in Tp(M1), therefore we get
Tp(M1) = S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )\bot (S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ))
\bot ,
where (S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ))
\bot is the complementary subspace of S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) in Tp(M1). Since S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) and
(S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ))
\bot are nondegenerate therefore we have
(S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ))
\bot = S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot \bot (S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot )\bot .
Then from [3], we can construct a quasiorthonormal basis of M1 along \mathrm{K}\mathrm{e}\mathrm{r} f\ast therefore we obtain
g(\xi i, \xi j) = g(Ni, Nj) = 0, g(\xi i, Nj) = \delta ij ,
g(W\alpha , \xi j) = g(W\alpha , Nj) = 0, g(W\alpha ,W\beta ) = \epsilon \alpha \delta \alpha \beta ,
(1)
where \{ Ni\} are smooth lightlike vector fields of (S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot )\bot , \{ \xi i\} is a basis of \bigtriangleup and \{ W\alpha \} is
a basis of S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot . Denote the set of vector fields \{ Ni\} by \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) and consider
\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) = \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )\bot S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot .
Using (1), it is clear that \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) and \mathrm{K}\mathrm{e}\mathrm{r}(f\ast ) are not orthogonal to each other. Denote \scrV =
= \mathrm{K}\mathrm{e}\mathrm{r} f\ast , the vertical space of Tp(M1) and \scrH = \mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ), the horizontal space then we have
Tp(M1) = \scrV p \oplus \scrH p.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
SLANT LIGHTLIKE SUBMERSIONS FROM AN INDEFINITE ALMOST HERMITIAN MANIFOLD . . . 965
Definition 2.1 [11]. Let (M1, g1) be a semi-Riemannian manifold and (M2, g2) an r-lightlike
manifold. Let f : M1 \rightarrow M2 be a submersion such that:
(a) \mathrm{d}\mathrm{i}\mathrm{m}\bigtriangleup = \mathrm{d}\mathrm{i}\mathrm{m}\{ (\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) \cap (\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot \} = r, 0 < r < \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
\mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ),\mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot \bigr\} ;
(b) f\ast preserves the length of horizontal vectors, that is, g1(X,Y ) = g2(f\ast X, f\ast Y ) for X,Y \in
\in \Gamma (\scrH ).
Then f is called an r-lightlike submersion.
Case 2: \mathrm{d}\mathrm{i}\mathrm{m}\bigtriangleup = \mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) < \mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot . Then \scrV = \bigtriangleup and \scrH = S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot \bot
\bot \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) and f is called an isotropic submersion.
Case 3: \mathrm{d}\mathrm{i}\mathrm{m}\bigtriangleup = \mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot < \mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ). Then \scrV = S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )\bot \bigtriangleup and \scrH = \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
and f is called a coisotropic submersion.
Case 4: \mathrm{d}\mathrm{i}\mathrm{m}\bigtriangleup = \mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) = \mathrm{d}\mathrm{i}\mathrm{m}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot . Then \scrV = \bigtriangleup and \scrH = \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) and f is
called a totally lightlike submersion.
We need the following theorem to define slant lightlike submersion from an indefinite almost
Hermitian manifold onto a lightlike manifold.
Theorem 2.1. Let f : M1 \rightarrow M2 be an r-lightlike submersion from an indefinite almost Her-
mitian manifold (M1, g1, J1), where g1 is a semi-Riemannian metric of index 2r to an r-lightlike
manifold (M2, g2). Let J\bigtriangleup be a distribution on M such that \bigtriangleup \cap J\bigtriangleup = 0. Then any complementary
distribution to J\bigtriangleup \oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) in S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) is Riemannian.
Proof. Assume that J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) is invariant with respect to J therefore 1 = g(\xi ,N) =
= g(J\xi , JN) = 0, for any \xi \in \Gamma (\mathrm{R}\mathrm{a}\mathrm{d}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) and N \in \Gamma (\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )), which leads to a
contradiction. Also J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) does not belong to S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot , since S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot is orthogonal
to S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ), therefore 0 = g(J\xi , JN) = g(\xi ,N) = 1. Thus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) is distribution on M.
Moreover J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) does not belong to \bigtriangleup , if JN \in \Gamma (\bigtriangleup ) then J2N = - N \in \Gamma (J\bigtriangleup ) which is
a contradiction. Similarly J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) does not belong to J\bigtriangleup . Hence J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) \subset S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
such that J\bigtriangleup \cap J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) = \{ 0\} .
