CUBO, A Mathematical Journal

Vol. 24, no. 01, pp. 105–114, April 2022

DOI: 10.4067/S0719-06462022000100105

Some results on the geometry of warped product
CR-submanifolds in quasi-Sasakian manifold

Shamsur Rahman

Department of Mathematics, Maulana

Azad National Urdu University

Polytechnic Satellite Campus Darbhanga

Bihar- 846002, India.

shamsur@rediffmail.com

ABSTRACT

The present paper deals with a study of warped product

submanifolds of quasi-Sasakian manifolds and warped prod-

uct CR-submanifolds of quasi-Sasakian manifolds. We have

shown that the warped product of the type M = D⊥×yDT

does not exist, where D⊥ and DT are invariant and anti-

invariant submanifolds of a quasi-Sasakian manifold M̄, re-

spectively. Moreover we have obtained characterization re-

sults for CR-submanifolds to be locally CR-warped products.

RESUMEN

El presente art́ıculo trata de un estudio de subvariedades

producto alabeadas de variedades cuasi-Sasakianas y CR-

subvariedades producto alabeadas de variedades cuasi-

Sasakianas. Hemos mostrado que el producto alabeado de

tipo M = D⊥×yDT no existe, donde D⊥ y DT son subva-

riedades invariantes y anti-invariantes de una variedad cuasi-

Sasakiana M̄, respectivamente. Más aún, hemos obtenido re-

sultados de caracterización para que CR-subvariedades sean

localmente CR-productos alabeados.

Keywords and Phrases: Warped product, CR-submanifolds, quasi Sasakian manifold, canonical structure.

2020 AMS Mathematics Subject Classification: 53C25, 53C40.

Accepted: 18 January, 2022

Received: 02 May, 2021

c©2022 S. Rahman. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462022000100105
https://orcid.org/0000-0003-0995-2860
mailto:shamsur@rediffmail.com


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1 Introduction

If (D, gD) and (E, gE) are two semi-Riemannian manifolds with metrics gD and gE respectively

and y a positive differentiable function on D, then the warped product of D and E is the manifold

D×yE = (D×E, g), where g = gD +y
2gE. Further, let T be tangent to M = D×E at (p, q). Then

we have

‖T ‖2 = ‖dπ1T ‖
2 + y2‖dπ2T ‖

2

where πi(i = 1, 2) are the canonical projections of D×E onto D and E.

A warped product manifold D×yE is said to be trivial if the warping function y is constant. In a

warped product manifold, we have

∇U V = ∇V U = (U ln y)V (1.1)

for any vector fields U tangent to D and V tangent to E [5].

The idea of a warped product manifold was introduced by Bishop and O’Neill [5] in 1969. Chen [2]

has studied the geometry of warped product submanifolds in Kaehler manifolds and showed that the

warped product submanifold of the type D⊥×yDT is trivial where DT and D⊥ are φ-invariant and

anti-invariant submanifolds of a Sasakian manifold, respectively. Many research articles appeared

exploring the existence or nonexistence of warped product submanifolds in different spaces [1, 10, 6].

The idea of CR-submanifolds of a Kaehlerian manifold was introduced by A. Bejancu [9]. Later, A.

Bejancu and N. Papaghiue [11], introduced and studied the notion of semi-invariant submanifolds of

a Sasakian manifold. These submanifolds are closely related to CR-submanifolds in a Kaehlerian

manifold. However the existence of the structure vector field implies some important changes.

Later on, Binh and De [4] studied CR-warped product submanifolds of a quasi-Saskian manifold.

The purpose of this paper is to study the notion of a warped product submanifold of quasi-Sasakian

manifolds. In the second section we recall some results and formulae for later use. In the third

section, we prove that the warped product in the form M = D⊥×yDT does not exist except

for the trivial case, where DT and D⊥ are invariant and anti-invariant submanifolds of a quasi-

Sasakian manifold M̄, respectively. Also, we obtain a characterization result of the warped product

CR-submanifolds of the type M = D⊥×yDT .

