Invariant Submanifolds of Trans-Sasakian Manifolds
We prove the equivalence of total geodesicity, recurrence, birecurrence, generalized birecurrence, Riccigeneralized birecurrence, parallelism, biparallelism, pseudoparallelism, bipseudoparallelism of σσ for the invariant submanifold $M$ of the trans-Sasakian manifold $\tilde{M}$.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507994214629376 |
|---|---|
| author | Anitha, B. S. Bagewadi, C. S. Анітха, В. С. Багеваді, С. С. |
| author_facet | Anitha, B. S. Bagewadi, C. S. Анітха, В. С. Багеваді, С. С. |
| author_sort | Anitha, B. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:49:57Z |
| description | We prove the equivalence of total geodesicity, recurrence, birecurrence, generalized birecurrence, Riccigeneralized birecurrence, parallelism, biparallelism, pseudoparallelism, bipseudoparallelism of σσ for the invariant submanifold $M$ of the trans-Sasakian manifold $\tilde{M}$. |
| first_indexed | 2026-03-24T02:18:09Z |
| format | Article |
| fulltext |
UDC 517.96
C. S. Bagewadi, B. S. Anitha (Kuvempu Univ., Karnataka, India)
INVARIANT SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS
IНВАРIАНТНI ПIДМНОГОВИДИ ТРАНС-МНОГОВИДIВ САСАКЯНА
We show the equivalence of totally geodesicity, recurrence, birecurrence, generalized birecurrence, Ricci-generalized
birecurrence, parallelism, biparallelism, pseudoparallelism, bipseudoparallelism of σ for the invariant submanifold M of
trans-Sasakian manifold M̃.
Показано еквiвалентнiсть повної геодезичностi, зворотностi, подвiйної зворотностi, узагальненої подвiйної зво-
ротностi, узагальненої подвiйної зворотностi Рiччi, паралелiзму, подвiйного паралелiзму, псевдопаралелiзму та
подвiйного псевдопаралелiзму σ для iнварiантного пiдмноговиду M транс-многовиду Сасакяна M̃.
1. Introduction. Let M be an almost contact Riemannian manifold with a contact form η, the
associated vector field ξ, a (1, 1)-tensor field φ and the associated Riemannian metric g. Further
an almost contact metric manifold is a contact metric manifold if g(X,φY ) = dη(X,Y ) for all
X,Y ∈ TM. A K-contact manifold is a contact metric manifold while converse is true if the Lie
derivative of φ in the character direction ξ vanishes. A Sasakian manifold is always a K-contact
manifold. A 3-dimensional K-contact manifold is a Sasakian manifold. A contact metric manifold is
Sasakian if (∇Xφ)Y = g(X,Y )ξ − η(Y )X. Odd dimensional spheres and C? ×R are examples of
Sasakian manifolds.
In 1972, K. Kenmotsu [4] studied a class of contact Riemannian manifolds called Kenmotsu
manifolds, which is not Sasakian. In fact Kenmotsu proved that a locally Kenmotsu manifold is a
warped product I×fN of an interval I and a Kahlerian manifold with a warping function f(t) = set,
where S is a non-zero contact. Hyperbolic space is an example of Kenmotsu manifold.
In the Gray – Hervella classification of almost Hermitian manifolds [10], there appears a class
W4 of Hermitian manifolds which are closely related to locally conformal Kaehler manifolds. An
almost contact metric structure on a manifold M is called a trans-Sasakian structure [11] if the
product manifold M × R belongs to the class W4. The class C5 ⊕ C6 [13] coincides with the class
of trans-Sasakian structure of (α, β). The monkey saddle is an example of trans-Sasakian manifold.
This class consists of both Sasakian and Kenmotsu structures. If α = 1, β = 0, then the class reduces
to Sasakian, where as if α = 0, β = 1 their reduces to Kenmotsu. J. C. Marrero [11] has shown that
trans-Sasakian manifolds for n ≥ 5 do not exist. If α 6= 0, β = 0 then it is α-Sasakian, if α = 0,
β 6= 0 then it is β-Kenmotsu and if α = β = 0 then it is cosympletic.
The geometry of invariant submanifolds of trans-Sasakian manifolds is carried out by Aysel
Turgut Vanli and Ramazan Sari [3] and they have shown that an invariant submanifold M carries
trans-Sasakian structure and established the equivalence of totally geodesicity of M, σ is parallel, σ
is 2-parallel, σ is semiparallel.
In this paper we extend the study and show that for invariant submanifolds of trans-Sasakian
manifolds the equivalence of M, totally geodesic, when σ is recurrent, 2-recurrent, generalized
2-recurrent, 2-semiparallel, pseudoparallel, 2-pseudoparallel, Ricci-generalized pseudoparallel, 2-
Ricci-generalized pseudoparallel their equivalence. Finally it is concluded that the result of Aysel
c© C. S. BAGEWADI, B. S. ANITHA, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10 1309
1310 C. S. BAGEWADI, B. S. ANITHA
Turgut Vanli and Ramazan Sari [3] and the above results proved are all equivalent to one another.
