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Abstract
In the present article, we derive an inequality in terms of slant immersions and well define warping function for the squared norm of second fundamental form for warped product semi-slant submanifold in a locally product Riemannian manifold. Moreover, the equality cases are verified and generalized the inequality for semi-invariant warped products in locally Riemannain product manifold.
PUBLIC INTEREST STATEMENT
The extrinsic geometry of such warped product submanifolds actually is the mathematization that explicates our awareness of different concrete shapes in given ambient spaces and the intrinsic geometry of such warped product submanifolds is proper and Riemannian geometry. The study of warped products from this extrinsic point of view was initiated around the beginning of this century. Since then the study of warped product submanifolds from the extrinsic point of view has become a very active research subject in differential geometry and many nice results on this subject have been obtained by many geometers. In similar, we obtained the relation between the second fundamental form, the main extrinsic invariant, the main intrinsic invariants are the warping function of a warped product semi-slant submanifolds and slant angle.
1. Introduction
The notion of warped product manifolds plays very important roles not only in differential geometry but also in general relativity theory in physics. For example, Robertson-Walker space-times, asymptotically flat spacetime, Schwarzschild spacetime, and Reissner-Nordstrom spacetime are warped product manifolds (Hiepko, Citation1979). The geometry of warped products has a crucial role in differential geometry, as well as physical sciences. Bishop and O’Neill (Citation1969) discovered the concept of warped product manifolds to derive an example of Riemannian manifolds of negative curvature, such manifolds are natural generalizations of Riemannian products manifolds. Therefore, many geometers are studied in Ali and Luarian (Citation2017), Ali, Othman, and Ozel (Citation2015), Ali and Ozel (Citation2017), Ali, Uddin, and Othman (Citation2017), Al-Solamy and Khan (Citation2012), Al-Solamy, Khan, and Uddin (Citation2017), Atceken (Citation2008, Citation2013), Chen (Citation2001), Sahin (Citation2006a, Citation2006b, Citation2006c). It is interesting to see that there exist no warped product semi-slant submanifolds of the forms and
in a Kaehler manifold
such that
and
are holomorphic and slant submanifolds, respectively (see Sahin, Citation2006b). While, Atceken (see examples 3.1 (Atceken, Citation2008)) has given an example on the existence of warped product semi-slant submanifold of the form
in a locally product Riemannian manifold such that
and
are invariant and slant submanifolds, respectively. Hence, the geometry of warped product submanifolds in a locally product Riemannian manifold is different from the geometry of warped product submanifolds in Kaehler manifold. Therefore, we consider such a warped product semi-slant submanifold as mixed totally geodesic of locally product Riemannian manifold and obtain a geometric inequality for the length of the second fundamental form in terms of slant immersion and warping functions.
2. Preliminaries
Assume that be a manifold of dimension
with a tensor field of such that
where is a one-one tensor field and
represent the identity transformation. Thus,
is an almost product manifold with almost product structure
. If an almost product manifold
admits a Riemannian metric
satisfying
2010 Mathematics Subject Classification. 53C40 Primary 53C20 53C42 secondary.
Key words and phrases. Mean curvature, warped products, Riemannian manifolds, semi-slant immersions. For any , where
denotes the set of all vector fields of
then
is said to be an almost product Riemannian metric manifold. Denote
the Levi-Civita connection on
with respect to
. If
, for all
, then
is a locally product Riemannian manifold with Riemannian metric
(see Sahin, Citation2006a).
Let be a submanifold of locally product Riemannian manifold
with an induced metric
. If
and
are induced Riemannian connections on normal bundle
and tangent bundle
and of
, respectively, then Gauss and Weingarten formulas are given by
for each and
, where
and
are the second fundamental form and shape operator for an immersion
into
. They are correlated as
For any , we can write
where and
are tangential and normal components of
, respectively. The covariant derivatives of the endomorphism
as
A submanifold of a locally product Riemannian manifold
is said to be totally umbilical (and totally geodesic respectively) if
for all . Then
is a mean curvature vector of
given by
, where
is the dimension of
and
is a local orthonormal frame of the tangent vector space
. Furthermore, if
, then
is minimal in
.
