A note on invariant submanifolds of $(k, μ)$-contact manifolds
The object of the present paper is to study invariant submanifolds of a $(k, μ)$-contact manifold and to find the necessary and sufficient conditions for an invariant submanifold of a $(k, μ)$-contact manifold to be totally geodesic.
Gespeichert in:
| Datum: | 2010 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2010
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2979 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508988239511552 |
|---|---|
| author | Avik, De Авік, Де |
| author_facet | Avik, De Авік, Де |
| author_sort | Avik, De |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:38Z |
| description | The object of the present paper is to study invariant submanifolds of a $(k, μ)$-contact manifold and to find the necessary and sufficient conditions for an invariant submanifold of a $(k, μ)$-contact manifold to be totally geodesic. |
| first_indexed | 2026-03-24T02:33:57Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 515.12
Avik De (Univ. Calcutta, India)
A NOTE ON INVARIANT SUBMANIFOLDS
OF (k, µ)-CONTACT MANIFOLDS
ПРО ДЕЯКI ПIДМНОГОВИДИ
(k, µ)-КОНТАКТНИХ МНОГОВИДIВ
The object of the present paper is to study invariant submanifolds of a (k, µ)-contact manifold and to find
the necessary and sufficient conditions for an invariant submanifold of a (k, µ)-contact manifold to be totally
geodesic.
Метою статтi є вивчення iнварiантних пiдмноговидiв (k, µ)-контактного многовиду та встановлення
необхiдних i достатнiх умов для того, щоб iнварiантний пiдмноговид (k, µ)-контактного многовиду був
цiлком геодезичним.
1. Introduction. It is well known [1, 2] that the tangent sphere bundle of a flat Rie-
mannian manifold admits a contact metric structure satisfying R(X,Y )ξ = 0, where R
is the curvature tensor. On the other hand, on a manifold M equipped with a Sasakian
structure (φ, ξ, η, g), one has
R(X,Y )ξ = η(Y )X − η(X)Y, X, Y ∈ Γ(TM). (1)
As a generalization of both R(X,Y )ξ = 0 and the Sasakian case (1), Blair, Koufogior-
gos and Papantoniou [3] introduced the case of contact metric manifolds with contact
metric structure (φ, ξ, η, g) which satisfy
R(X,Y )ξ = k(η(Y )X − η(X)Y ) + µ(η(Y )hX − η(X)hY ) (2)
for all X, Y ∈ Γ(TM), where k and µ are real constants and 2h is the Lie derivative
of φ in the direction ξ. A contact metric manifold belonging to this class is called a
(k, µ)-contact manifold. In fact, there are many motivations for studying (k, µ)-contact
manifolds: the first is that, in the non-Sasakian case (that is, for k 6= 1) the condition (2)
determines the curvature completely; moreover, while the values of k and µ change,
the form of (2) is invariant under D-homothetic deformations [3]; finally there is a
complete classification of these manifolds, given in [4] by Boeckx, who proved also that
any non-Sasakian (k, µ)-contact manifold is locally homogeneous and strongly locally
φ-symmetric [5, 6]. There are also non-trivial examples of (k, µ)-contact manifolds,
the most important being the unit tangent sphere bundle of a Riemannian manifold of
constant sectional curvature with the usual contact metric structure.
An odd dimensional invariant submanifold of a (k, µ)-contact manifold is a subman-
ifold for which the structure tensor field φ maps tangent vectors into tangent vectors.
Such a submanifold inherits a contact metric stucture from the ambient space and it is
in fact a (k, µ)-contact manifold [16].
c© AVIK DE, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11 1555
1556 AVIK DE
In [11] Kon proved that an invariant submanifold of a Sasakian manifold is to-
tally geodesic, provided the second fundamental form of the immersion is covariantly
constant. Generalising this result of Kon the authors [16] proved that if the second fun-
damental form of an invariant submanifold in a (k, µ)-contact manifold is covariantly
constant then either k = 0 or the submanifold is totally geodesic.