We provide an example of trans-Sasakian manifold which is not totally geodesic.
2. Basic concepts. The covariant differential of the pth order, p ≥ 1 of a (0, k)-tensor field T,
k ≥ 1 denoted by ∇pT, defined on a Riemannian manifold (M, g) with the Levi – Civita connection
∇. The tensor T is said to be recurrent [15], if the following condition holds on M :
(∇T )(X1, . . . , Xk;X)T (Y1, . . . , Yk) = (∇T )(Y1, . . . , Yk;X)T (X1, . . . , Xk) (2.1)
and
(∇2T )(X1, . . . , Xk;X,Y )T (Y1, . . . , Yk) = (∇2T )(Y1, . . . , Yk;X,Y )T (X1, . . . , Xk)
respectively, where X,Y,X1, Y1, . . . , Xk, Yk ∈ TM. From (2.1) it follows that at a point x ∈ M,
if the tensor T is non-zero, then there exists a unique 1-form φ, a (0, 2)-tensor ψ, defined on a
neighborhood U of x such that
∇T = T ⊗ φ, φ = d(log ‖T‖) (2.2)
and
∇2T = T ⊗ ψ (2.3)
respectively, hold on U, where ‖T‖ denotes the norm of T and ‖T‖2 = g(T, T ). The tensor T is said
to be generalized 2-recurrent if
((∇2T )(X1, . . . , Xk;X,Y )− (∇T ⊗ φ)(X1, . . . , Xk;X,Y ))T (Y1, . . . , Yk) =
= ((∇2T )(Y1, . . . , Yk;X,Y )− (∇T ⊗ φ)(Y1, . . . , Yk;X,Y ))T (X1, . . . , Xk),
holds on M, where φ is a 1-form on M. From this it follows that at a point x ∈M if the tensor T is
non-zero, then there exists a unique (0, 2)-tensor ψ, defined on a neighborhood U of x, such that
∇2T = ∇T ⊗ φ+ T ⊗ ψ, (2.4)
holds on U.
Let f : (M, g)→ (M̃, g̃) be an isometric immersion from an n-dimensional Riemannian manifold
(M, g) into (n + d)-dimensional Riemannian manifold (M̃, g̃), n ≥ 2, d ≥ 1. We denote by ∇ and
∇̃ as Levi – Civita connection of Mn and M̃n+d respectively. Then the formulas of Gauss and
Weingarten are given by
∇̃XY = ∇XY + σ(X,Y ), (2.5)
∇̃XN = −ANX +∇⊥XN, (2.6)
for any tangent vector fields X, Y and the normal vector field N on M, where σ, A and ∇⊥ are the
second fundamental form, the shape operator and the normal connection respectively. If the second
fundamental form σ is identically zero then the manifold is said to be totallygeodesic. The second
fundamental form σ and AN are related by
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INVARIANT SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS 1311
g̃(σ(X,Y ), N) = g(ANX,Y ),
for tangent vector fields X, Y. The first and second covariant derivatives of the second fundamental
form σ are given by
(∇̃Xσ)(Y,Z) = ∇⊥X(σ(Y, Z))− σ(∇XY,Z)− σ(Y,∇XZ), (2.7)
(∇̃2σ)(Z,W,X, Y ) = (∇̃X∇̃Y σ)(Z,W ) =
= ∇⊥X((∇̃Y σ)(Z,W ))− (∇̃Y σ)(∇XZ,W )−
−(∇̃Xσ)(Z,∇YW )− (∇̃∇XY σ)(Z,W ) (2.8)
respectively, where ∇̃ is called the van der Waerden – Bortolotti connection ofM [7]. If ∇̃σ = 0, then
M is said to have parallel second fundamental form [7]. We next define endomorphisms R(X,Y )
and X ∧B Y of χ(M) by
R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z,
(X ∧B Y )Z = B(Y, Z)X −B(X,Z)Y
(2.9)
respectively, where X,Y, Z ∈ χ(M) and B is a symmetric (0, 2)-tensor.
Now, for a (0, k)-tensor field T, k ≥ 1 and a (0, 2)-tensor field B on (M, g), we define the tensor
Q(B, T ) by
Q(B, T )(X1, . . . , Xk;X,Y ) = −(T (X ∧B Y )X1, . . . , Xk)− . . .
. . .− T (X1, . . . , Xk−1(X ∧B Y )Xk). (2.10)
Putting into the above formula T = σ, ∇̃σ and B = g, B = S, we obtain the tensors Q(g, σ),
Q(S, σ), Q(g, ∇̃σ) and Q(S, ∇̃σ).
