1. Introduction
The theory of Chen invariants [
1] is presently one of the most interesting research topics in differential geometry of submanifolds. He established some sharp inequalities, well-known as Chen’s inequalities, for a submanifold in a real space form using the scalar curvature, the sectional curvature, Ricci curvature and the squared mean curvature. In other words, he gave simple relationships between the main intrinsic invariants and the extrinsic invariants of a submanifold in a real space form. It is well known that theorems which relate intrinsic and extrinsic curvatures of submanifolds always play an important role in differential geometry. So the study of this topic has attracted a lot of attention in the last two decades. Many Chen invariants and inequalities exist for the different classes of submanifolds in various ambient spaces; see [
2,
3,
4,
5,
6,
7,
8] and reference therein.
On the other hand, Hayden [
9] introduced the notion of a semi-symmetric metric connection on a Riemannian manifold. Yano [
10] studied some properties of a Riemannian manifold with a semi-symmetric metric connection. Nakao [
11] studied submanifolds in a Riemannian manifold with a semi-symmetric metric connection. Agashe and Chafle [
12,
13] introduced the notion of a semi-symmetric non-metric connection on a Riemannian manifold and studied submanifolds in a Riemannian manifold with a semi-symmetric non-metric connection.
Mihai and Özgür [
14,
15] proved Chen’s inequalities for submanifolds in a real space with a semi-symmetric metric connection, a complex space with a semi-symmetric metric connection and a Sasakian space form with a semi-symmetric metric connection. They also studied Chen’s inequalities for submanifolds in a real space form endowed with a semi-symmetric non-metric connection [
16]. By using two new algebraic lemmas Zhang et al. [
17] obtained Chen’s inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature endowed with a semi-symmetric non-metric connection.
Instead of the extrinsic squared mean curvature, the Casorati curvature of a submanifold in a Riemannian manifold was considered as an extrinsic invariant defined as the normalized square of the length of the second fundamental form of the submanifold. The notion of Casorati curvature extends the concept of the principle direction of a hypersurface in a Riemannian manifold. Therefore, it is of great interest to obtain optimal inequalities for the Casorati curvatures of submanifolds in different manifolds. Decu et al. [
18] obtained some optimal inequalities involving the scalar curvature and the Casorati curvature of a submanifold in a real space form. Some optimal inequalities involving Casorati curvatures were proved in [
19,
20,
21] for slant submanifolds in quaternionic space forms. Recently, Lee et al. [
22,
23,
24] proved optimal inequalities involving the Casorati curvature of submanifols in real and generalized space forms endowed with a semi-symmetric metric connection. Using a different algebra approach, Zhang et al. [
25] established optimal inequalities involving the Casorati curvature of submanifols in a Riemannian manifold of quasi-constant curvature with a semi-symmetric metric connection. But optimal inequalities involving the Casorati curvature of submanifolds in an ambient space with a semi-symmetric non-metric connection haven’t been established.
In this paper, we will study some optimal inequalities involving the Casorati curvature of submanifols in a generalized space forms endowed with semi-symmetric non-metric connections.
2. Preliminaries
Let
be an
-dimensional Riemannian manifold with a Riemannian metric g and a linear connection
on
. If the torsion tensor
of
, defined by
for any smooth vector fields
and
on
, satisfies
for a 1-form
ϕ, then the linear connection
is called a semi-symmetric connection. The vector field
U is defined by
for any vector field
on
. If
satisfies
,
is called a semi-symmetric metric connection. If
satisfies
, then
is called a semi-symmetric non-metric connection.
Let
denote the Levi-Civita connection with respect to the Riemannian metric
g on
. Agashe and Chafle [
12] introduced a semi-symmetric non-metric connection
which is given by
for any smooth vector fields
and
on
.
We will consider the Riemannian manifold
endowed with a semi-symmetric non-metric connection
and the Levi-Civita connection
. Let
and
be curvature tensors of the Riemannian manifold
with respect to
and
, respectively. Then
can be written as [
12]
for any smooth vector fields
on
, where
-tensor field
s is given by
Denote by λ the trace of s.
Let be an n-dimensional submanifold in the Riemannian manifold . On the submanifold we consider the induced semi-symmetric non-metric connection denoted by ∇ and the induced Levi-Civita connection denoted by . We also denote by R and the curvature tensor on with respect to ∇ and , respectively.
The Gauss formulas with respect to
and
, respectively, can be written as
for any smooth vector fields
on
, where
is the second fundamental form of
in
and
h is a (0,2)-tensor on
. From [
13], we know
In [
13], the Gauss equation for the submanifold
into
with respect to the semi-symmetric non-metric connection is
for any smooth vector fields
on
.
Let
,
, be a 2-plane section. Denote by
the sectional curvature of
with respect to the induced semi-symmetric non-metric connection ∇. For any orthonormal basis
of the tangent space
the scalar curvature
τ at
x with respect to the semi-symmetric non-metric connection is defined by
and the normalized scalar curvature
ρ with respect to the semi-symmetric non-metric connection is defined by
Let
be an orthormal basis of the normal space
. We denote by
H the mean curvature vector of
with respect to the semi-symmetric non-metric connection, that is
Then the squared norm of
h over dimension
n is called the Casorati curvature of
with respect to the semi-symmetric non-metric connection, which is denoted by
. That is,
Suppose that
L is an
l-dimensional subspace of
,
, and
is an orthonormal basis of
L. Then the Casorati curvature of the
l-plane section
L with respect to the semi-symmetric non-metric connection is defined by
We define the normalized
δ-Casorati curvatures
and
with respect to the semi-symmetric non-metric connection as the following:
and
The submanifold
is called invariantly quasi-umbilical if there exist
p mutually orthogonal unit normal vectors
such that the shape operators with respect to all directions
have an eigenvalue of multiplicity
and that for each
the distinguished eigendirection is the same [
26].