2 Preliminaries

If M̄ is a real (2n+ 1) dimensional differentiable manifold, endowed with an almost contact metric

structure (f, ξ, η, g), then

f2U = −U + η(U)ξ, η(ξ) = 1, f(ξ) = 0, η(fU) = 0, (2.1)



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Some results on the geometry of warped product ... 107

η(U) = g(U, ξ), g(fU, fV ) = g(U, V ) − η(U)η(V ), (2.2)

for any vector fields U, V tangent to M̄, where I is the identity on the tangent bundle ΓM̄ of M̄.

Throughout the paper, all manifolds and maps are differentiable of class C∞. We denote by ̥M̄

the algebra of the differentiable functions on M̄ and by Γ(E) the ̥M̄ module of the sections of a

vector bundle E over M̄.

The Nijenhuis tensor field, denoted by Nf , with respect to the tensor field f, is given by

Nf(U, V ) = [fU, fV ] + f
2[U, V ] − f[fU, V ] + f[U, fV ],

and the fundamental 2-form Λ is given by

Λ(U, V ) = g(U, fV ), ∀U, V ∈ Γ(T M̄).

The curvature tensor field of M̄, denoted by R̄ with respect to the Levi-Civita connection ∇̄, is

defined by

R̄(U, V )W = ∇̄U ∇̄V W − ∇̄V ∇̄UW − ∇̄[U,V ]W, ∀U, V ∈ Γ(T M̄),

Definition 2.1.

(a) An almost contact metric manifold M̄(f, ξ, η, g) is called normal if

Nf (U, V ) + 2dη(U, V )ξ = 0, ∀U, V ∈ Γ(T M̄),

or equivalently

(∇̄fU f)V = f(∇̄Uf)V − g(∇̄Uξ, V )ξ, ∀U, V ∈ Γ(T M̄).

(b) The normal almost contact metric manifold M̄ is called cosympletic if dΛ = dη = 0.

If M̄ is an almost contact metric manifold, then M̄ is a quasi-Sasakian manifold if and only if ξ is

a Killing vector field [7] and

(∇̄U f)V = g(∇̄fUξ, V )ξ − η(V )∇̄fU ξ, ∀U, V ∈ Γ(T M̄). (2.3)

Next we define a tensor field F of type (1, 1) by

FU = −∇̄Uξ, ∀U ∈ Γ(T M̄). (2.4)



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Lemma 2.1. For a quasi-Sasakian manifold M̄, we have

(i) (∇̄ξf)U = 0, ∀U ∈ Γ(T M̄),

(ii) f ◦ F = F ◦ f,

(iii) Fξ = 0,

(iv) g(FU, V ) + g(U, FV ) = 0,

(v) η ◦ F = 0,

(vi) (∇̄U F)V = R̄(ξ, U)V ,

for all U, V ∈ Γ(T M̄).

The tensor field f defines on M̄ an f-structure in sense of K. Yano [12], that is

f3 + f = 0.

If M is a submanifold of a quasi-Sasakian manifold M̄ and denote by N the unit vector field normal

to M. Denote by the same symbol g the induced tensor metric on M, by ∇ the induced Levi-

Civita connection on M and by T M⊥ the normal vector bundle to M. The Gauss and Weingarten

methods are

∇̄U V = ∇U V + σ(U, V ), (2.5)

∇̄U λ = −AλU + ∇
⊥
U λ, ∀U, V ∈ Γ(T M), (2.6)

where ∇⊥ is the induced connection in the normal bundle, σ is the second fundamental form of

M and Aλ is the Weingarten endomorphism associated with λ. The second fundamental form σ

and the shape operator A are related by

g(AλU, V ) = g(h(U, V ), λ), (2.7)

where g denotes the metric on M̄ as well as the induced metric on M [7].

For any U ∈ T M, we write

fU = rU + sU, (2.8)

where rU is the tangential component of fU and sU is the normal component of fU, respectively.

Similarly, for any vector field λ normal to M, we put

fλ = Jλ + Kλ (2.9)

where Jλ and Kλ are the tangential and normal components of fλ, respectively.