Definition 2.1. The immersion f is said to be
semiparallel [9] if R̃ · σ = 0, (2.11)
2-semiparallel [14] if R̃ · ∇̃σ = 0, (2.12)
pseudoparallel [2] if R̃ · σ = L1Q(g, σ), (2.13)
2-pseudoparallel [14] if R̃ · ∇̃σ = L1Q(g, ∇̃σ) (2.14)
and
Ricci-generalized pseudoparallel [12] if R̃ · σ = L2Q(S, σ) (2.15)
respectively, where R̃ denotes the curvature tensor with respect to connection ∇̃ and
R̃(X,Y )σ(U, V ) = (∇̃X∇̃Y − ∇̃Y ∇̃X − ∇̃[X,Y ])σ(U, V ) and (R̃(X,Y )∇̃σ)(U, V,W ) =
= R̃(X,Y )(∇̃Uσ)(V,W ). Here L1, L2 are functions depending on σ and ∇̃σ.
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1312 C. S. BAGEWADI, B. S. ANITHA
Now we introduce the definition of 2-Ricci-generalized pseudoparallel.
Definition 2.2. The immersion f is said to be 2-Ricci-generalized pseudoparallel if
R̃ · ∇̃σ = L2Q(S, ∇̃σ), (2.16)
where L2 is a function depending on ∇̃σ.
From the Gauss and Weingarten formulas, we obtain
(R̃(X,Y )Z)T = R(X,Y )Z +Aσ(X,Z)
Y −Aσ(Y,Z)
X. (2.17)
By (2.11), we have
(R̃(X,Y ) · σ)(U, V ) = R⊥(X,Y )σ(U, V )− σ(R(X,Y )U, V )− σ(U,R(X,Y )V ), (2.18)
for all vector fields X, Y, U and V tangent to M, where
R⊥(X,Y ) = [∇⊥X ,∇⊥Y ]−∇⊥[X,Y ]. (2.19)
Similarly, we obtain
(R̃(X,Y ) · ∇̃σ)(U, V,W ) = R⊥(X,Y )(∇̃σ)(U, V,W )− (∇̃σ)(R(X,Y )U, V,W )−
−(∇̃σ)(U,R(X,Y )V,W )− (∇̃σ)(U, V,R(X,Y )W ), (2.20)
for all vector fields X, Y, U, V, W tangent to M, where (∇̃σ)(U, V,W ) = (∇̃Uσ)(V,W ) [1].
3. Preliminaries. Let M be a n = (2m+ 1)-dimensional almost contact metric manifold with
an almost contact metric structure (φ, ξ, η, g), where φ is a (1, 1)-tensor field, ξ is a vector field, η is
a 1-form and g is the associated Riemannian metric such that [5],
φ2 = −I + η ⊗ ξ, η(ξ) = 1, η ◦ φ = 0, φξ = 0, (3.1)
g(φX, φY ) = g(X,Y )− η(X)η(Y ), g(X, ξ) = η(X), g(φX, Y ) = −g(X,φY ), (3.2)
for all vector fields X, Y on M̃.
An almost contact metric structure (φ, ξ, η, g) on M is called a trans-Sasakian structure [13] if
(M × R, J,G) belongs to the class W4 [10], where J is the almost complex structure on M × R
defined by J(X,λd/dt) = (φX − λξ, η(X)d/dt) for all vector fields X on M and smooth function
λ on M ×R and G is the product metric on M ×R. This may be expressed by the condition [6]
(∇Xφ)Y = α(g(X,Y )ξ − η(Y )X) + β(g(φX, Y )ξ − η(Y )φX), (3.3)
for some smooth functions α and β on M and we say that the trans-Sasakian structure is of type
(α, β).
Let M be a trans-Sasakian manifold. From (3.3), it is easy to see that
∇Xξ = −αφX + β(X − η(X)ξ). (3.4)
If α = 1, β = 0 it reduces to Sasakian manifold.
If α = 0, β = 1 it reduces to Kenmotsu manifold.
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INVARIANT SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS 1313
In an n-dimensional trans-Sasakian manifold, we have
R(X,Y )ξ = (α2 − β2) {η(Y )X − η(X)Y }+ 2αβ {η(Y )φ(X)− η(X)φ(Y )}+
+
{
(Y α)φX − (Xα)φY + (Y β)φ2X − (Xβ)φ2Y
}
, (3.5)
R(ξ,X)Y = (α2 − β2) {g(X,Y )ξ − η(Y )X}+ (ξβ)η(Y ) {−X + η(X)ξ} , (3.6)
R(ξ,X)ξ = (α2 − β2 − ξβ) {η(X)ξ −X} , (3.7)
2αβ + ξα = 0, (3.8)
S(X, ξ) = ((n− 1)(α2 − β2)− ξβ)η(X)− (n− 2)Xβ − (φX)α, (3.9)
Qξ = ((n− 1)(α2 − β2)− ξβ)ξ − (n− 2) gradβ + φ(gradα). (3.10)
Further, in a trans-Sasakian manifold of type (α, β), we have
φ(gradα) = (n− 2) gradβ. (3.11)
Using (3.11) the equations (3.5) – (3.7), (3.9) and (3.10) reduce to
R(X,Y )ξ = (α2 − β2) {η(Y )X − η(X)Y } , (3.12)
R(ξ,X)Y = (α2 − β2) {g(X,Y )ξ − η(Y )X} , (3.13)
R(ξ,X)ξ = (α2 − β2) {η(X)ξ −X} , (3.14)
S(X, ξ) = (n− 1)(α2 − β2)η(X), (3.15)
Qξ = (n− 1)(α2 − β2)ξ (3.16)
respectively.
