Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 231416, 11 pages
doi:10.1155/2012/231416
Research Article
A Basic Inequality
for the Tanaka-Webster Connection
Dae Ho Jin1 and Jae Won Lee2
1
2
Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Correspondence should be addressed to Jae Won Lee, leejaewon@sogang.ac.kr
Received 7 November 2011; Accepted 28 November 2011
Academic Editor: C. Conca
Copyright q 2012 D. H. Jin and J. W. Lee. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
For submanifolds tangent to the structure vector field in Sasakian space forms, we establish a
Chen’s basic inequality between the main intrinsic invariants of the submanifold namely, its
pseudosectional curvature and pseudosectional curvature on one side and the main extrinsic
invariant namely, squared pseudomean curvature on the other side with respect to the TanakaWebster connection. Moreover, involving the pseudo-Ricci curvature and the squared pseudomean curvature, we obtain a basic inequality for submanifolds of a Sasakian space form tangent to
the structure vector field in terms of the Tanaka-Webster connection.
1. Introduction
One of the basic interests in the submanifold theory is to establish simple relationship
between intrinsic invariants and extrinsic invariants of a submanifold. Gauss-Bonnet
Theorem, Isoperimetric inequality, and Chern-Lashof Theorem are those such kind of study.
Chen 1 established a nice basic inequality-related intrinsic quantities and extrinsic
ones of submanifolds in a space form with arbitrary codimension. Moreover, he studied the
basic inequalities of submanifolds of complex space forms and characterize submanifolds
when the equality holds.
In this paper, we introduce pseudosectional curvatures and pseudo-Ricci curvature
for the Tanaka-Webster connection in a Sasakian space form. After then, we study basic
inequalities for submanifolds of a Sasakian space form of a constant pseudosectional
curvature and a pseudo-Ricci curvature in terms of the Tanaka-Webster connection.
2
Journal of Applied Mathematics
2. Preliminaries
be an odd-dimensional Riemannian manifold with a Riemannian metric g satisfying
Let M
ηξ 1,
ϕ2 −I η ⊗ ξ,
ηX g X, ξ,
g ϕX, ϕY g X, Y − ηXηY .
2.1
Let Φ denote the
Then ϕ, ξ, η, g is called the almost contact metric structure on M.
given by ΦX, Y g X, ϕY for all X, Y ∈ T M,
the set of vector
fundamental 2 form in M
fields of M. If Φ dη, then M is said to be a contact metric manifold. Moreover, if ξ is a
Killing vector field with respect to g , and the contact metric structure is called a K-contact
structure. Recall that a contact metric manifold is K-contact if and only if
X ξ −ϕX
∇
2.2
is the Levi-Civita connection of M.
where ∇
The structure of M
is said to be
for any X ∈ T M,
normal if ϕ, ϕ 2dη ⊗ ξ − 0, where ϕ, ϕ is the Nijenhuis torsion of ϕ. A Sasakian manifold
is a normal contact metric manifold. In fact, an almost contact metric structure is Sasakian if
and only if
X ϕ Y gX, Y ξ − ηY X
∇
2.3
for all vector fields X and Y . Every Sasakian manifold is a K-contact manifold.
is called a ϕ-section if it is
a plane section π in Tp M
Given a Sasakian manifold M,
spanned by X and ϕX, where X is a unit tangent vector field orthogonal to ξ. The sectional
has
curvature Kπ
of a ϕ-section π is called ϕ-sectional curvature. If a Sasakian manifold M
is called a Sasakian space form, denoted by Mc.
constant ϕ-sectional curvature c, M
For
more details, see 2.
ϕ, ξ, η, g. We also denote by g the
Now let M be a submanifold immersed in M,
induced metric on M. Let T M be the Lie algebra of vector fields in M and T ⊥ M the set of all
vector fields normal to M. We denote by h the second fundamental form of M and by Av the
Weingarten endomorphism associated with any v ∈ T ⊥ M. We put hrij ghei , ej , er for any
orthonormal vector ei , ej ∈ T M and er ∈ T ⊥ M. The mean curvature vector field H is defined
by H 1/ dim Mtraceh. M is said to be totally geodesic if the second fundamental form
vanishes identically.