For all U, V ∈ Γ(T M) the covariant derivatives of the tensor fields r and s are defined as

(∇̄U r)V = ∇UrV − r∇U V, (2.10)

(∇̄U s)V = ∇
⊥
U sV − s∇U V. (2.11)



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Some results on the geometry of warped product ... 109

3 Warped Product Submanifolds

If DT and D⊥ are invariant and anti-invariant submanifolds of a quasi-Sasakian manifold M̄, then

their warped product CR-submanifolds are one of the following forms:

(i) M = D⊥×yDT ,

(ii) M = DT ×yD⊥.

For case (i), when ξ ∈ T DT , we have the following theorem.

Theorem 3.1. There do not exist warped product CR-submanifolds M = D⊥×yDT in a quasi-

Sasakian manifold M̄ such that DT is an invariant submanifold, D⊥ is an anti-invariant subman-

ifold of M̄ and ξ is tangent to M.

Proof. If M = D⊥×yDT is a warped product CR-submanifold of a quasi-Sasakian manifold M̄

such that DT is an invariant submanifold tangent to ξ and D⊥ is an anti-invariant submanifold of

M̄, then from (1.1), we have

∇U W = ∇W U = (W ln y)U,

for any vector fields W and U tangent to D⊥ and DT , respectively.

In particular,

∇W ξ = (W ln y)ξ, (3.1)

using (2.4), (2.5) and ξ is tangent to D⊥, we have

∇W ξ = −FW, h(W, ξ) = 0. (3.2)

It follows from (3.1) and (3.2) that W ln y = 0, for all W ∈ T D⊥, i. e., y is constant for all

W ∈ T D⊥.

Now, the other case, when ξ tangent to D⊥ is dealt in the following two results.

Lemma 3.1. Let M = D⊥×yDT be a warped product CR-submanifold of a quasi-Sasakian man-

ifold such that ξ is tangent to D⊥, where D⊥ and DT are any Riemannian submanifolds of M̄.

Then

(i) ξ ln y = −F,

(ii) g(σ(U, fU), sW) = −{η(W)F + (W ln y)}‖U‖2,

for any U ∈ T DT and W ∈ T D⊥.



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Proof. Let ξ ∈ T D⊥ then for any U ∈ T DT , we have

∇U ξ = (ξ ln y)U, (3.3)

From (2.4) and the fact that ξ is tangent to D⊥, we have ∇̄U ξ = −FU. With the help of (2.5),

we have

∇W ξ = −FW, h(W, ξ) = 0. (3.4)

From (3.3) and (3.4), we have ξ ln y = −F . Now, for any U ∈ T DT and W ∈ T D⊥, we have

∇̄UfW = (∇̄U f)W + f(∇̄UW). Using (2.3), (2.6), (2.8), (2.9) and by the orthogonality of the two

distributions, we derive

−η(W)∇̄fU ξ = −AsW U + ∇
⊥
U sW − r∇U W − s∇U W − Jh(U, W) − Kh(U, W).

Equating the tangential components, we get

−η(W)FfU = AsW U + r∇U W + Jh(U, W).

Taking the product with fU and using (2.2) and (2.3), we derive

−η(W)Fg(fU, fU) = g(AsW U, fU) + (W ln y)g(rU, fU) + g(Jh(U, W), fU)

= g(h(fU, fU), sW) + (W ln y)g(fU, fU) + g(fh(U, W), fU).

Using (2.2), we obtain

g(σ(U, fU), sW) = −{η(W)F + (W ln y)}‖U‖2. (3.5)

Theorem 3.2. If M = D⊥×yDT is a warped product CR-submanifold of a quasi-Sasakian man-

ifold M̄ such that ξ is tangent to D⊥ and if σ(U, fU) ∈ µ the invariant normal subbundle of M,

then W ln y = −η(W)F, for all U ∈ T DT and Z ∈ T N⊥.

Proof. The affirmation follows from formula (3.5) by means of the known truth.

The warped product M = DT ×yD⊥, we have the following theorem.

Theorem 3.3. There do not exist warped product CR-submanifolds M = DT ×yD⊥ in a quasi-

Sasakian manifold M̄ such that ξ is tangent to D⊥.