Denote the complementary distribution to J\bigtriangleup \oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) in S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) by D. Then for a
local quasiorthonormal frames on M1, \{ \xi 1, . . . , \xi r, J\xi 1, . . . , J\xi r, N1, . . . Nr, JN1, . . . , JNr\} form an
orthonormal basis of \bigtriangleup \oplus J\bigtriangleup \oplus \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )\oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{k}\mathrm{e}\mathrm{r} f\ast ). Now define \{ U1, . . . , U2r, V1, . . . , V2r\}
as
U1 =
1\surd
2
(\xi 1 +N1), U2 =
1\surd
2
(\xi 1 - N1),
U3 =
1\surd
2
(\xi 2 +N2), U4 =
1\surd
2
(\xi 2 - N2),
. . . . . . . . . . . . . . . . . .
U2r - 1 =
1\surd
2
(\xi r +Nr), U2r =
1\surd
2
(\xi r - Nr),
V1 =
1\surd
2
(J\xi 1 + JN1), V2 =
1\surd
2
(J\xi 1 - JN1),
V3 =
1\surd
2
(J\xi 2 + JN2), V4 =
1\surd
2
(J\xi 2 - JN2),
. . . . . . . . . . . . . . . . . .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
966 R. SACHDEVA, R. KUMAR, S. S. BHATIA
V2r - 1 =
1\surd
2
(J\xi r + JNr), V2r =
1\surd
2
(J\xi r - JNr).
Hence \mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{ \xi i, Ni, J\xi i, JNi\} is a nondegenerate space of constant \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x} 2r, that is, \bigtriangleup \oplus J\bigtriangleup \oplus
\oplus \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )\oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) is nondegenerate and of constant \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x} 2r on M1. Since
\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(TM1) =
= \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(\bigtriangleup \oplus \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) + \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(J\bigtriangleup \oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) + \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(D\bot S(\mathrm{K}\mathrm{e}\mathrm{r} f\bot
\ast )),
therefore we have 2r = 2r + \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(D \bot S(\mathrm{K}\mathrm{e}\mathrm{r} f\bot
\ast )), this implies that D \bot S(\mathrm{K}\mathrm{e}\mathrm{r} f\bot
\ast ) is Rieman-
nian and hence D is Riemannian.
Theorem 2.1 is proved.
3. Slant lightlike submersion. Since geometry of lightlike submanifolds has extensive uses
in mathematical physics therefore as a generalization of holomorphic and totally real submanifolds,
slant lightlike submanifolds of indefinite Hermitian manifolds were introduced by Sahin in [9] as
below.
Definition 3.1. Let M be an r-lightlike submanifold of an indefinite Hermitian manifold \=M of
index 2r. Then M is a slant lightlike submanifold of \=M if the following conditions are satisfied:
(A) \mathrm{R}\mathrm{a}\mathrm{d}(TM) is a distribution on M such that J \mathrm{R}\mathrm{a}\mathrm{d}TM \cap \mathrm{R}\mathrm{a}\mathrm{d}(TM) = \{ 0\} .
(B) For each nonzero vector field tangent to D at p \in U \subset M, the angle \theta (X) between JX and
the vector space Dp is constant, that is, it is independent of the choice of p \in U \subset M and X \in Dp,
where D is complementary distribution to J \mathrm{R}\mathrm{a}\mathrm{d}TM\oplus J \mathrm{l}\mathrm{t}\mathrm{r}(TM) in the screen distribution S(TM).
This constant angle \theta (X) is called slant angle of the distribution D. A slant lightlike submanifold is
said to be proper if D \not = \{ 0\} and \theta \not = 0,
\pi
2
.
Thus using Theorem 2.1 and the definition of slant lightlike submanifolds, we can define slant
lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold as
below.
Definition 3.2. Let (M1, g1, J) be a real 2m-dimensional indefinite almost Hermitian manifold,
where g1 is semi-Riemannian metric of index 2r, 0 < r < m and (M2, g2) an r-lightlike manifold.
Let f : M1 \rightarrow M2 be an r-lightlike submersion. We say that f is a slant lightlike submersion if the
following conditions are satisfied:
(C) J\bigtriangleup is a distribution in \mathrm{K}\mathrm{e}\mathrm{r} f\ast such that \bigtriangleup \cap J\bigtriangleup = \{ 0\} .