From now on, we assume that the dimension of M is n 1, and that of the ambient
is 2m 1 m ≥ 2. We also assume that the structure vector field ξ is tangent to
manifold M
M. Hence, if we denote by D the orthogonal distribution to ξ in T M, we have the orthogonal
direct decomposition of T M by T M D ⊕ span{ξ}. For any X ∈ T M, we write ϕX T X NX, where T X NX, resp. is the tangential normal, resp. component of ϕX. If ϕM is a
K-contact manifold, 2.2 gives
hX, ξ −NX,
2.4
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3
for any X in T M. Given a local orthonormal frame {e1 , . . . , en } of D, we can define the squared
norms of T and N by
T 2 n
2
g ei , T ej ,
n
2
g ei , Nej ,
N2 i,j1
2.5
i,j1
resepectively. It is easy to show that both T 2 and N2 are independent of the choice of
the orthonormal frames. The submanifold M is said to be invariant if N is identically zero,
that is, ϕX ∈ T M for any X ∈ T M. On the other hand, M is said to be an anti-invariant
submanifold if T is identically zero, that is, ϕX ∈ T ⊥ M for any X ∈ T M.
3. The Tanaka-Webster Connection for Sasakian Space Form
The Tanaka-Webster connection 3, 4 is the canonical affine connection defined on a
nondegenerate pseudo-Hermitian CR-manifold. Tanno 5 defined the Tanaka-Webster
connection for contact metric manifolds by the canonical connection which coincides with
the Tanaka-Webester connection if the associated CR-structure is integrable. We define
the Tanaka-Webster connection for submanifolds of Sasakian manifolds by the naturally
extended affine connection of Tanno’s Tanaka-Webster connection. Now we recall the Tanaka for contact metric manifolds
Webster connection ∇
X Y ηXϕY ∇X η Y ξ − ηY ∇X ξ,
XY ∇
3.1
∇
is written by
Together with 2.1, ∇
for all vector fields X, Y ∈ T M.
XY ∇
X Y ηXϕY ηY ϕX − g Y, ϕX ξ.
∇
3.2
Also, by using 2.1 and 2.3, we can see that
0,
∇η
0,
∇ξ
0,
∇ϕ
g 0.
∇
3.3
by
in terms of ∇
We define the Tanaka-Webster curvature tensor of R
X∇
YZ − ∇
YZ − ∇
Y∇
X,Y Z,
RX,
Y Z ∇
3.4
for all vector fields X, Y , and Z in M.
Let Mc
be a Sasakian space form of constant sectional curvature c and M a sub
manifold of Mc.
Then, we have the following Gauss’ equation:
c 3 gY, Z − ηY ηZ X − gX, Z − ηXηZ Y
RX,
Y Z 4
gX, ZηY − gY, ZηX ξ 2g X, ϕY ϕZ
c 7 g Z, ϕY ϕX − g Z, ϕX ϕY
4
for any tangent vector fields X, Y, Z tangent to M.
3.5
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◦
Let us define the connection ∇ on M induced from the Tanaka-Webster connection ∇
on M given by
◦
◦
3.6
∇X Y ∇X Y hX,
Y ,
is called the lightlike second fundamental form of M with
for any X, Y ∈ ΓT M, where h
◦
respect to the induced connection ∇. In the view of 3.2 and 3.6,
◦
∇X Y hX,
Y ∇X Y hX, Y ηXϕY ηY ϕX − g Y, ϕX ξ.
3.7
From 3.7, we obtain
◦
∇X Y ∇X Y ηXT Y ηY T X − g Y, ϕX ξ,
3.8
hX,
Y hX, Y ηXNY ηY NX,
3.9
where ϕX T X NX.
From 3.3, 3.8, and 3.9 it is easy to verify the following:
◦
∇η 0,
◦
∇ξ 0,
◦
∇ϕ 0,
◦
∇g 0.
3.10
Moreover, for the induced connection ∇, we have the following
∇X ξ −T X,
hX, ξ −NX.
3.11
together with 3.5, we have
From the definition of R,
◦
c 3 gY, Z − ηY ηZ gX, W − gX, Z − ηXηZ gY, W
g RX, Y Z, W 4
gX, ZηY − gY, ZηX gξ, W 2g X, ϕY g ϕZ, W
c 7 g Z, ϕY g ϕX, W − g Z, ϕX g ϕY, W
4
Z − g hX,
W ,
g hX,
W, hY,
Z, hY,
3.12
for any X, Y, Z, W ∈ T M.
