1. Introduction
One of the main topics of Information Geometry, which is regarded as a combination of differential geometry and statistics, deals with families of probability distributions, more exactly with their invariant properties. Information Geometry has many applications in image processing, physics, computer science, machine learning, etc.
In [
1], Amari defined a statistical manifolds and presented some applications in Information Geometry. Such a manifold deals with dual connections (or conjugate connections), and, consequently, is closely related to an affine manifold.
Let
be an affine connection on a Riemannian manifold
. A pair
is a
statistical structure on
if
for any
[
2]. A Riemannian manifold
on which a pair of torsion-free affine connections
and
satisfying
is defined for any
and
Z ∈
is called a
statistical manifold; one says that the connections
and
are
dual connections (see [
1,
3]).
Any torsion-free affine connection
always has a dual connection given by
where
denotes the Levi–Civita connection of
[
1].
One challenge in submanifold theory is to obtain relations between the intrinsic and extrinsic invariants of a submanifold. An important new step in this topic is due to B.-Y. Chen, starting from 1993 [
4]; he established such inequalities in a real space form, known as Chen inequalities. Since then, many geometers have studied this problem for different kind of submanifolds in certain ambient spaces (for example, see [
5,
6,
7,
8,
9,
10]). For the collections of the results related to Chen inequalities see also [
11] and references therein.
The squared mean curvature is the main extrinsic invariant; the classical curvature invariants, namely the scalar curvature and the Ricci curvature, represent the main intrinsic invariants. A relation between the Ricci curvature and the main extrinsic invariant squared mean curvature for a submanifold in a real space form was given in [
7] by B.-Y. Chen and now known as the Chen–Ricci inequality. In [
12,
13], K. Matsumoto and I. Mihai found relations between Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. In [
14], A. Mihai and I. N. Rădulescu proved a Chen inequality involving the scalar curvature and a Chen–Ricci inequality for special contact slant submanifolds of Sasakian space forms.
Furthermore, in [
15], M. E. Aydın, A. Mihai and I. Mihai established relations between the extrinsic and intrinsic invariants for submanifolds in statistical manifolds of constant curvature. In [
16], A. Mihai and I. Mihai considered statistical submanifolds of Hessian manifolds of constant Hessian curvature. As generalizations of the results given in [
15], H. Aytimur and C. Özgür studied same problems for submanifolds in statistical manifolds of quasi constant curvature [
17].
Recently, in [
18], B.-Y. Chen, A. Mihai and I. Mihai gave the Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature.
In [
19], H. Aytimur, M. Kon, A. Mihai, C. Özgür and K. Takano established a Chen first inequality and a Chen inequality for the invariant
for statistical submanifolds of Kähler-like statistical manifolds, under a curvature condition. Very recently, in [
20], A. Mihai and I. Mihai proved a Chen inequality for the
-invariant; also, the
-invariant was studied in other ambient spaces by G. Macsim, A. Mihai and I. Mihai (see [
21]), for example for Lagrangian submanifolds in quaternionic space forms.
Motivated by the above mentioned studies, as a continuation of the results obtained in [
19], in the present paper we prove Chen first inequality and a Chen inequality for the invariant
for statistical submanifolds of Sasaki-like statistical manifolds, under a natural curvature condition.
2. Sasaki-Like Statistical Manifolds and Their Submanifolds
Let
be an odd dimensional manifold and
be a tensor field of type
, a vector field and a 1-form on
, respectively. If
and
satisfy the following conditions
for
, then
is said to have an almost contact structure
and it is called an
almost contact manifold.
In [
22], K. Takano started with a semi-Riemannian manifold
with the almost contact structure
, on which another tensor field
of type
satisfying
for vector fields
X and
Y on
is considered.
is called an
almost contact metric manifold of certain kind [
22,
23].
One has
and the following important relation holds:
From (
4), it follows that the tensor field
is not symmetric with respect to
. This means that
does not vanish everywhere. On the almost contact manifold, we have
and
; then, on the almost contact metric manifold of certain kind, one has
and
.
In [
22], Takano defined a statistical manifold on the almost contact metric manifold of certain kind.
is called a
Sasaki-like statistical manifold if
Suppose that the curvature tensor
with respect to
satisfies
where
c is a constant (see [
22]).
By interchanging
and
in (
8), one obtains the similar condition for curvature tensor
.
If is a Sasaki manifold, then the previous relation represents the curvature condition of being a Sasakian space form (i.e., the -sectional curvature is constant, c).
On a statistical manifold, the curvature tensor fields of
and
, respectively, denoted by
and
satisfy the relation
(see [
2]).
Let
be an immersion, where
is a statistical manifold. One considers a pair
on
M, defined by
for any
, where the connection induced from
by
f on the induced bundle
is denoted by the same symbol
. Then
is a statistical structure on
, called the one
induced by f from
[
2].