(D) For each nonzero vector field X tangent to D, the angle \theta (X) between JX and D is
constant, where D is complementary distribution to J\bigtriangleup \oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) in S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ).
Hence we have
TpM1 = \scrV p \oplus \scrH p =
=
\bigl\{
\bigtriangleup \bot (J\bigtriangleup \oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{k}\mathrm{e}\mathrm{r} f\ast )\bot D\} \oplus \{ f(D)\bot \mu \bot \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bigr\}
,
where \mu is the orthogonal complementary subbundle to f(D) in S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ). Let f be a slant lightlike
submersion from an indefinite almost Hermitian manifold (M1, g1, J) onto an r-lightlike manifold
(M2, g2), then any X \in \scrV p can be written as
JX = \phi X + \omega X, (2)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
SLANT LIGHTLIKE SUBMERSIONS FROM AN INDEFINITE ALMOST HERMITIAN MANIFOLD . . . 967
where \phi X and \omega X are the tangential and the transversal components of JX, respectively. Similarly
for any V \in \scrH p, we get
JV = BV + CV, (3)
where BV and CV are the tangential and the transversal components of JV, respectively. Denote by
P1, P2, Q1 and Q2 the projections on the distributions \bigtriangleup , J\bigtriangleup , J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{k}\mathrm{e}\mathrm{r} f\ast ) and D, respectively.
Then we can write
X = P1X + P2X +Q1X +Q2X (4)
for any X \in \scrV p. Applying J to (4) we obtain
JX = JP1X + JP2X + \phi Q2X + \omega Q2X + \omega Q1X (5)
for any X \in \scrV p. Then clearly
JP1X = \phi P1X \in \Gamma (J\bigtriangleup ), JP2X = \phi P2X \in \Gamma (\bigtriangleup ), \omega P1X = 0, \omega P2X = 0,
\phi Q2X \in \Gamma (D), \omega Q2X \in \Gamma (f(D)), \phi Q1X = 0, \omega Q1X \in \Gamma (\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )).
Therefore we can write
\phi X = \phi P1X + \phi P2X + \phi Q2X.
Since the geometry of Riemannian submersions is characterized by O’Neill’s tensors \scrT and \scrA .
Therefore Sahin [11] defined these tensors for lightlike submersions as
\scrT XY = h\nabla \nu X\nu Y + \nu \nabla \nu XhY, (6)
\scrA XY = \nu \nabla hXhY + h\nabla hX\nu Y, (7)
where \nabla is the Levi – Civita connection of g1. It should be noted that \scrT and \scrA are skew-symmetric
in Riemannian submersions but not in lightlike submersions because the horizontal and vertical
subspaces are not orthogonal to each other. \scrT and \scrA both reverses the horizontal and vertical
subspaces and moreover \scrT has symmetry property, that is
\scrT XY = \scrT Y X. (8)
Using (6) and (7), we have the following lemma.
Lemma 3.1. Let f be a slant lightlike submersion form an indefinite almost Hermitian manifold
(M1, g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2).
Then
(i) \nabla UV = \scrT UV + \nu \nabla UV,
(ii) \nabla V X = h\nabla V X + \scrT V X,
(iii) \nabla XV = \scrA XV + \nu \nabla XV,
(iv) \nabla XY = h\nabla XY +\scrA XY
for any X,Y \in \Gamma (\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) and U, V \in \Gamma (\mathrm{K}\mathrm{e}\mathrm{r} f\ast ).
Using (2) and (3) with Lemma 3.1, we obtain the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
968 R. SACHDEVA, R. KUMAR, S. S. BHATIA
Lemma 3.2. Let f be a slant lightlike submersion form an indefinite almost Hermitian manifold
(M1, g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2).
Then
(\nabla X\omega )Y = C\scrT XY - \scrT X\phi Y, (\nabla X\phi )Y = B\scrT XY - \scrT X\omega Y, (9)
where
(\nabla X\omega )Y = h\nabla X\omega Y - \omega \nabla XY, (\nabla X\phi )Y = \scrV \nabla X\phi Y - \phi \scrV \nabla XY
for any X,Y \in (\mathrm{k}\mathrm{e}\mathrm{r} f\ast ).
Theorem 3.1. Let f be a lightlike submersion from an indefinite almost Hermitian manifold
(M1, g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2).