A submanifold M of a trans-Sasakian manifold M̃ is called an invariant submanifold of M̃, if
for each x ∈ M, φ(TxM) ⊂ TxM. As a consequence, ξ becomes tangent to M. In an invariant
submanifold of a trans-Sasakian manifold
σ(X, ξ) = 0, (3.17)
for any vector X tangent to M.
4. Recurrent invariant submanifolds of trans-Sasakian manifolds. We consider invariant
submanifold of a trans-Sasakian manifold satisfying the conditions σ is recurrent, 2-recurrent, gen-
eralized 2-recurrent and M has parallel third fundamental form. As a result of this we state the
following theorem.
Theorem 4.1. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then σ is
recurrent if and only if it is totally geodesic.
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1314 C. S. BAGEWADI, B. S. ANITHA
Proof. Let σ be recurrent, from (2.2) and we get
(∇̃Xσ)(Y,Z) = φ(X)σ(Y,Z),
where φ is a 1-form on M and in view of (2.7) and taking Z = ξ in the above equation, we have
∇⊥Xσ(Y, ξ)− σ(∇XY, ξ)− σ(Y,∇Xξ) = φ(X)σ(Y, ξ). (4.1)
Using (3.4), (3.17) in (4.1), we obtain (α2 + β2)σ(X,Y ) = 0. Since α and β are not simultaneously
zero. Hence (α2 + β2) 6= 0 and σ(X,Y ) = 0. Thus M is totally geodesic. The converse statement
is trivial.
Theorem 4.1 is proved.
Theorem 4.2. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then M
has parallel third fundamental form if and only if it is totally geodesic.
Proof. Let M has parallel third fundamental form. Then we obtain
(∇̃X∇̃Y σ)(Z,W ) = 0.
Taking W = ξ and using (2.8) in the above equation, we have
∇⊥X((∇̃Y σ)(Z, ξ))− (∇̃Y σ)(∇XZ, ξ)− (∇̃Xσ)(Z,∇Y ξ)− (∇̃∇XY σ)(Z, ξ) = 0. (4.2)
By virtue of (2.7) in (4.2) and using (3.17), we get
2∇⊥Xασ(Z, φY )− 2∇⊥Xβσ(Z, Y )− 2ασ(∇XZ, φY ) + 2βσ(∇XZ, Y )− σ(Z,∇XαφY ) +
+ σ(Z,∇XβY )− σ(Z,∇Xβη(Y )ξ)− ασ(Z, φ∇XY ) + βσ(Z,∇XY ). (4.3)
Putting Y = ξ and using (3.4), (3.17) in (4.3), we get (α2 + β2)2σ(X,Z) = 0. Since (α2 + β2) 6= 0,
then σ(X,Z) = 0. Thus M is totally geodesic. The converse statement is trivial.
Theorem 4.2 is proved.
Corollary 4.1. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then σ is
2-recurrent if and only if it is totally geodesic.
Proof. Let σ be 2-recurrent, from (2.3), we have
(∇̃X∇̃Y σ)(Z,W ) = σ(Z,W )φ(X,Y ). (4.4)
Taking W = ξ in (4.4) and using the proof of the Theorem 4.2, we get (α2 + β2)2σ(X,Z) = 0.
Since (α2 + β2) 6= 0, then σ(X,Z) = 0. Thus M is totally geodesic. The converse statement is
trivial.
Corollary 4.1 is proved.
Theorem 4.3. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then σ is
generalized 2-recurrent if and only if it is totally geodesic.
Proof. Let σ be generalized 2-recurrent, from (2.4), we obtain
(∇̃X∇̃Y σ)(Z,W ) = ψ(X,Y )σ(Z,W ) + φ(X)(∇̃Y σ)(Z,W ), (4.5)
where ψ and φ are 2-recurrent and 1-form respectively. Taking W = ξ in (4.5) and using (3.17), we
get
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INVARIANT SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS 1315
(∇̃X∇̃Y σ)(Z, ξ) = φ(X)(∇̃Y σ)(Z, ξ).