Motivated by these works we have studied the possible necessary and sufficient con-
ditions of an invariant submanifold of a (k, µ)-contact manifold to be totally geodesic.
In this paper we have generalized the results of [16]. In the present paper we have
proved that the recurrency, 2-recurrency and generalised 2-recurrency of the second
fundamental form of an invariant submanifold of a (k, µ)-contact manifold are equiva-
lent. And any one of these three conditions can be taken as a necessary and sufficient
condition of the submanifold to be totally geodesic. Since N(k)-contact metric mani-
fold is a special case of (k, µ)-contact manifold, therefore the above results also hold
in any N(k)-contact metric manifold. Finally we have studied the semiparallelity of an
invariant submanifold of a (k, µ)-contact manifold.
2. Preliminaries. An n-dimensional manifold Mn(n is odd) is said to admit an
almost contact structure [1, 15, 18] if it admits a tensor field φ of type (1, 1), a vector
field ξ and a 1-form η satisfying
φ2X = −X + η(X)ξ, η(ξ) = 1, (3)
φξ = 0, η(φX) = 0. (4)
An almost contact metric structure is said to be normal if the induced almost complex
structure J on the product manifold Mn × R defined by
J
(
X, f
d
dt
)
=
(
φX − fξ, η(X)
d
dt
)
is integrable, where X is tangent to M, t is the coordinate of R and f is a smooth
function on Mn × R. Let g be the compatible Riemannian metric with almost contact
structure (φ, ξ, η), that is,
g(φX, φY ) = g(X,Y )− η(X)η(Y ).
Then Mn becomes an almost contact metric manifold equipped with an almost contact
metric structure (φ, ξ, η, g). From (3) it can be easily seen that
g(X,φY ) = −g(φX, Y ), g(X, ξ) = η(X),
for any vector fields X, Y on the manifold. An almost contact metric structure becomes
a contact metric structure if g(X,φY ) = dη(X,Y ), for all vector fields X, Y.
Let f : (M, g) −→ (M̄, ḡ) be an isometric immersion of an n-dimensional Rieman-
nian manifold (M, g) into (n + d)-dimensional Riemannian manifold (M̄, ḡ), n ≥ 2,
d ≥ 1. We denote by ∇ and ∇ the Levi-Civita connections of M and M̄ respectively,
and by T⊥M its normal bundle. Then for vector fields X, Y ∈ TM, the second fun-
damental form σ is given by the formula σ(X,Y ) = ∇XY − ∇XY. Furthermore, for
N ∈ T⊥(M), AN : TM −→ TM will denote the Weingarten operator in the direction
of N, ANX = ∇⊥
XN−∇XN, where∇⊥ denotes the normal connection of M. The sec-
ond fundamental form σ and AN are related by ḡ(σ(X,Y ), N) = g(ANX,Y ), where
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
A NOTE ON INVARIANT SUBMANIFOLDS OF (k, µ)-CONTACT MANIFOLDS 1557
g is the induced metric of ḡ for any vector fields X,Y tangent to M. The covariant
derivative ∇σ and second covariant derivative ∇2
σ of σ are defined by(
∇Xσ
)
(Y, Z) = ∇⊥
X(σ(Y,Z))− σ(∇XY,Z)− σ(Y,∇XZ), (5)
(∇2
σ)(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 ), (6)
respectively, where ∇σ is a normal bundle valued tensor of type (0, 3) and ∇ is called
the van der Waerden – Bortolotti connection of M.
The basic equation of Gauss is given by [7]
R̄(X,Y, Z,W ) =
= R(X,Y, Z,W )− g(σ(X,W ), σ(Y,Z)) + g(σ(X,Z), σ(Y,W )).
However, for a (k, µ)-contact metric manifold Mn of dimension n, we have [2]
(∇Xφ)Y = g(X + hX, Y )ξ − η(Y )(X + hX),
where h =
1
2
£ξφ. From the above equation we also have
∇Xξ = −φX − φh(X).
Now from the Gauss formula we have
∇Xξ = ∇Xξ + σ(X, ξ).