Then f is a proper slant lightlike submersion if and only if
(i) J(\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) is a distribution on M1;
(ii) for any X \in \Gamma (\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) there exists a constant \lambda \in [ - 1, 0] such that
\phi 2Q2X = \lambda Q2X. (10)
Moreover, in this case, \lambda = - \mathrm{c}\mathrm{o}\mathrm{s}2 \theta .
Proof. Let f be a slant lightlike submersion then J\bigtriangleup is a distribution on S(TM). Hence using
Theorem 2.1, J(\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) is a distribution on M1. Next the slant angle between JQ2X and Dp
is constant and given by
\mathrm{c}\mathrm{o}\mathrm{s} \theta (Q2X) =
g(JQ2X,\phi Q2X)
| JQ2X| | \phi Q2X|
= - g(Q2X,\phi 2Q2X)
| Q2X| | \phi Q2X|
. (11)
On the other hand, \mathrm{c}\mathrm{o}\mathrm{s} \theta (Q2X) is also given by
\mathrm{c}\mathrm{o}\mathrm{s} \theta (Q2X) =
| \phi Q2X|
| JQ2X|
. (12)
Hence using (11) and (12), we obtain
\mathrm{c}\mathrm{o}\mathrm{s}2 \theta (Q2X) = - g(Q2X,\phi 2Q2X)
| Q2X| 2
.
Since the angle \theta (Q2X) is constant on D therefore we have \phi 2Q2X = \lambda Q2X, where \lambda = - \mathrm{c}\mathrm{o}\mathrm{s}2 \theta .
Conversely (i) implies that J\bigtriangleup is a distribution on S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )). Hence using Theorem 2.1, any
complementary distribution to J\bigtriangleup \oplus J \mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) in S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast ) is Riemannian.
Theorem 3.1 is proved.
Corollary 3.1. Let f be a proper slant lightlike submersion from an indefinite almost Hermitian
manifold (M1, g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold
(M2, g2) with slant angle \theta . Then, for any X,Y \in (\mathrm{K}\mathrm{e}\mathrm{r} f\ast ), we have
g1(\phi X, \phi Y ) = \mathrm{c}\mathrm{o}\mathrm{s}2 \theta g1(X,Y ), (13)
g1(\omega X,\omega Y ) = \mathrm{s}\mathrm{i}\mathrm{n}2 \theta g1(X,Y ). (14)
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SLANT LIGHTLIKE SUBMERSIONS FROM AN INDEFINITE ALMOST HERMITIAN MANIFOLD . . . 969
Theorem 3.2. Let f be a lightlike submersion from an indefinite almost Hermitian manifold
(M1, g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2).
Then f is a proper slant lightlike submersion if and only if
(i) J(\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) is a distribution on M ;
(ii) for any vector field X tangent to M1, there exists a constant \nu \in [ - 1, 0] such that
B\omega Q2X = \nu Q2X, (15)
where \nu = - \mathrm{s}\mathrm{i}\mathrm{n}2 \theta .
Proof. Let f be a slant lightlike submersion, then J(\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{k}\mathrm{e}\mathrm{r} f\ast )) is a distribution on M1. Next,
applying J to (5) and using (2) to (4), we obtain
- X = - P1X - P2X + \phi 2Q2X + \omega \phi Q2X +B\omega Q1X +B\omega Q2X,
comparing the components of the distribution D both sides of the above equation we get
- Q2X = \phi 2Q2X +B\omega Q2X, (16)
hence using (10), we obtain (15). Conversely, using (15) and (16), we have \phi 2Q2X = - \mathrm{c}\mathrm{o}\mathrm{s}2 \theta Q2X.
Hence proof follows from Theorem 3.1.
Further, we prove that the orthogonal complement subbundle \mu of fD in S(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )
\bot is holo-
morphic with respect to J and we obtain its dimension.
Theorem 3.3. Let f be a lightlike submersion from an indefinite almost Hermitian manifold
(M1, g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2).
Then \mu is invariant with respect to J.
Proof. Using (2), for any V \in \Gamma (\mu ) and \omega X \in \Gamma (f(D)), we have g1(JV, \omega X) = - g1(JV, \phi X).
By virtue of Theorem 3.1, we get g1(JV, \omega X) = - \mathrm{c}\mathrm{o}\mathrm{s}2 \theta g1(V,X) + g1(V, \omega \phi X) = 0. Similarly
g1(JV, Y ) = - g1(V, JY ) = 0 for any Y \in \Gamma (\mathrm{K}\mathrm{e}\mathrm{r} f\ast ). Also for any N \in \Gamma (\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )) we have
g1(JV,N) = - g1(V, JN) = 0. Hence the proof follows.