By virtue of (2.7) and (2.8) in above equation and in view of (3.17), we have
2∇⊥Xασ(Z, φY )− 2∇⊥Xβσ(Z, Y )− 2ασ(∇XZ, φY ) + 2βσ(∇XZ, Y )−
−σ(Z,∇XαφY ) + σ(Z,∇XβY )− σ(Z,∇Xβη(Y )ξ)− ασ(Z, φ∇XY ) + βσ(Z,∇XY ) =
= {ασ(Z, φY )− βσ(Z, Y )} .
Putting Y = ξ and using (3.4), (3.17) in the above equation, we obtain (α2+β2)2σ(X,Z) = 0. Since
(α2 + β2) 6= 0, then σ(X,Z) = 0. Thus M is totally geodesic. The converse statement is trivial.
Theorem 4.3 is proved.
5. 2-Semiparallel, pseudoparallel, 2-pseudoparallel, Ricci-generalized pseudoparallel and
2-Ricci-generalized pseudoparallel invariant submanifolds of trans-Sasakian manifolds. We
consider invariant submanifolds of trans-Sasakian manifolds satisfying the conditions R̃ · ∇̃σ = 0,
R̃ · σ = L1Q(g, σ), R̃ · ∇̃σ = L1Q(g, ∇̃σ) R̃ · σ = L2Q(S, σ) and R̃ · ∇̃σ = L2Q(S, ∇̃σ).
Theorem 5.1. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then the
submanifold M is 2-semiparallel if and only if it is totally geodesic.
Proof. Let M be 2-semiparallel R̃ · ∇̃σ = 0. Put X = V = ξ in (2.20), we get
R⊥(ξ, Y )(∇̃σ)(U, ξ,W )− (∇̃σ)(R(ξ, Y )U, ξ,W )−
−(∇̃σ)(U,R(ξ, Y )ξ,W )− (∇̃σ)(U, ξ,R(ξ, Y )W ) = 0. (5.1)
In view of (2.7), (3.4), (3.13), (3.14) and (3.17), we have the following equalities:
(∇̃σ)(U, ξ,W ) = (∇̃Uσ)(ξ,W ) =
= ∇⊥Uσ(ξ,W )− σ (∇Uξ,W )− σ (ξ,∇UW ) =
= ασ(φU,W )− βσ(U,W ), (5.2)
(∇̃σ)(R(ξ, Y )U, ξ,W ) = (∇̃R(ξ,Y )Uσ)(ξ,W ) =
= ∇⊥R(ξ,Y )Uσ(ξ,W )− σ(∇R(ξ,Y )Uξ,W )− σ(ξ,∇R(ξ,Y )UW ) =
= −α(α2 − β2)η(U)σ(φY,W ) + β(α2 − β2)η(U)σ(Y,W ), (5.3)
(∇̃σ)(U,R(ξ, Y )ξ,W ) = (∇̃Uσ)(R(ξ, Y )ξ,W ) =
= ∇⊥Uσ(R(ξ, Y )ξ,W )− σ(∇UR(ξ, Y )ξ,W )− σ(R(ξ, Y )ξ,∇UW ) =
= ∇⊥Uσ
(
(α2 − β2) {η(Y )ξ − Y } ,W
)
− σ
(
∇U (α2 − β2) {η(Y )ξ − Y } ,W
)
+
+(α2 − β2)σ(Y,∇UW ) (5.4)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1316 C. S. BAGEWADI, B. S. ANITHA
and
(∇̃σ)(U, ξ,R(ξ, Y )W ) = (∇̃Uσ)(ξ,R(ξ, Y )W ) =
= ∇⊥Uσ(ξ,R(ξ, Y )W )− σ(∇Uξ,R(ξ, Y )W )− σ(ξ,∇UR(ξ, Y )W ) =
= −α(α2 − β2)η(W )σ(φU, Y ) + β(α2 − β2)η(W )σ(U, Y ). (5.5)
Substituting (5.2) – (5.5) into (5.1), we obtain
R⊥(ξ, Y ) {ασ(φU,W )− βσ(U,W )}+ α(α2 − β2)η(U)σ(φY,W )−
−β(α2 − β2)η(U)σ(Y,W )−∇⊥Uσ
(
(α2 − β2) {η(Y )ξ − Y } ,W
)
+
+σ
(
∇U (α2 − β2) {η(Y )ξ − Y } ,W
)
− (α2 − β2)σ(Y,∇UW )+
+α(α2 − β2)η(W )σ(φU, Y )− β(α2 − β2)η(W )σ(U, Y ) = 0. (5.6)
Taking W = ξ and using (3.4), (3.17) in (5.6), we get (α2 − β2)(α2 + β2)σ(U, Y ) = 0. Since
(α2 + β2) 6= 0, hence if α 6= ±β and then σ(U, Y ) = 0, i.e., M is totally geodesic. The converse
statement is trivial.
