1. Introduction
A Reimannian manifold is a statistical manifold of probability distributions possessing a Riemannian metric and two dual (conjugate) affine connections without torsion [
1]. A statistical framework of a Riemannian metric and its extension are a Riemannian connection. The theory of statistical submanifolds and statistical manifolds is a recent geometry that plays a crucial role in several fields of mathematics. Various results have been derived by distinguished geometers in this area.
K. Kenmotsu [
2] found interesting results and studied the warped product spaces of the type
, where
is a Kaehlerian manifold with a maximal dimension that falls under Tanno’s categorization of connected nearly contact metric manifolds (called the third class). Then, the author examined the characteristics of
and described it using tensor relations. A manifold of this type is referred to as a Kenmotsu manifold. A new notion in the statistical manifold, the Kenmotsu statistical manifold, was initiated by Furuhata et al. in [
3]. Locally, it is the warped product of a holomorphic statistical manifold and a line. By establishing a natural affine connection to a Kenmotsu manifold, they developed a Kenmotsu statistical manifold in the same publication. Recently, Murathan et al. [
4] talked about the term
-
.
On the other hand, the concept of submersion in differential geometry was first reported by O’Neill [
5] and Gray [
6], and Watson [
7] later brought the concept of almost Hermitian submersions by using Riemannian submersions (
) from almost Hermitian manifolds.
Afterwards, there have been several subclasses of almost Hermitian manifolds between which almost Hermitian submersions have been found. Additionally, under the heading of contact
, Ṣahin in [
8] extended
to a wide variety of subclasses of virtually contact metric manifolds. In [
9] the majority of studies on Riemannian, almost Hermitian, or contact
are contained.
Barndroof-Nielsen and Jupp [
10] discussed
from the viewpoint of statistics. Abe and Hasegawa introduced and studied the
between statistical manifolds in [
11]. The
of the space of the multivariate normal distribution, statistical manifolds with virtually contact structures, and statistical manifolds with almost complex structures were among the topics that K. Takano found intriguing to research (see [
12,
13,
14]). Remarkable statistical submersions have recently been studied, including para-Kähler-like statistical submersions [
15], cosymplectic-like statistical submersions [
16], and quaternionic Kähler-like statistical submersions [
17]. Most of the research related to the various submersion can be found in [
18,
19,
20,
21,
22,
23,
24].
Inspired by the affirmative works, we consider with some examples. Then, we study Kenmotsu-like statistical submersions () and give many results for such submersions with new examples. This study contributes to developing the literature.
2. Kenmotsu-like Statistical Manifolds ()
Let
be a semi-Riemannian manifold and nondegenerate metric
, and a torsion-free affine connection by
. Triplet
is a
statistical manifold with symmetric
[
12]. For a statistical manifold
, we describe a second connection
as
for any
,
. Here, affine connection
is referred to as a
conjugate (or
dual) of the connection
with respect to
. Affine connection
is torsion-free with symmetric
and obeys
where in the Levi-Civita connection
on
.
A statistical manifold is
. For example, let
be a semi-Riemannian manifold along its Riemannian connection
is a
trivial statistical manifold. In this case,
(
) stands for the curvature tensor on
with respect to affine connection
(its conjugate
). Now, we produce
for any
[
12].
Let
be a
-dimensional semi-Riemannian manifold that admits the almost contact structure
that contains another tensor field,
, of type
that fulfils
for any
. Then,
is a metric manifold with almost contact structure
of a specific sort [
14]. Then,
In fact,
is a nonsymmetric tensor field, which shows that
everywhere. The almost contact manifold also entertain the following equations:
We also obtained the almost contact metric manifold of a specific sort [
14], such that
Murathan et al. [
4] produced a method of how to construct
relying on the idea of a statistical manifold similar to that of the Kähler-like statistical manifold. They defined
-
and said that an almost contact metric such as statistical manifold
is referred to as a
-Kenmotsu-like statistical manifold if
where
is differentiable function on
. They proved the following theorem [
4]:
Theorem 1. Let be a Kähler-like statistical manifold, and be trivial statistical manifold. . Under Proposition 2.2 (see [4]), is a . Now,
is called a
if the following conditions hold:
Consequently, we have the following lemma:
Lemma 1. is a if and only if is a .
In [
25], certain bounds for statistical curvatures of submanifolds with any codimension of
were obtained. Now, we give the following examples on
-
:
Example 1. Let us assume a Kähler-like statistical manifold , where and the flat affine connection . Also, is a trivial statistical manifold with constant curvature 0. From Theorem 1, warped product manifold is a β-. Example 2. A Euclidean space with local coordinate system that admits the following almost complex structure J:the metric with a flat affine connection is referred as a Kähler-like statistical manifold (see [14]). If is a trivial statistical manifold. In view of [4], the product manifold is called a . Let us define and η byand We also find Example 3. From [26], we know that the half upper space with that was described as in [26], is a Kähler-like statistical manifold. So, If is a trivial statistical manifold, It is recognised by [4] that the product manifold is a . We examine curvature tensor
on a statistical manifold similar to that of
with respect to
, such that
where
. Afterwards, shifting
to
in (
11), we produce the expression for the curvature tensor
in terms of
.