For an orthonormal basis {e1 , . . . , en1 } of the tangent space Tp M, p ∈ M, the pseudoscalar curvature τ at p is defined by
τ ei ∧ ej ,
K
i<j
3.13
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5
i ∧ej denotes the pseudosectional curvature of M associated with the plane section
where Ke
In particular, if we put en1 ξp ,
spanned by ei and ej for the Tanaka-Webster connection ∇.
then 3.13 implies that
2
τ
n
ei ∧ ej 2 Ke
i ∧ ξ.
K
i
/j
3.14
i1
Moreover, from 3.9, we have
r hij ,
h
ij
r 0,
h
in1
i, j ∈ {1, . . . , n},
j ∈ {1, . . . , n 1}.
3.15
M is said to
1/ dim Mtraceh.
The pseudomean curvature vector field H is defined by H
be totally pseudogeodesic if the second fundamental h form vanishes identically. From 2.5,
3.12 and 3.14, we obtain the following relationship between the pseudoscalar curvature
and the pseudomean curvature of M,
2 2
c 3 3c 13
2
τ n 12 H
T 2 .
− h nn − 1
4
4
3.16
We now recall the Chen’s lemma.
Lemma 3.1 see 6. Let a1 , . . . , an , c be n 1 n ≥ 2 real numbers such that
n
ai
i1
2
n
2
n − 1
ai c .
3.17
i1
Then, 2a1 a2 ≥ c, with the equality holding if and only if a1 a2 a3 · · · an .
Let p ∈ M and let π be a plane section of Tp M which is generated by orthonormal
vectors X and Y . We can define a function απ of tangent space Tp M into 0, 1 by
απ gT X, Y 2 ,
3.18
which is well defined.
Now, we prove the following.
Theorem 3.2. Let M be an n 1-dimensional n ≥ 2 submanifold isometrically immersed in a m
dimensional Sasakian space form Mc
such that the structure vector field ξ is tangent to M in terms
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Journal of Applied Mathematics
Then, for each point p ∈ M and each plane section π ⊂ Tp M, we
of the Tanaka-Wester connection ∇.
have the following:
τ − Kπ
≤
n 12 n − 1 2 1
H n 1n 2c 3
2n
8
3c 13
3c 13
2
απ.
T −
8
4
3.19
Equality in 3.19 holds at p ∈ M if and only if there exist an orthonormal basis {e1 , . . . , en1 } of
Tp M and an orthonormal basis {en2 , . . . , em } of Tp⊥ M such that a π = Span{e1 , e2 } and b the
shape operators Ar Aer , r n 2, . . . , m, take the following forms:
⎞
0
0 ⎟
⎠,
⎛
a 0
⎜ 0 −a
A⎝
0 0 0n−1
⎛
hr11 hr12
⎜
A ⎝hr12 −hr11
0
0
3.20
⎞
0
⎟
0 ⎠,
r n 3, . . . , m.
0n−1
Proof. Let Mn1 be a submanifold of Mc.
We introduce
ρ 2
τ−
c 3 3c 13
n 12 n − 1 2
−
T 2 .
H − n 1n 2
2n
4
4
3.21
Then, from 3.16 and 3.21, we get
2
2
2c 3
.
n 1H
nh
n ρ −
4
3.22
Let p be a point of M and let π ⊂ Tp M be a plane section at p. We choose an orthonormal
basis {e1 , . . . , en1 } for Tp M and {en2 , . . . , em } for Tp⊥ M such that en1 ξ, π Span{e1 , e2 },
is parallel to en2 . Then, from 3.22, we get
and the pseudomean curvature vector H
⎛
⎞
2
n1
n1 m 2 2 2
2c
3
n2
n2 n2 r ρ −
⎠
n⎝
h
h
h
h
ii
ii
ij
ij
4
rn3 i,j
i/
j
i1
i1
3.23
and so, by applying Lemma 3.1, we obtain
n2 h
n2 ≥
2h
22
11
i/
j
n2
h
ij
2
m r
h
ij
rn3 i,j
2
ρ −
2c 3
.