Theorem 5.1 is proved.
Theorem 5.2. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then the
submanifold M is pseudoparallel if and only if it is totally geodesic.
Proof. Let M be pseudoparallel R̃ · σ = L1Q(g, σ). Put X = V = ξ in (2.10), (2.18) and
adding, we get
R⊥(ξ, Y )σ(U, ξ)− σ(R(ξ, Y )U, ξ)− σ(U,R(ξ, Y )ξ) =
= −L1
{
g(ξ, ξ)σ(U, Y )− g(ξ, U)σ(ξ, Y ) + g(ξ, Y )σ(ξ, U)− g(Y,U)σ(ξ, ξ)
}
. (5.7)
Using (3.14) and (3.17) in (5.7), we get [(α2 − β2) + L1]σ(U, Y ) = 0. If L1 6= −(α2 − β2) and
α 6= ±β, then σ(U, Y ) = 0, i.e., M is totally geodesic. The converse statement is trivial.
Theorem 5.2 is proved.
Theorem 5.3. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then the
submanifold M is 2-pseudoparallel if and only if it is totally geodesic.
Proof. Let M be 2-pseudoparallel R̃ · ∇̃σ = L1Q(g, ∇̃σ). Put X = V = ξ in (2.10), (2.20) and
adding, in view of (3.1) and (3.17), we get
R⊥(ξ, Y )(∇̃σ)(U, ξ,W )− (∇̃σ)(R(ξ, Y )U, ξ,W )−
−(∇̃σ)(U,R(ξ, Y )ξ,W )− (∇̃σ)(U, ξ,R(ξ, Y )W ) =
= −L1
[
η(W )
{
∇⊥ξ σ(Y,U)− σ(∇ξY, U)− σ(Y,∇ξU)
}
−
−∇⊥Wσ(Y,U) + σ(∇WY,U) + σ(Y,∇WU)− η(Y )
{
∇⊥ξ σ(W,U)− σ(∇ξW,U) −
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
INVARIANT SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS 1317
−σ(W,∇ξU)} − η(U)
{
∇⊥ξ σ(Y,W )− σ(∇ξY,W )− σ(Y,∇ξW )
}]
. (5.8)
Substituting (5.2) – (5.5) into (5.8), we obtain
R⊥(ξ, Y ) {ασ(φU,W )− βσ(U,W )}+ α(α2 − β2)η(U)σ(φY,W )−
−β(α2 − β2)η(U)σ(Y,W )−∇⊥Uσ
(
(α2 − β2) {η(Y )ξ − Y } ,W
)
+
+σ
(
∇U (α2 − β2) {η(Y )ξ − Y } ,W
)
− (α2 − β2)σ(Y,∇UW )+
+α(α2 − β2)η(W )σ(φU, Y )− β(α2 − β2)η(W )σ(U, Y ) =
= −L1
[
η(W )
{
∇⊥ξ σ(Y, U)− σ(∇ξY,U)− σ(Y,∇ξU)
}
−
−∇⊥Wσ(Y,U) + σ(∇WY, U) + σ(Y,∇WU)−
−η(Y )
{
∇⊥ξ σ(W,U)− σ(∇ξW,U)− σ(W,∇ξU)
}
−
−η(U)
{
∇⊥ξ σ(Y,W )− σ(∇ξY,W )− σ(Y,∇ξW )
}]
. (5.9)
Taking W = ξ and using (3.4), (3.17) in (5.9), we get (α2 − β2)(α2 + β2)σ(U, Y ) = 0. Since
(α2 + β2) 6= 0, hence if α 6= ±β and then σ(U, Y ) = 0, i.e., M is totally geodesic. The converse
statement is trivial.
Theorem 5.3 is proved.
Theorem 5.4. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then the
submanifold M is Ricci-generalized pseudoparallel if and only if it is totally geodesic.
Proof. Let M be Ricci-generalized pseudoparallel R̃ · ∇̃σ = L2Q(S, σ). Put X = V = ξ in
(2.10), (2.18) and adding, we get
R⊥(ξ, Y )σ(U, ξ)− σ(R(ξ, Y )U, ξ)− σ(U,R(ξ, Y )ξ) =
= −L2
{
S(ξ, ξ)σ(U, Y )− S(ξ, U)σ(ξ, Y ) + S(ξ, Y )σ(ξ, U)− S(Y,U)σ(ξ, ξ)
}
. (5.10)
Using (3.14), (3.15) and (3.17) in (5.10), we have (α2− β2)[1 +L2(n− 1)]σ(U, Y ) = 0. If α 6= ±β
and L2 6= −
1
n− 1
, then σ(U, Y ) = 0, i.e., M is totally geodesic. The converse statement is trivial.