Definition 2.1. A submanifold of a locally product Riemannian manifold
, then for each non zero vector
tangent to
at a point
, the angle
between
and
is called a Wirtinger angle of
. Hence,
is said to be a slant submanifold if the Wirtinger angle is constant and it is independent from the choice of
and
. The holomorphic and totally real submanifolds are slant submanifolds with slant angle
and
, respectively. A slant submanifold is said to be proper if it is neither holomorphic nor totally real. More generally, a distribution
on
is called a slant distribution if the angle
between
and
has same value of
for each
and a non zero vector
.
Thus for a slant submanifold , a normal bundle
can be expressed as
where is an invariant normal bundle with respect to
orthogonal to
. We recall following result for a slant submanifold of a locally product Riemannian manifold given by H. Li (cf. Li & Li, Citation2005).
Theorem 2.1. If is a submanifold of a locally product Riemannian manifold
, then
is a slant submanifold if and only if there exists a constant
such that
. In this case,
is a slant angle of
, and then it satisfies
.
Therefore, the following identities which are consequences from the Theorem 2.1
for any . Now let
be an orthonormal basis of the tangent space
and
belonging to the orthonormal basis
of the normal bundle
. Then we define
As a consequence for a differentiable function , we have
where gradient is defined by
, for any
.
3. Semi-slant submanifolds
Semi-slant submanifolds were described by Papaghiuc (Citation1994). These submanifolds are generalizations of CR-submanifolds with slant angle .
Definition 3.1. A submanifold of an almost complex manifold
is called a semi-slant submanifold if there exist two orthogonal distributions
and
such that
is holomorphic, i.e.,
is slant distribution with slant angle
The dimensions of holomorphic distribution and slant distribution
of semi-slant submanifold of a locally product Riemannian manifold
are denoted by
and
respectively. Then
is holomorphic if
and slant if
. It is called proper semi-slant if the slant angle different from
and
. Moreover, if
is an invariant subspace under the endomorphism
of normal bundle
, then, in case of semi-slant submanifold, the normal bundle
can be decomposed as
. A semi-slant submanifold is said to be a mixed totally geodesic, if
, for any
and
.
4. Warped product submanifolds with the form ![](//:0)
![](//:0)
Let and
be two Riemannian manifolds with a
, a positive differentiable function on
, we define on the product manifold
with metric
, where
and
are natural projections on
and
. Under these condition the product manifold is called warped product of
and
, it is denoted by
and
is called warping function. So we have the following lemma
Lemma 4.1 Let be a warped product manifold. Then for any
and
, we have
where and
are the Levi-Civita connections on
and
respectively. Thus
is the gradient of
is defined as
. If the warping function
is constant, then the warped product manifold
is called trivial, otherwise non-trivial. Furthermore, in a warped product manifold
,
is totally geodesic and
is totally umbilical submanifold in
, respectively (cf. Bishop & O’Neill, Citation1969). There are two types of warped product semi-slant submanifolds
and
. For the second case, we have following non-existence theorem from Atceken (Citation2008).
Theorem 4.1. Assume that is a locally Riemannian product manifold and
is a submanifold of
. Then there exists no a warped product semi-slant submanifold
in
such that
is an invariant submanifold and
is a proper slant submanifold of
.
Now, we develop some important lemmas for first type warped product for later use in the inequality and we refer for example to see their existence, Example 4.1 in Atceken (Citation2008).
Lemma 4.2. Let be a warped product semi-slant submanifold of a locally product Riemannian manifold
. Then
for any and
.