Theorem 3.4. Let f be a proper slant lightlike submersion from an almost Hermitian manifold
(Mm
1 , g1, J) onto an r-lightlike manifold (Mn
2 , g2), where g1 is a semi-Riemannian metric of index
2r. Then \mathrm{d}\mathrm{i}\mathrm{m}(\mu ) = 2n - m+ 2r. If \mu = \{ 0\} , then n =
m - 2r
2
.
Proof. Since \mathrm{d}\mathrm{i}\mathrm{m}D = m - n - 3r and \mathrm{d}\mathrm{i}\mathrm{m}S(\mathrm{K}\mathrm{e}\mathrm{r} f\bot
\ast ) = n - r. Therefore \mathrm{d}\mathrm{i}\mathrm{m}\mu = 2n - m+2r.
Moreover M1 is almost Hermitian manifold so its dimension m is even and hence dimension of \mu is
even.
Lemma 3.3. Let f be a lightlike submersion from an indefinite almost Hermitian manifold
(Mm
1 , g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold
(Mn
2 , g2). Let \{ e1, . . . , em - n - 3r\} be a local orthonormal basis of D, then \{ \mathrm{c}\mathrm{s}\mathrm{c} \theta \omega e1, . . .
. . . , \mathrm{c}\mathrm{s}\mathrm{c} \theta \omega em - n - 3r\} is a local orthonormal basis of fD.
Proof. Since \{ e1, . . . , em - n - 3r\} be a local orthonormal basis of D and D is Riemannian
therefore using (14) we have
g1(\mathrm{c}\mathrm{s}\mathrm{c} \theta \omega ei, \mathrm{c}\mathrm{s}\mathrm{c} \theta \omega ej) = \mathrm{c}\mathrm{s}\mathrm{c}2 \theta \mathrm{s}\mathrm{i}\mathrm{n}2 \theta g1(ei, ej) = \delta ij ,
this proves the lemma.
Since for any Q2X \in \Gamma (D), \phi Q2X \in \Gamma (D) therefore the distribution D is even dimensional.
Hence we have the following result similar to the above lemma.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
970 R. SACHDEVA, R. KUMAR, S. S. BHATIA
Lemma 3.4. Let f be a lightlike submersion from an indefinite almost Hermitian manifold
(Mm
1 , g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold
(Mn
2 , g2). If \{ e1, . . . , em - n - 3r
2
\} are unit vector fields in D, then \{ e1, \mathrm{s}\mathrm{e}\mathrm{c} \theta \phi e1, e2, \mathrm{s}\mathrm{e}\mathrm{c} \theta \phi e2, . . .
. . . , em - n - 3r
2
, \mathrm{s}\mathrm{e}\mathrm{c} \theta \phi em - n - 3r
2
\} is a local orthonormal basis of D.
Theorem 3.5. Let f be a lightlike submersion from an indefinite almost Hermitian manifold
(M1, g1, J), where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2).
If \omega is parallel with respect to \nabla , then we have
\scrT \phi X\phi X = - cos2\theta \scrT XX, \scrT \phi X\phi X = - \scrT XX, and \scrT \phi X\phi X = 0, (17)
for any X \in \Gamma (D), X \in \Gamma (\bigtriangleup \bot J\bigtriangleup ) and X \in \Gamma (J(\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )), respectively.
Proof. Let \omega be parallel, then from (9) we have C\scrT XY = \scrT X\phi Y for X,Y \in \Gamma (TM1).
Interchanging the role of X and Y, we get C\scrT Y X = \scrT Y \phi X. Thus we obtain
C\scrT XY - C\scrT Y X = \scrT X\phi Y - \scrT Y \phi X.
Using (8), we derive
\scrT X\phi Y = \scrT Y \phi X.
Then substituting Y by \phi X we get \scrT X\phi 2X = \scrT \phi X\phi X. Thus using Theorem 3.1 with the fact that
\phi 2X = - X, for any X \in \Gamma (\bigtriangleup \bot J\bigtriangleup ) and \phi X = 0 for any X \in \Gamma (J(\mathrm{l}\mathrm{t}\mathrm{r}(\mathrm{K}\mathrm{e}\mathrm{r} f\ast )), (17) follows.