Theorem 5.5. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then the
submanifold M is 2-Ricci-generalized pseudoparallel, if and only if it is totally geodesic.
Proof. Let M be 2-Ricci-generalized pseudoparallel R̃ · ∇̃σ = L2Q(S, ∇̃σ). Put X = V = ξ in
(2.10), (2.20) and adding, in view of (3.15) and (3.17) we obtain
R⊥(ξ, Y )(∇̃σ)(U, ξ,W )− (∇̃σ)(R(ξ, Y )U, ξ,W )−
−(∇̃σ)(U,R(ξ, Y )ξ,W )− (∇̃σ)(U, ξ,R(ξ, Y )W ) =
= −L2
[
(n− 1)(α2 − β2)η(W )
{
∇⊥ξ σ(Y,U)− σ(∇ξY,U)− σ(Y,∇ξU)
}
−
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1318 C. S. BAGEWADI, B. S. ANITHA
−(n− 1)(α2 − β2)
{
∇⊥Wσ(Y,U)− σ(∇WY,U)− σ(Y,∇WU)
}
−
−(n− 1)(α2 − β2)η(Y )
{
∇⊥ξ σ(W,U)− σ(∇ξW,U)− σ(W,∇ξU)
}
−
−(n− 1)(α2 − β2)η(U)
{
∇⊥ξ σ(Y,W )− σ(∇ξY,W )− σ(Y,∇ξW )
}]
. (5.11)
Substituting (5.2) – (5.5) into (5.11), we have
R⊥(ξ, Y ) {ασ(φU,W )− βσ(U,W )}+ α(α2 − β2)η(U)σ(φY,W )−
−β(α2 − β2)η(U)σ(Y,W )−∇⊥Uσ
(
(α2 − β2) {η(Y )ξ − Y } ,W
)
+
+σ
(
∇U (α2 − β2) {η(Y )ξ − Y } ,W
)
− (α2 − β2)σ(Y,∇UW )+
+α(α2 − β2)η(W )σ(φU, Y )− β(α2 − β2)η(W )σ(U, Y ) =
= −L2
[
(n− 1)(α2 − β2)η(W )
{
∇⊥ξ σ(Y, U)− σ(∇ξY,U)− σ(Y,∇ξU)
}
−
−(n− 1)(α2 − β2)
{
∇⊥Wσ(Y,U)− σ(∇WY, U)− σ(Y,∇WU)
}
−
−(n− 1)(α2 − β2)η(Y )
{
∇⊥ξ σ(W,U)− σ(∇ξW,U)− σ(W,∇ξU)
}
−
−(n− 1)(α2 − β2)η(U)
{
∇⊥ξ σ(Y,W )− σ(∇ξY,W )− σ(Y,∇ξW )
}]
. (5.12)
Taking W = ξ and using (3.4), (3.15), (3.17) in (5.12), we get (α2 − β2)(α2 + β2)σ(U, Y ) = 0.
Since (α2 + β2) 6= 0, hence if α 6= ±β and then σ(U, Y ) = 0. i.e., M is totally geodesic. The
converse statement is trivial.
Theorem 5.5 is proved.
Using Theorems 4.1 to 4.3, 5.1 to 5.5, Corollary 4.1 and the result of [3], we have the following
result.
Corollary 5.1. Let M be an invariant submanifold of a trans-Sasakian manifold M̃. Then the
following statements are equivalent:
(1) σ is parallel;
(2) σ is 2-parallel;
(3) σ is recurrent;
(4) σ is 2-recurrent;
(5) σ is generalized 2-recurrent;
(6) M has parallel third fundamental form;
(7) M is semiparallel;
(8) M is 2-semiparallel, if α 6= ±β;
(9) M is pseudoparallel, if L1 6= −(α2 − β2) and α 6= ±β;
(10) M is 2-pseudoparallel, if α 6= ±β;
(11) M is Ricci-generalized pseudoparallel, if L2 6= −
1
n− 1
and α 6= ±β;
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
INVARIANT SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS 1319
(12) M is 2-Ricci-generalized pseudoparallel, if α 6= ±β;
(13) M is totally geodesic.
Example of trans-Sasakian manifold. We consider the 3-dimensional manifold M =
= {(x, y, z) ∈ R3 : x 6= 0, y 6= 0}, where (x, y, z) are the standard coordinates in R3. Let
{E1, E2, E3} be linearly independent global frame field on M given by
E1 =
ez
x
(
∂
∂x
+ y
∂
∂z
)
, E2 =
ez
y
∂
∂y
, E3 =
∂
∂z
.
Let g be the Riemannian metric defined by
g(E1, E2) = g(E2, E3) = g(E1, E3) = 0,
g(E1, E1) = g(E2, E2) = g(E3, E3) = 1.
The (φ, ξ, η) is given by
η = dz − ydx, ξ = E3 =
∂
∂z
,
φE1 = E2, φE2 = −E1, φE3 = 0.