Let be a statistical manifold and be a submanifold of . Then is also a statistical manifold with the induced statistical structure on from and we call as a statistical submanifold in .
In the statistical setting, Gauss and Weingarten equations are respectively specified by [
27]
for any
and
, where
and
are the dual connections on
. Similarly, on
, we denote them with ∇ and
. For
and
, the symmetric and bilinear imbedding curvature tensor of
in
are indicated by
h and
, respectively. The finest relation between
h (
) and
A (
) is [
27]:
We indicate the curvature tensor fields of
and ∇ as
and
, respectively. Then, for any
, the corresponding Gaussian equations are [
27]
and
Thus, the statistical curvature tensor fields of
and
are, respectively, specified by
For
, we put
where
(
) and
(
) indicate the tangential and normal components of
(
), respectively. Likewise, we can write
3. Background of Statistical Submersions
This segment provides the prior knowledge required for .
Let us consider two semi-Riemannian manifolds,
and
, and let a semi-Riemannian submersion
such that
maintains the lengths of horizontal vectors, and all the fibers are semi-Riemannian submanifolds of
(for more details, see [
9,
21]). Abe and Hasegawa [
11] investigated affine submersions with horizontal distribution from a statistical manifold. Furthermore,
was discussed by Takano in [
12,
13].
Let a semi-Riemannian submersion
between the semi-Riemannian manifolds
and
. The semi-Riemannian submanifold
has
dimensions and an induced metric
known as a
fiber and denoted by
for any point
. The vertical and horizontal distributions in the tangent bundle
of
are indicated by
and
, respectively. Thus, we have
If there is a vector field
X on
, we refer to it as projectable. Vector field
on
, such that
, for each
. In this instance,
X and
are referred to as
-related. A vector field
X on
if it is projectable, it is referred to as basic [
5]. We have the following information if
X and
Y are the fundamental vector fields,
-related to
,
:
,
is a fundamental vector field is , and . vector field and ,
For any vertical vector field U, is vertical.
O’Neill’s law describes the geometry of semi-Riemannian submersions. Tensors
and
are defined as follows using [
5]:
with respect to any vector fields
E and
F on
. It is clear that skew-symmetric operators
and
on the tangent bundle of
reverse the vertical and horizontal distributions. We provide a summary of the characteristics of tensor fields
and
. If
are vertical vector fields on
, and
are horizontal vector fields, we possess
Let
be a semi-Riemannian submersion from a statistical manifold
. Let us use symbols
and
to represent the affine connections on
. It is obvious that
for vertical vector fields
E and
F on
. It is simple to observe that
and
are conjugate to each other and torsion-free with respect to
.
Let submersion
between two statistical manifolds be a
statistical submersion if
obeys
for basic vector field
and
. Shifting ∇ for
in the aforementioned expressions, we derive
and
[
12].
and
vanish if and only if
is integrable with respect to ∇ and
, respectively. For
and
, we produce
4. Properties of Statistical Submersions
In this section, we discuss some useful properties of statistical submersion proposed by Takano [
12]. First, we have the following lemmas for this study. Therefore, for a statistical submersion
, we have [
5,
12]
Lemma 2 ([
12]).
If X and Y are horizontal vector fields, then . Lemma 3 ([
12]).
For and . Then we haveFurthermore, if X is basic, then and .
Moreover, let
(resp.
is a horizontal vector field like that
at each point
, where
(resp.
be the curvature tensor with respect to the induced affine connection
(resp.
). Thus we have the following theorem [
12].
Theorem 2 ([
12,
14]).
If is a statistical submersion then for and Now, we describe with
,
and
the orthonormal frame of
,
and
, respectively, such that
,
and
,
. With
and
, we jointly define the connection forms in terms of local coordinates
with respect to the affine connection
and its conjugate
. Adopting (
1), we produce
and
for any
. The horizontal vector fields accordingly determine the fiber’s mean curvature vector field with regard to the affine connection
and its conjugate connection
,
5. Kenmotsu-like Statistical Submersion ()
Assume that
is an almost contact metric manifold. If
is a semi-Riemannian submersion, each fiber is a
-invariant semi-Riemannian submersion of
and vector field
is tangent to
; therefore,
is an
almost contact metric submersion. If
U is basic on
, which is
-related to
on
, then
(resp.
) is basic and
-related to
(resp.
) [
14].
Analogous to the Sasaki-like statistical submersion [
14], we describe
as follows:
Definition 1. A isif is a , if each fiber is a φ-invariant semi-Riemannian submanifold of and tangent to vector field ξ.
Therefore, we produced the following results:
Lemma 4. Let be a then for and , we have Proof. In light of Lemma 3, one produces the above relations. □
Lemma 5. Let be a ; then, we have for and . Proof. Since vertical and horizontal distributions are
-invariant for
, in view of Lemma 3 and (
10), we obtain (
35). Now, (
36)–(
38) follows for
and
with using Lemma 3 and (
10). Similarly, we produce (
39) and (
40) for
. Immediately, this also gives us (
41). □
Adopting Lemmas 4 and 5, the following results entail:
Theorem 3. Let be a . Then, is a Kähler-like statistical manifold and a .