Theorem 3.6. Let f be a lightlike submersion from an indefinite Kaehler manifold (M1, g1),
where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2). Then the
distribution \scrV defines a totally geodesic foliation on M1 if and only if
\omega (\nu \nabla X\phi Y + \scrT X\omega Y ) + C(\scrT X\phi Y + h\nabla X\omega Y ) = 0
for any X,Y \in \scrV .
Proof. Let X,Y \in \Gamma (\scrV ) then using Lemma 3.1 with (2) and (3) we obtain
\nabla XY = - J\nabla XJY = - J(\scrT X\phi Y + \nu \nabla X\phi Y + \scrT X\omega Y + h\nabla X\omega Y ) =
= - (B\scrT X\phi Y + C\scrT X\phi Y + \phi \nu \nabla X\phi Y + \omega \nu \nabla X\phi Y + \phi \scrT X\omega Y + \omega \scrT X\omega Y+
+Bh\nabla X\omega Y + Ch\nabla X\omega Y ).
Hence \nabla XY \in \Gamma (\scrV ) if and only if \omega (\nu \nabla X\phi Y + \scrT X\omega Y ) + C(\scrT X\phi Y + h\nabla X\omega Y ) = 0.
Similarly, we can prove the following theorem.
Theorem 3.7. Let f be a lightlike submersion from an indefinite Kaehler manifold (M1, g1),
where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2). Then the
distribution \scrH defines a totally geodesic foliation on M1 if and only if
\phi (\nu \nabla UBV +\scrA UCV ) +B(\scrA UBV + h\nabla UCV ) = 0
for any U, V \in \scrH .
Corollary 3.2. Let f be a lightlike submersion from an indefinite Kaehler manifold (M1, g1),
where g1 is a semi-Riemannian metric of index 2r, onto an r-lightlike manifold (M2, g2). Then M1
is a locally product Riemannian manifold if and only if
\omega (\nu \nabla X\phi Y + \scrT X\omega Y ) + C(\scrT X\phi Y + h\nabla X\omega Y ) = 0,
\phi (\nu \nabla UBV +\scrA UCV ) +B(\scrA UBV + h\nabla UCV ) = 0
for X,Y \in \Gamma (\scrV ) and U, V \in \Gamma (\scrH ).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
SLANT LIGHTLIKE SUBMERSIONS FROM AN INDEFINITE ALMOST HERMITIAN MANIFOLD . . . 971
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Received 23.01.13
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
|
| id | umjimathkievua-article-1894 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T02:14:45Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/36/661bd15f2f52da6a5ee25d36114d4d36.pdf |
| spelling | umjimathkievua-article-18942019-12-05T09:30:55Z Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold Похилi свiтлоподiбнi занурення з невизначеного майже ермiтового многовиду в свiтлоподiбний многовид Bhatia, S. S. Kumar, R. Sachdeva, R. Бхатія, С. С. Кумар, Р. Сачдєва, Р. We introduce slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold. We establish existence theorems for these submersions. We also investigate the necessary and sufficient conditions for the leaves of the distributions to be totally geodesic foliations in indefinite almost Hermitian manifolds. Введено поняття похилих свiтлоподiбних занурень з невизначеного майже ермiтового многовиду в свiтлоподiбний многовид. Доведено теореми iснування для таких занурень. Вивчено необхiднi та достатнi умови для того, щоб листки розподiлiв були повнiстю геодезичними розшаруваннями в невизначених майже ермiтових многовидах. Institute of Mathematics, NAS of Ukraine 2016-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1894 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 7 (2016); 963-971 Український математичний журнал; Том 68 № 7 (2016); 963-971 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/1894/876 Copyright (c) 2016 Bhatia S. S.; Kumar R.; Sachdeva R. |
| spellingShingle | Bhatia, S. S. Kumar, R. Sachdeva, R. Бхатія, С. С. Кумар, Р. Сачдєва, Р. Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold |
| title | Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold |
| title_alt | Похилi свiтлоподiбнi занурення з невизначеного майже ермiтового многовиду в свiтлоподiбний многовид |
| title_full | Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold |
| title_fullStr | Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold |
| title_full_unstemmed | Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold |
| title_short | Slant lightlike submersions from an indefinite almost Hermitian manifold onto a lightlike manifold |
| title_sort | slant lightlike submersions from an indefinite almost hermitian manifold onto a lightlike manifold |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1894 |
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