The linearity property of φ and g yields that
η(E3) = 1, φ2U = −U + η(U)E3,
g(φU, φW ) = g(U,W )− η(U)η(W ),
for any vector fields U,W on M. By definition of Lie bracket, we have
[E1, E2] = y
ez
x
E2 −
e2z
xy
E3, [E1, E3] = −E1, [E2, E3] = −E2.
Let ∇ be the Levi – Civita connection with respect to above metric g is given by Koszula formula
2g(∇XY, Z) = X(g(Y,Z)) + Y (g(Z,X))− Z(g(X,Y ))−
−g(X, [Y, Z])− g(Y, [X,Z]) + g(Z, [X,Y ]).
Then we get
∇E1E1 = E3, ∇E1E2 = −
e2z
2xy
E3, ∇E1E3 = −E1 +
e2z
2xy
E2,
∇E2E1 = −y
ez
x
E2 +
e2z
2xy
E3, ∇E2E2 = y
ez
x
E1 + E3, ∇E2E3 = −
e2z
2xy
E1 − E2,
∇E3E1 =
e2z
2xy
E2, ∇E3E2 = −
e2z
2xy
E1, ∇E3E3 = 0.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1320 C. S. BAGEWADI, B. S. ANITHA
The tangent vectors X and Y to M are expressed as linear combination of E1, E2, E3, i.e., X =
= a1E1+a2E2+a3E3 and Y = b1E1+b2E2+b3E3, where ai and bj are scalars. Clearly (φ, ξ, η, g)
and X, Y satisfy equations (3.1), (3.2), (3.3) and (3.4) with α = − e
2z
2xy
and β = −1. Thus M is a
trans-Sasakian manifold. In particular we consider the example of monkey saddle given by
M =
{
(x, y, z) ∈ R3 : z = x3 − 3xy2
}
.
By the above x 6= 0, y 6= 0 ⇒ z 6= 0 and M = R3 − {0}. We show that though α 6= −β, M is
not totally geodesic. For if X is a patch defined by X(u, v) = (u, v, u3 − 3uv2) then any tangent
vector V to the monkey saddle is given by V = C1Xu + C2Xv, where Xu = (1, 0, 3u − 3v2) and
Xv = (0, 1,−6uv). M will not be totally geodesic, if ∇V V 6= 0. On verification we can see that
∇V V 6= 0. Hence M is not totally geodesic.
Conclusion. From the above discussion we conclude that α 6= ±β is only a necessary condition
but not a sufficient condition. Hence it needs further investigation.
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Received 20.02.12,
after revision — 02.10.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
|
| id | umjimathkievua-article-2068 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:18:09Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5b/d1ec7a1933616f0736a16c547678025b.pdf |
| spelling | umjimathkievua-article-20682019-12-05T09:49:57Z Invariant Submanifolds of Trans-Sasakian Manifolds Інваріантні підмноговиди транс-многовидів Сасакяна Anitha, B. S. Bagewadi, C. S. Анітха, В. С. Багеваді, С. С. We prove the equivalence of total geodesicity, recurrence, birecurrence, generalized birecurrence, Riccigeneralized birecurrence, parallelism, biparallelism, pseudoparallelism, bipseudoparallelism of σσ for the invariant submanifold $M$ of the trans-Sasakian manifold $\tilde{M}$. Показано еквiвалентнiсть повної геодезичності, зворотності, подвійної зворотності, узагальненої подвійної зво-ротності, узагальненої подвійної зворотності Річчі, паралелізму, подвійного паралелізму, псевдопаралелізму та подвійного псевдопаралелізму a для інваріантного підмноговиду $M$ транс-многовиду Сасакяна $\tilde{M}$. Institute of Mathematics, NAS of Ukraine 2015-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2068 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 10 (2015); 1309-1320 Український математичний журнал; Том 67 № 10 (2015); 1309-1320 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2068/1151 https://umj.imath.kiev.ua/index.php/umj/article/view/2068/1152 Copyright (c) 2015 Anitha B. S.; Bagewadi C. S. |
| spellingShingle | Anitha, B. S. Bagewadi, C. S. Анітха, В. С. Багеваді, С. С. Invariant Submanifolds of Trans-Sasakian Manifolds |
| title | Invariant Submanifolds of Trans-Sasakian Manifolds |
| title_alt | Інваріантні підмноговиди транс-многовидів Сасакяна |
| title_full | Invariant Submanifolds of Trans-Sasakian Manifolds |
| title_fullStr | Invariant Submanifolds of Trans-Sasakian Manifolds |
| title_full_unstemmed | Invariant Submanifolds of Trans-Sasakian Manifolds |
| title_short | Invariant Submanifolds of Trans-Sasakian Manifolds |
| title_sort | invariant submanifolds of trans-sasakian manifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2068 |
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