Proof. If ξ ∈ T N⊥, then from (1.1), we have

∇U ξ = (U ln y)ξ, (3.6)

for any U ∈ T DT . While using (2.4), (2.5) and ξ ∈ T D⊥, we have

∇Uξ = −FU, h(U, ξ) = 0. (3.7)

From (3.6) and (3.7), it follows that U ln y = 0, for all U ∈ T DT , and this means that y is constant

on NT .



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The remaining case, when ξ ∈ T DT is dealt with the following two theorems.

Theorem 3.4. Let M = DT ×yD⊥ be a warped product CR-submanifold of a quasi-Sasakian

manifold M̄ such that ξ is tangent to DT . Then (∇̄U F)W ∈ µ, for each U ∈ T DT and W ∈ T D⊥,

where µ is an invariant normal subbundle of T M.

Proof. For any U ∈ T DT and W ∈ T D⊥, we have

g(f∇̄UW, fW) = g(∇̄UW, W) = g(∇U W, W).

Using (1.1), we get

g(f∇̄UW, fW) = (U ln y)‖W‖
2
. (3.8)

On the other hand, we have

∇̄U fW = (∇̄U f)W + f(∇̄UW),

for any U ∈ T DT and W ∈ T D⊥. Using (2.3) and the fact that ξ is tangent to DT , the left-hand

side of the above equation is identically zero, that is

∇̄UfW = f(∇̄UW). (3.9)

Taking the product with fW in (3.9) and making use of formula (2.6), we obtain

g(f∇̄UW, fW) = g(∇
⊥
U sW, sW).

Then from (2.10), we derive g(f∇̄U W, fW) = g((∇̄U s)W, sW) + g(s∇UW, sW).

From (1.1) we have

g(f∇̄UW, fW) = (U ln y)g(sW, sW) + g((∇̄U s)W, sW)

= (U ln y)g(fW, fW) + g((∇̄U s)W, sW).

Therefore by (2.2), we obtain

g(f∇̄UW, fW) = (U ln y)‖W‖
2 + g((∇̄U s)W, sW). (3.10)

Thus (3.8) and (3.9) imply

g((∇̄U s)W, sW) = 0. (3.11)

Also, as DT is an invariant submanifold then fQ ∈ T DT , for any Q ∈ T DT , thus on using (2.11)

and the fact that the product of tangential components with normal is zero, we obtain

g((∇̄Us)W, fQ) = 0. (3.12)

Hence from (3.11) and (3.12), it follows that (∇̄U s)W ∈ µ, for all U ∈ T DT and W ∈ T D⊥.



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Theorem 3.5. A CR-submanifold M of a quasi-Sasakian manifold (M̄, f, ξ, g) is a CR-warped

product if and only if the shape operator of M satisfies

AfW U = (fUµ)W, U ∈ B ⊕ 〈ξ〉, W ∈ B
⊥
, (3.13)

for some function µ on M, fulfilling C(µ) = 0, for each C ∈ B⊥.

Proof. If M = DT ×yD⊥ is a CR-warped product submanifold of a quasi-Sasakian manifold M̄,

with ξ ∈ T DT , then for any U ∈ T DT and W, Q ∈ T D⊥, we have

g(AfW U, Q) = g(σ(U, Q), fW) = g(∇̄QU, fW) = g(f∇̄QU, W)

= g(∇̄QfU, W) − g((∇̄Qf)U, W).

By equations (1.1), (2.3) and the fact that ξ is tangent to DT , we derive

g(AfW U, Q) = (fU ln y)g(W, Q). (3.14)

On the other hand, we have g(σ(U, V ), sW) = g(f∇̄U V, W) = −g(fV, ∇̄UW), for each U, V ∈ T DT

and W ∈ T N⊥. Using (1.1), we obtain g(σ(U, V ), sW) = 0. Taking into account this fact in (3.14),

we obtain (3.13).

Conversely, suppose that M is a proper CR-submanifold of a quasi-Sasakian manifold M satisfying

(3.13), then for any U, V ∈ B ⊕ 〈ξ〉,

g(σ(U, V ), fW) = g(AfW U, V ) = 0.