Let us recall the following two lemmas in [
25].
Lemma 1. Let be a function in defined byIf , then we havewith the equality holding if and only if Lemma 2. Let be a function in defined byIf , then we havewith the equality holding if and only if 3. Optimal Inequalities for the Casorati Curvatures of Submanifolds in a Generalized Complex Space form Endowed with a Semi-Symmetric Non-Metric Connection
A
-dimensional almost Hermitian manifold
is said to be a generalized complex space form [
27], if there exists two functions
and
on
N such that
for any smooth vector fields
on
N, where
is the curvature tensor with respect to the Levi-Civita connection
. In such a case, we will write
.
We endow the generalized complex space form
with a semi-symmetric non-metric connection
. Let
be an
n-dimensional submanifold of
,
. For any vector field
X tangent to
M, we decompose
as
where
and
are tangential and normal components of
, respectively. We also set
For submanifolds in the generalized complex space form with a semi-symmetric non-metric connection, we establish the following inequalities involving the normalized δ-curvatures and .
Theorem 1. Let , be an n-dimensional submanifold in a 2m-dimensional generalized complex space form endowed with a semi-symmetric non-metric connection . Then
(i) The normalized δ-curvature satisfiesMoreover, the equality holds if and only if is an invariantly quasi-umbilical submanifold. (ii) The normalized δ-curvature satisfiesMoreover, the equality holds if and only if is an invariantly quasi-umbilical submanifold. Proof. Let and be orthonormal bases of and , respectively, .
For
,
,
, from (
2), (
4) and (
5), we get
By summation over
, it follows that
(i) Without loss of generality, we can assume that
satisfies
We define the following function, denoted by
, which is a quadratic polynomial in the components of the second fundamental form:
Setting
we consider the problem as following:
where
.
By Lemma 1, we have
with equality holding if and only if
Therefore, we have
with equality holding if and only if
and
From (
9) and (
10), we get
By the definition of
, we can obtain
And the equality holds if and only if
where we used the relation (
3) of
h and
.
From (
11), we know that
is invariantly quasi-umbilical.
(ii) Without loss of generality, we can also assume that
satisfies
Considering the following quadratic polynomial in the components of the second fundamental form
Setting
we consider the problem as following:
where
.
By Lemma 2, we have
with equality holding if and only if
Therefore, we have
with equality holding if and only if
and
Then by (
12) and (
13) and the definition of
, we can easily derive the inequality (
7). And the equality can be also easily verified. ☐
Remark 1. For , where c is a constant, then from Theorem 1 we can get optimal inequalities for the Casorati curvatures of submanifolds in the complex space form endowed with a semi-symmetric non-metric connection.
4. Optimal Inequalities for the Casorati Curvatures of Submanifolds in a Generalized Sasakian Space form Endowed with a Semi-Symmetric Non-Metric Connection
Let
N be a
-dimensional almost contact metric manifold (see [
28]) with an almost contact metric structure
consisting of a
-tensor field
φ, a vector field
ξ, a 1-form
η and a Riemannian metric
g on
N satisfying
for all vector fields
on
N. Such a manifold is said to be a contact metric manifold if
, where
is called the fundamental 2-form of
N [
28].
Given an almost contact metric manifold
N with an almost contact metric structure
,
N is called generalized Sasakian space form [
29] if there exists three functions
,
and
on
N such that
for any smooth vector fields
on
N, where
is the curvature tensor with respect to the Levi-Civita connection
. In such a case, we will write
. If
,
, where
c is a constant, then
N is a Sasakian space form.
Now we endow the generalized Sasakian space form
with a semi-symmetric non-metric connection
. Let
be an
n-dimensional submanifold of
,
. We set
for any vector field X tangent to
, where
and
are tangential and normal components of
, respectively. We also set
and decompose
where
and
denote the tangential and normal components of
ξ.
For submanifolds in a generalized Sasakian space form with the semi-symmetric non-metric connection, we establish the following inequalities involving the normalized δ-curvatures and .
Theorem 2. Let , be an n-dimensional submanifold in a (2m + 1)-dimensional generalized Sasakian space form endowed with a semi-symmetric non-metric connection . Then
(i) The normalized δ-curvature satisfiesMoreover, the equality holds if and only if is an invariantly quasi-umbilical submanifold. (ii) The normalized δ-curvature satisfiesMoreover, the equality holds if and only if is an invariantly quasi-umbilical submanifold. Proof. Let and be orthonormal bases of and , respectively, .
For
,
,
, from (
2), (
4) and (
14), we get
By summation over
, it follows that
The rest of the proof is the same as Theorem 1. So we will no longer describe here. ☐
Remark 2. For ,, from Theorem 2 we can get the optimal inequalities for the Casorati curvatures of submanifolds in the Sasakian space form endowed with a semi-symmetric non-metric connection.