4
3.24
Journal of Applied Mathematics
7
On the other hand, from 3.12, we have
m 2 2 c 3 3c 13 n2 n2
n2
r r
r
h11 h22 − h
Kπ h11 h22 − h12
g 2 e1 , ϕe2 .
12
4
4
rn3
3.25
Then, from 3.24 and 3.25, we get
m 2 2 1 2
ρ 3c 13 2 n2
r
r
g e1 , ϕe2 Kπ
h
h
h
2j
1j
2
4
2 i / j>2 ij
rn2 j>2
m m 2
2
1 r
r 1
r h
h
h
22
ij
11
2 rn3 i,j>2
2 rn3
3.26
3.27
ρ 3c 13
απ.
≥ 2
4
Combining 3.21 and 3.27, the inequality 3.19 yields. If the equality in 3.19 holds, then
the inequalities given by 3.24 and 3.27 become equalities. In this case, we have
n2 h
n2 h
n2 0,
h
2j
ij
1j
r hij
r 0,
r h
h
2j
1j
i/
j > 2,
r ∈ {n 3, . . . , m}; i, j ∈ {3, . . . , n 1},
3.28
n3 · · · h
m h
m 0.
n3 h
h
22
22
11
11
Moreover, choosing e1 and e2 such that hn2
12 0, from 3.11, we also have the following
n2 h
n2 · · · h
n2
n2 h
h
22
33
11
n1n1 0.
3.29
Thus, with respect to the chosen orthonormal basis {e1 , . . . , em }, the shape operators of M
take the forms.
We now define a well-defined function δM on M by using inf Kp
inf{Kπ
|
π is a plane section ⊂ Tp M} in the following manner:
δM τ − inf K.
3.30
If c −13/3, then we obtain directly from 3.19 the following result.
Corollary 3.3. Let M be an n 1-dimensional n ≥ 2 submanifold isometrically immersed in a
m-dimensional Sasakian space form M−13/3
such that the structure vector field ξ is tangent to
Then, for each point p ∈ M and each plane section
M in terms of the Tanaka-Wester connection ∇.
π ⊂ Tp M, we have the following:
n 12 n − 1 2 1
δM ≤
H − n 1n 2.
2n
6
3.31
The equality in 3.31 holds if and only if M is a anti-invariant submanifold with rankT 2.
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Journal of Applied Mathematics
Proof. In order to estimate δM , we minimize T 2 − 2απ in 3.19. For an orthonormal basis
{e1 , . . . , en1 } of Tp M with π Span{e1 , e2 }, we write
T 2 − 2απ n1
n1 g 2 e1 , ϕej g 2 e2 , ϕej .
g 2 ei , ϕej 2
i,j3
3.32
j3
Thus, we see that the minimum value of T 2 − 2απ is zero, provided that π Span{e1 , e2 }
is orthogonal to ξ, and span{ϕej | j 3, . . . , n} is orthogonal to Tp M. Thus we have 3.31
with equality case holding if and only if M is anti-invariant such that rankT 2.
4. A Pseudo-Ricci Curvature for Sasakian Space Form
We denote the set of unit vectors in Tp M by Tp1 M by
Tp1 M X ∈ Tp M | gX, X 1 .
4.1
Let {e1 , . . . , ek }, 2 ≤ k ≤ n, be an orthonormal basis of a k-place section Πk of Tp M. If k n,
then Πk Tp M, and if k 2, then Π2 is a plane section of Tp M. For a fixed i ∈ {1, . . . , k}, a
Πk ei , is defined by 7
k-pseudo-Ricci curvature of Πk at ei , denoted by Ric
Πk ei Ric
k
ij ,
K
j
/i
4.2
of
ij is the pseudosectional curvature in terms of the Tanaka-Webster connection ∇
where K
the plane section spanned by ei and ej . We note that an n-pseudo-Ricci curvature RicTp M ei i . Thus, for any orthonormal basis
is the usual pseudo-Ricci curvature of ei , denoted by Rice
{e1 , . . . , en1 } for Tp M and for a fixed i ∈ {1, . . . , n 1}, we have the following:
Tp M ei Rice
i Ric
n1
ij .
K
j/
i
4.3
The pseudoscalar curvature τΠk of the k-plane section Πk is given by
τΠk ij .