This implies that g(∇̄U fV, W) = 0, that is, g(∇U V, W) = 0. This means B ⊕ 〈ξ〉 is integrable and

its leaves are totally geodesic in M. Now, for any W, Q ∈ B⊥ and U ∈ B ⊕ 〈ξ〉, we have

g(∇W Q, fU) = g(∇̄W Q, fU) = g(f∇̄W Q, U) = g(∇̄W fQ, U) − g((f∇̄W f)Q, U).

By equations (2.3) and (2.6), it follows that g(∇W Q, fU) = −g(AfQW, U). Thus from (2.6), we

arrive at g(∇W Q, fU) = −g(σ(W, U), fQ). Again using (2.7) and (3.13), we obtain

g(∇W Q, fU) = −g(AfQU, W) = −(fUµ)g(W, Q). (3.15)

If N⊥ is a leaf of B
⊥ and σ⊥ is the second fundamental form of the immersion of D⊥ into M, then

for any W, Q ∈ B⊥, we have

g(σ⊥(W, Q), fU) = g(∇W Q, fU). (3.16)

Hence, from (3.15) and (3.16), we find that

g(σ⊥(W, Q), fU) = −(fUµ)g(W, Q).



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Some results on the geometry of warped product ... 113

This means that the integral manifold D⊥ of B
⊥ is totally umbilical in M. Since C(µ) = 0 for

each C ∈ B⊥, which implies that the integral manifold of B⊥ is an extrinsic sphere in M, this

means that the curvature vector field is nonzero and parallel along N⊥. Hence by virtue of a result

in [7], M is locally a warped product DT ×yD⊥, where DT and N⊥ denote the integral manifolds

of the distributions B ⊕ 〈ξ〉 and B⊥, respectively and y is the warping function.

Acknowledgements

The authors grateful the referee(s) for the corrections and comments in the revision of this paper.



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References

[1] K. Arslan, R. Ezentas, I. Mihai and C. Murathan, “Contact CR-warped product submanifolds

in Kenmotsu space forms”, J. Korean Math. Soc., vol. 42, no. 5, pp. 1101–1110, 2005.

[2] A. Bejancu, “CR-submanifold of a Kaehler manifold. I”, Proc. Amer. Math. Soc., vol. 69, no.

1, 135–142, 1978.

[3] A. Bejancu and N. Papaghiuc, “Semi-invariant submanifolds of a Sasakian manifold.”, An.

Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.), vol 27, no. 1, pp. 163–170, 1981.

[4] T.-Q. Binh and A. De, “On contact CR-warped product submanifolds of a quasi-Sasakian

manifold”, Publ. Math. Debrecen, vol. 84, no. 1-2, pp. 123–137, 2014.

[5] R. L. Bishop and B. O’Neill, “Manifolds of negative curvature”, Trans. Amer. Math. Soc.,

vol. 145, pp. 1–49, 1969.

[6] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., vol. 509,

Berlin-New York: Springer-Verlag, 1976.

[7] C. Calin, “Contributions to geometry of CR-submanifold”, PhD Thesis, University of Ias,i,

Ias,i, Romania, 1998.

[8] B.-Y. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds”, Monatsh.

Math., vol. 133, no. 3, pp. 177–195, 2001.

[9] I. Hasegawa and I. Mihai, “Contact CR-warped product submanifolds in Sasakian manifolds”,

Geom. Dedicata, vol. 102, pp. 143–150, 2003.

[10] S. Hiepko, “Eine innere Kennzeichnung der verzerrten Produkte”, Math. Ann., vol. 241, no.

3, pp. 209–215, 1979.

[11] M.-I. Munteanu, “A note on doubly warped product contact CR-submanifolds in trans-

Sasakian manifolds”, Acta Math. Hungar., vol 116, no. 1-2, pp. 121–126, 2007.

[12] K. Yano, “On structure defined by a tensor field f of type (1, 1) satisfying f3 +f = 0”, Tensor

(N.S.), vol. 14, pp. 99–109, 1963.

[13] K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, vol. 3, Singapore:

World Scientific Publishing Co., 1984.


	Introduction
	Preliminaries
	Warped Product Submanifolds