Proof. The above lemmas show that each fiber is
. Now, we prove that
is a Kähler-like statistical manifold. Let
be a basic vector field and
related to
. Now, we have
Since
is a
. From the above expression, we produce
which shows that the base manifold is a Kähler-like statistical manifold. □
Lemma 6. Let a . Thenif . Proof. Consider
is a
. Thus,
Setting
in the above expression, we find
Adopting Lemma 3, we produce
Hence, the vertical parts from (
43) hold
Because
,
. Because
, we obtain the required results. □
By virtue of Lemma 4, we obtain . This entails the following.
Theorem 4. Let be a . Then, for , we haveif rank . Again, in view of Lemma 1 and using , we obtain the following corollary:
Corollary 1. Let be a . Then, for , we haveif . 6. Curvature-Based Characteristics of Kenmotsu-like Statistical Submersion
Statistical manifolds on almost Hermite-like manifolds were proposed by Takano in [
12]. if
J is parallel with respect to the
, then
is called a
Kähler-like statistical manifold [
12]. Moreover, curvature tensor
on a Kähler-like manifold
with respect to
is given by
Let
be a
. Then, the expression for the curvature tensor of
is given by (
11). Adopting Theorem 2, we produce
where
. We also produce
similarly
Now, from Theorem 2, we produce
for
and
.
Using Lemma 5, and Theorems 4 and (
55) together, we produce the following results:
Theorem 5. Let be a . Let the total manifold and base manifold be holds of the curvature tensor of the (11) kind with and (44) with , respectively; then, rank . Corollary 2. Let be a . If rank and the total manifold holds the curvature tensor of the (11) kind with , the base manifold obeys the curvature tensor of the (44) kind with . Once again, using Lemma 1 and Theorem 4, Equation (
49) can be reconstructed as below:
thus, in light of Lemma 1, we obtain
If , then we obtain or . Thus one obtain
Theorem 6. Let be a and the total manifold holds the curvature tensor of kind (11) with c. Let the rank and for . Then, - 1.
each fiber is totally geodesic submanifold of and the base manifold is flat if , such that the curvature holds the kind (11) with 3. - 2.
here and ,
- (i)
if g is positive definite, then ,
- (ii)
and X is spacelike (timelike) or and X is timelike (spacelike) if and only if is spacelike (timelike),
- (iii)
horizontal vector X is null if and only if is null.
Corollary 3. Let be a and the total manifold hold the curvature tensor of kind (11) with c. If rank and N is a constant, the result is identical to that of Theorem 7. In addition, (
50) clearly shows that
which implies that, from Lemma 1, we obtain
If , we obtain or . Thus, we produce
Theorem 7. Let be a and the total manifold hold the curvature tensor of kind (11) with c. Let rank and for . Then - 1.
each fiber is totally geodesic submanifold of and the base manifold is flat if , such that the curvature hold the (11) kind with 3. - 2.
in the case of and ,
- (i)
if g is positive definite, then ,
- (ii)
and X is spacelike (timelike) or and X is timelike (spacelike) if and only if is spacelike (timelike),
- (iii)
horizontal vector X is null if and only if is null.
Corollary 4. Let be a and the total manifold hold the curvature tensor of (11) kind with c. Let the rank and is constant, the result is identical to that of Theorem 7. 7. Kenmotsu-like Statistical Submersion with Conformal Fibers
This section is devoted to the with conformal fibers ().
Let us assume that , like a , admits . For if () satisfies, then is with isometric fibers (). Then, from Lemma 1, we can obtain .
Lemma 7. If be a with ; then, ω has isometric fibers.
Theorem 8. Let be a with . Let the total manifold and each fiber that is a totally geodesic submanifold of hold the curvature tensor of the (11) kind with c. Theorem 9. Let be a with and the total manifold hold the curvature tensor of the (11) kind with c. Let the rank ; then, - 1.
if the total manifold satisfies the (11) kind; - 2.
the base manifold is flat;
- 3.
if , each fiber holds the (11) kind.
Example 4. Let be a obtained in Example 1. Then, the as the projection mapping is defined by From this, and . It is easy to verify that dim and . Hence, is integrable with respect to .
Example 5. Let be the given in Example 2. Next, we describe the as the projective mapping Then, we produce and . It is trivial that . Since , we obtain
Example 6. Let be the given in Example 3. Next we describe the as the projective mapping. 8. Discussion
This subject is from differential geometry, which is a traditional yet very active branch of pure mathematics with notable applications in a number of areas of physics. Until recently, applications in the theory of statistics were fairly limited, but within the last few years, there has been intensive interest in the subject. So, the geometric study of is new and has many research problems.
In this discourse, we defined and exhibited that, for a , the base manifold is a Kähler-like statistical manifold, and the fibers are . Moreover, we characterized the total space and the base space of such submersions. We presented a along conformal fibers having isometric fibers. Using these results, different spaces can be studied for these issues, and many new relationships between intrinsic and extrinsic curvatures can be discussed.