K
1≤i<j≤n1
4.4
The relative null spae of M at p is defined by 8
Np X ∈ Tp M | hX,
Y 0, ∀Y ∈ Tp M .
4.5
Journal of Applied Mathematics
9
Theorem 4.1. Let Mc
be a m-dimensional Sasakian space form and M an n 1-dimensional
Then,
submanifold tangent to ξ with respect to the Tanaka-Webster connection ∇.
i for each unit vector X ∈ Tp M orthogonal to ξ, we have
2
4 RicX
≤ n 12 H
n − 1c 3 3c 13T X2 ,
4.6
ii if Hp
0, then a unit tanget vector X ∈ Tp M orthogonal to ξ satisfies the equality case
of 4.6 if and only of X ∈ Np .
iii the equality case of 4.6 holds identically for all unit tangent vectors orthogonal to ξ at p
if and only if p is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.
Proof. i Let X ∈ Tp M be a unit tangent vector at p, orthogonal to ξ. We choose an
orthonormal basis {e1 , . . . , en1 } for Tp M and {en2 , . . . , em } for Tp⊥ M such that e1 X and
en1 ξ. Then, from 3.16, we have
2
2
c 3 3c 13
−
τ h
n 12 H
T 2 .
2
− nn − 1
4
4
4.7
From 4.7, we get
⎡
⎤
m
2
2 2
2
r
2
r ⎦
r · · · h
⎣ h
τ
h
2
h
n 12 H
2
11
22
ij
n1n1
rn2
m
i<j
c 3 3c 13
r h
r
−
h
T 2
ii jj − nn − 1
4
4
rn2 2≤i<j≤n
m !
2 2 "
1 r
r
r
r
r
r
2
τ
h11 h22 · · · hn1n1 h11 − h22 − · · · − hn1n1
2 rn2
m m
2
c 3 3c 13
r h
r
r − 2
2
−
h
h
T 2 .
ij
ii jj − nn − 1
4
4
n2 i<j
rn2 2≤i<j≤n
−2
4.8
From 3.12, we have
ij K
m !
rn2
2 " c 3 3c 13 hr h
r − h
r
g 2 ei , T ej ,
ii jj
ij
4
4
4.9
and consequently
ij K
2≤i<j≤n1
m !
2 " n − 1n − 2c 3 3c 13 r h
r − h
r
h
T 2 − 2T e1 2 .
ii jj
ij
8
8
rn2
4.10
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Journal of Applied Mathematics
Substituting 4.10 into 4.8, one gets
m 2
2
n 12 2
r
ij
τ
−2
K
h
n 12 H
≥ 2
H 2
1j
2
rn2 j2
2≤i<j≤n1
n − 1c 3 3c 13
−
−
T e1 2 .
2
2
4.11
Therefore,
n − 1c 3 3c 13
n 12 2
−
−
T X2 ,
H ≥ 2 RicX
2
2
2
4.12
which is equivalent to 4.6
ii Assume that Hp
0. Equality holds in 4.6 if and only if
r
r · · · h
h
12
1n1 0,
r · · · h
r
r h
h
22
11
n1n1 ,
r ∈ {n 2, . . . , m}.
4.13
r 0 for each j ∈ {1, . . . , n 1}, r ∈ {n 2, . . . , m}, that is, X ∈ Np .
Then, h
1j
iii The equality case of 4.6 holds for all unit tangent vectors at p if and only if
r 0,
h
ij
r
h
11
··· r
h
n1n1
−
r
2h
ii
i/
j, r ∈ {n 2, . . . , m},
0,
i ∈ {1, . . . , n 1}, r ∈ {n 2, . . . , m}.
4.14
i , en1 ξ 0 from 3.10, p is a totally pseudogeodesic point, and, hence, ϕTp M ⊂
Since he
Tp M. The converse is trivial.
Corollary 4.2. Let M be an n 1-dimensional invariant submanifold of a Sasakian space form Mc.
Then,
i for each unit vector X ∈ Tp M orthogonal to ξ, we have
4 RicX
≤ n − 1c 3 3c 13.
4.15
ii A unit tanget vector X ∈ Tp M orthogonal to ξ satisfies the equality case of 4.6 if and
only if X ∈ Np .
iii The equality case of 4.6 holds identically for all unit tangent vectors orthogonal to ξ at p
if and only if p is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.
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