Since the submanifold M is invariant, we have from the above two equations,
∇Xξ = −φX − φh(X) and σ(X, ξ) = 0. (7)
3. Immersions of recurrent type. We denote by ∇pT the covariant differential of
the pth order, p ≥ 1, of a (0, k)-tensor field T, k ≥ 1, defined on a Riemannian manifold
(M, g) with the Levi-Civita connection ∇. According to [14], the tensor T is said to be
recurrent and 2-recurrent, if the following conditions hold on M
(∇T )(X1, . . . , Xk;X)T (Y1, . . . , Yk) = (∇T )(Y1, . . . , Yk;X)T (X1, . . . , Xk), (8)
(∇2T )(X1, . . . , Xk;X,Y )T (Y1, . . . , Yk) = (∇2T )(Y1, . . . , Yk;X,Y )T (X1, . . . , Xk),
(9)
respectively, where X, Y, X1, Y1, . . . , Xk, Yk ∈ TM. From (8) it follows that at a point
x ∈ M if the tensor T is non-zero, then there exists a unique 1-form θ, respectively, a
(0, 2)-tensor ψ, defined on a neighborhood U of x, such that
∇T = T ⊗ θ, θ = d(log ‖T‖), (10)
respectively
∇2T = T ⊗ ψ,
holds on U, where ‖T‖ denotes the norm of T.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1558 AVIK DE
The tensor T is said to be generalized 2-recurrent if
(∇2T )(X1, . . . , Xk;X,Y )− (∇T ⊗ θ)(X1, . . . , Xk;X,Y )T (Y1, . . . , Yk) =
= (∇2T )(Y1, . . . , Yk;X,Y )− (∇T ⊗ θ)(Y1, . . . , Yk;X,Y )T (X1, . . . , Xk)
holds on M, where θ 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 ⊗ θ + T ⊗ ψ,
holds on U.
The notion of generalized 2-recurrent tensors in Riemannian spaces is introduced by
Ray [13].
J. Deprez defined the immersion to be semiparallel if
R̄(X,Y ).σ = (∇X∇Y −∇Y∇X −∇[X,Y ])σ = 0,
holds for all vector fields X, Y tangent to M. J. Deprez mainly paid attention to the
case of semiparallel immersions in real space forms [8, 9]. Later, Lumiste showed that
a semiparallel submanifold is the second order envelope of the family of parallel sub-
manifolds [12]. In [10] H. Endo studied semiparallelity condition for a contact metric
manifold. He showed that a semiparallel contact metric manifold is totally geodesic
under certain conditions.
4. Recurrent submanifolds of (k, µ)-contact manifolds. To prove the main theo-
rem we first state two lemmas.
Lemma 1 [19]. Let M be a submanifold of a contact metric manifold M. If ξ is
orthogonal to M, then M is anti-invariant.
Lemma 2 [17]. We know that if (M,φ, ξ, η, g) be a contact Riemannian manifold
and ξ belong to the (k, µ)-nullity distribution, then k ≤ 1. If k < 1, then (M,φ, ξ, η, g)
admits three mutually orthogonal and integrable distributions D(0), D(λ), D(−λ),
defined by the eigenspaces of h, where λ =
√
1− k.
Now, if X ∈ D(λ), then hX = λX and if X ∈ D(−λ), then hX = −λX.
Theorem 1. Let M be an invariant submanifold of a (k, µ)-contact manifold, with
k 6= 0. Then the following conditions are equivalent:
(i) M is totally geodesic;
(ii) the second fundamental form σ is recurrent;
(iii) the second fundamental form σ is 2-recurrent;
(iv) the second fundamental form σ is generalized 2-recurrent.
Proof. Suppose M is totally geodesic, then (ii), (iii) and (iv) are trivially true.