PROOF. If and
, we have
From (2.2) and (2.5) (i), we get
From the fact that and
are orthogonal, we obtain
Then from (2.3) (i), we derive
Using Lemma 4.1 (ii), we arrive at
As and
are orthogonal to each other by the definition of
tensor field
, the second term of last equation should be zero. Then we get
Replacing by
in the above equation and from the property of linearity, we get the first result of lemma. Now interchanging
by
, we obtain
It completes the proof of the lemma. □
Lemma 4.3. Let be a warped product semi-slant submanifold of a locally product Riemannian manifold
. Then
for any and
PROOF. Suppose that and (2.5) (i), we have
for . Then from Theorem 2.1, implies that
Since and
are orthogonal then, we obtain
From Lemma 4.1 (ii), we arrive at
Finally, we obtain
If interchanging by
and using Riemannian metric property in the above equation we get the second assertion of the first part of the lemma. Now replacing
by
in (4.3), then we get
Thus using Theorem 2.1, in left hand side of the above equation fora slant submanifold, we reach the second part of lemma. Again replacing by
then we get final result of lemma. It completes the proof of the lemma.□
5. An inequality for semi-slant warped product submanifolds
In this section, we obtain a geometric inequality for a warped product semi-slant submanifold in terms of the second fundamental form and the warping function with mixed totally geodesic submanifold. Now, we describe an orthonormal frame for a semi-slant submanifold, which we shall use in the proof of inequality theorem.
Let be an
-dimensional warped product semi-slant submanifold of
-dimensional locally product Riemannian manifold
such that the dimension of
is
and the dimension of
is
, where
and
are the integral manifolds of
and
, respectively. We consider
and
which are orthonormal frames of
and
respectively. Thus the orthonormal frames of the normal sub bundles,
and invariant sub bundle
, respectively are
.
Theorem 5.1. Let be a
-dimensional mixed totally geodesic warped product semi-slant submanifold of
-dimensional locally product Riemannian manifold
such that
is holomorphic submanifold of dimension
and
is a proper slant submanifold of dimension
of
. Then
(i) The squared norm of the second fundamental form of is given by
(ii) The equality holds in (5.1), if and
is totally geodesic in
. Moreover,
can not be minimal.
PROOF. By the definition of second fundamental form, we have
Since, is mixed totally geodesic, then we get
Leaving second term and using (2.11) in first term, we obtain
The above expression can be written as in the components of and
, then we derive
We will remove the last term and using the adapted frame for , we derive
Again using the adapted frame for and the fact that second fundamental form is symmetric, then we get
Then using Lemma 4.2 and Lemma 4.3, we arrive at
Thus combining first and second terms and using the property of trigonometric identities in the third and fourth terms, we get
Last the above equation can be modified as
From definition of adapted frame for , finally, we obtain
If the equality holds, from the leaving terms in (5.2) and (5.3), we obtain the following conditions, i.e., is totally geodesic in
and
. So the equality case holds. It is completed proof of the theorem. □
6. Conclusion remark
If we assume that the slant angle then warped product semi-slant submanifold
becomes a warped product semi-invariant submanifold of type
of a locally product Riemannian manifold, in this case, Theorem 5.1 is generalized to the inequality theorem which was obtained by Sahin (Citation2006a). Therefore, we say that Theorem 5.1 in Sahin (Citation2006c) is trivial case of our derived Theorem 5.1, that is
Theorem 6.1. Let be a
-dimensional mixed totally geodesic warped product semi-invariant submanifold of
-dimensional locally product Riemannian manifold
such that
is holomorphic submanifold of dimension
and
is a anti-invariant submanifold of dimension
of
. Then
(i) The squared norm of the second fundamental form of is given by
(ii) The equality holds in (5.1), if and
is totally geodesic in
. Moreover,
can not be minimal.
Additional information
Notes on contributors
![](/cms/asset/69e48313-a757-4915-9835-7c2a94a5d5a3/oama_a_1602017_ilg0001.jpg)
Rifaqat Ali
Rifaqat Ali is an Assistant Professor, Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia. He completed his PhD from Aligarh Muslim University, Aligarh India in 2012. He has qualified graduate aptitude test in engineering (GATE) All India rank-270. His research interests are Complex Analysis, Approximation Theory and Differential Geometry. He is also working as principal and co-principal investigator in several ongoing projects of King Khalid University.
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