Let
and
be two statistical manifolds. Then
is a
statistical immersion if
coincides with the induced statistical structure, i.e., if (
1) holds [
2]. Recall that, for
M an
n-dimensional submanifold of
, the Gauss formulas are
where
h and
are symmetric and bilinear, called
the imbedding curvature tensors of
M in
for
and
, respectively. The connections ∇ and
are called the
induced connections of
and
, respectively. Since
h and
are symmetric and bilinear, we have the linear transformations
and
defined by
and
for any unit vector in the normal bundle
and
[
3]. It is known that when we use the Levi–Civita connection,
h and
are called the
second fundamental form and the
shape operator with respect to the unit
, respectively, [
24].
Let
and
be affine and dual connections on
. We denote the induced connections ∇ and
of
and
, respectively, on
M. Let
,
and
be the Riemannian curvature tensors of
and
, respectively. Then the Gauss equations are given by
and
where
[
3].
In the following, we recall an example of a Sasaki-like statistical manifold, for which the curvature tensor of satisfies the Equation (8) with .
Example 1 ([22]). Let be a -dimensional affine space with the standard coordinates . One defines a semi-Riemannian metric on by One considers the affine connection , given bywhere and . Its conjugate can be find by straightforward calculations.
One also defines and η byand . Then represents a Sasaki-like statistical manifold with the curvature tensor of satisfying the Equation (8) with . From here, it can be easily found that Moreover, this manifold is not Sasaki with respect to the Levi–Civita connection.
For
one decomposes
where
and
are the tangential and normal components of
, respectively.
Similarly, we can write
where
and
are the tangential and normal components of
, respectively.
Recall the following definitions from [
25]:
Let be a Sasaki-like statistical manifold and M a submanifold of . For , if , then M is called an anti-invariant submanifold of . On the other hand, for a submanifold M, if , then M is called an invariant submanifold of .
Remark 1. For some examples of invariant and anti-invariant submanifolds of Sasaki-like statistical manifolds and endowed with the structure from the previous example see [25]. We will use the following standard notations (see also [
19]):
and
Let
and
be orthonormal tangent and normal frames, respectively, on
. The mean curvature vector fields are given by
and
In [
26], B. Opozda introduced the
K-sectional curvature of the statistical manifold in the following way: let
be a plane in
; for an orthonormal basis
of
, the
K-sectional curvature was defined by
where
is the curvature tensor field of Levi–Civita connection
on
.
In next sections, we will use the same notation g for the metric on the ambient space, for the simplicity of writing.
3. Chen First Inequality
In the present section, we recall the following algebraic lemma which will be used in the proof of the main theorem.
Lemma 1 ([18,19]). Let be an integer and n real numbers. Then we have The equality case of the above inequality holds if and only if .
Let be a -dimensional Sasaki-like statistical manifold satisfying (8), M an n-dimensional statistical submanifold of , and a plane section at . We consider an orthonormal basis of and , orthonormal basis of and , respectively.
We denote by
the sectional curvature of the Levi–Civita connection
on
M and by
the second fundamental form of
. From (
13), the sectional curvature
of the plane section
is
From (
8), (
9) and (
12),
and
The last equality can be written again as
By using the Gauss equation with respect to Levi–Civita connection, we find
where
the sectional curvature of the Levi–Civita connection
on
.
On the other hand, let
be the scalar curvature of
. Then, using (
13) and (
9), we get
where
is the scalar curvature of the Levi–Civita connection
on
. By using (
12) and (
8), we obtain
By similar calculations, we get
If we consider the last equality in (
15), we obtain
After straightforward calculations, we find
Using the last equality and (
5) in (
16), we get
The above equality can be written as
By using the Gauss equation with respect to the Levi–Civita connection, we have
By subtracting (
14) from (
17), we get
Using the above inequality, we get
Next, we can state the following main theorem.
Theorem 1. Let be a -dimensional Sasaki-like statistical manifold satisfying (8) and M an n-dimensional statistical submanifold of .
Assume that ξ is tangent to M.
If M is invariant, then If M is anti-invariant, then If ξ is normal to M and M is anti-invariant, then Moreover, one of the equality holds in the all cases if and only if for any we have 5. Conclusions
In Information Geometry, which is regarded as a combination of Differential Geometry and Statistics, one of the main topics and a modern one, at the same time, deals with families of probability distributions, more exactly with their invariant properties.
A challenge in submanifold theory is to obtain relations between extrinsic and intrinsic invariants of a submanifold. An important new step in this topic is due to B. Y. Chen, starting from 1993; new intrinsic invariants were introduced and such inequalities, known as Chen inequalities, were first established in a real space form. The introduction of Chen invariants was considered in the literature as one of the main contributions in classical Riemannian Geometry in the last decade of the 20-th century.
In this article, relations between extrinsic and intrinsic invariants of a submanifold, more precisely the Chen first inequality and a Chen inequality for the -invariant on statistical submanifolds of Sasaki-like statistical manifolds, under a curvature condition, are obtained.