Now suppose σ is recurrent, then from (10), we get
(∇̄Xσ)(Y,Z) = θ(X)σ(Y,Z),
where θ is a 1-form on M. Then in view of (5), we obtain
∇⊥
X(σ(Y,Z))− σ(∇XY,Z)− σ(Y,∇XZ) = θ(X)σ(Y,Z). (11)
By Lemma 1, ξ ∈ TM. So, taking Z = ξ in (11), we have
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
A NOTE ON INVARIANT SUBMANIFOLDS OF (k, µ)-CONTACT MANIFOLDS 1559
∇⊥
X(σ(Y, ξ))− σ(∇XY, ξ)− σ(Y,∇Xξ) = θ(X)σ(Y, ξ).
Then using (7), we obtain
σ(Y,∇Xξ) = 0.
Using (7) we get
σ(Y,X)− σ(Y, hX) = 0.
Therefore, Lemma 2 yields (1 ± λ)σ(Y,X) = 0, which implies σ(Y,X) = 0,
provided λ 6= ±1, or k 6= 0.
Thus M is totally geodesic, provided k 6= 0.
Proceeding in a similar manner we can prove that if σ is 2-recurrent or generalized
2-recurrent, then also M is totally geodesic.
Theorem 1 is proved.
Theorem 2. Let M be an invariant submanifold of a (k, µ)-contact manifold M̄.
Then M is totally geodesic if and only if M is semiparallel, provided k 6= ±µ
√
1− k.
Proof. We have
(R̄(X,Y ).σ)(V,W ) =
= R⊥(X,Y )(σ(V,W ))− σ(R(X,Y )V,W )− σ(V,R(X,Y )W ).
Suppose M is semiparallel. Then R̄(X,Y ).σ = 0, that is, R̄(X, ξ).σ = 0. Therefore,
we have
R⊥(X, ξ)(σ(V,W )) = σ(R(X, ξ)V,W ) + σ(V,R(X, ξ)W ).
Putting V = ξ, and using (7) we obtain
σ(R(X, ξ)ξ,W ) = 0. (12)
Using (2) in (12) we obtain
(k ± µ
√
1− k)σ(X,W ) = 0.
Therefore, σ(X,W ) = 0, provided k 6= ±µ
√
1− k. Hence M is totally geodesic. The
converse statement is trivial. This completes the proof of the theorem.
The corollary follows immediately:
Corollary. Let M be an invariant submanifold of a (k, µ)-contact manifold M̄.
Then the following conditions are equivalent:
(i) M is totally geodesic;
(ii) R̄(X, ξ).σ = 0;
(iii) R̄(X,Y ).σ = 0, where X and Y are arbitrary vector fields on M.
Acknowledgement. The author is thankful to the referee for his valuable comments
towards the improvement of the paper.
1. Blair D. E. Two remarks on contact metric structure // Tohoku Math J. – 1977. – 29. – P. 319 – 324.
2. Blair D. E. Riemannian geometry of contact and symplectic manifolds // Progr. Math. – 1979. – 203.
3. Blair D. E., Koufogiorgos T., Papantoniou B. J. Contact metric manifolds satisfying a nullity condition
// Isr. J. Math. – 1995. – 91. – P. 189 – 214.
4. Boeckx E. A full classification of contact metric (k, µ)-spaces // Ill. J. Math. – 2000. – 44. – P. 212 – 219.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1560 AVIK DE
5. Boeckx E. A class of locally φ-symmetric contact metric spaces // Arch. Math. (Basel). – 1999. – 72. –
P. 466 – 472.
6. Boeckx E. Contact-homogeneous locally φ-symmetric manifolds // Glasgow Math. J. – 2006. – 48. –
P. 93 – 109.
7. Chen B. Y. Geometry of submanifolds // Pure and Appl. Math. – 1973. – № 22.
8. Deprez J. Semi-parallel surfacess in the Euclidean space // J. Geom. – 1985. – 25. – P. 192 – 200.
9. Deprez J. Semi-parallel hypersurfaces // Rend. Semin. mat. Univ. e politecn. Torino. – 1986. – 44. –
P. 303 – 316.
10. Endo H. Certain submanifolds of contact metric manifolds // Tensor (N. S.). – 1988. – 47, № 2. –
P. 198 – 202.
11. Kon M. Invariant submanifolds of normal contact metric manifolds // Kodai Math. Sem. Rep. – 1973. –
27. – P. 330 – 336.
12. Lumiste U. Semi-symmetric submanifolds as the second order envelope of symmetric submanifolds //
Proc. Eston. Acad. Sci. Phys. Math. – 1990. – 39. – P. 1 – 8.
13. Ray A. K. On generalised 2-recurrents tensors in Riemannian spaces // Accad. Roy. Belg. Bull. Cl. Sci.
– 1972. – 5, № 58. – P. 220 – 228.
14. Roter W. On conformally recurrent Ricci-recurrent manifolds // Colloq. Math. – 1982. – 46, № 1. –
P. 45 – 57.
15. Sasaki S. Lecture notes on almost contact manifolds. Pt I. – Tohoku Univ., 1965.
16. Tripathi M. M., Sasahara T., Kim J.-S. On invariant submanifolds of contact metric manifolds // Tsukuba
J. Math. – 2005. – 29, №. 2. – P. 495 – 510.
17. Tanno S. Ricci curvatures of contact Riemannian manifolds // Tohoku Math. J. – 1988. – 40. – P. 441 – 448.
18. Yano K., Kon M. Structures on manifolds // Ser. Pure Math. – Singapore: World Sci., 1984.
19. Yano K., Kon M. Anti-invariant submanifolds // Lect. Notes Pure and Appl. Math. – 1976. – 21.
Received 05.02.10,
after revision — 30.06.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
|
| id | umjimathkievua-article-2979 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:33:57Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e4/342a6db4b5ba67f0de6155a0678fc6e4.pdf |
| spelling | umjimathkievua-article-29792020-03-18T19:41:38Z A note on invariant submanifolds of $(k, μ)$-contact manifolds Про деякі підмноговиди $(k, μ)$-контактних многовидів Avik, De Авік, Де The object of the present paper is to study invariant submanifolds of a $(k, μ)$-contact manifold and to find the necessary and sufficient conditions for an invariant submanifold of a $(k, μ)$-contact manifold to be totally geodesic. Метою статті є вивчення інваріантних підмноговидів $(k, μ)$-контактного многовиду та встановлення необхідних і достатніх умов для того, щоб інваріантний підмноговид $(k, μ)$-контактного многовиду був цілком геодезичним. Institute of Mathematics, NAS of Ukraine 2010-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2979 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 11 (2010); 1555–1560 Український математичний журнал; Том 62 № 11 (2010); 1555–1560 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2979/2708 https://umj.imath.kiev.ua/index.php/umj/article/view/2979/2709 Copyright (c) 2010 Avik De |
| spellingShingle | Avik, De Авік, Де A note on invariant submanifolds of $(k, μ)$-contact manifolds |
| title | A note on invariant submanifolds of $(k, μ)$-contact manifolds |
| title_alt | Про деякі підмноговиди $(k, μ)$-контактних многовидів |
| title_full | A note on invariant submanifolds of $(k, μ)$-contact manifolds |
| title_fullStr | A note on invariant submanifolds of $(k, μ)$-contact manifolds |
| title_full_unstemmed | A note on invariant submanifolds of $(k, μ)$-contact manifolds |
| title_short | A note on invariant submanifolds of $(k, μ)$-contact manifolds |
| title_sort | note on invariant submanifolds of $(k, μ)$-contact manifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2979 |
| work_keys_str_mv | AT avikde anoteoninvariantsubmanifoldsofkmcontactmanifolds AT avíkde anoteoninvariantsubmanifoldsofkmcontactmanifolds AT avikde prodeâkípídmnogovidikmkontaktnihmnogovidív AT avíkde prodeâkípídmnogovidikmkontaktnihmnogovidív AT avikde noteoninvariantsubmanifoldsofkmcontactmanifolds AT avíkde noteoninvariantsubmanifoldsofkmcontactmanifolds |