In this paper, we obtain the upper bounds for the normalized -Casorati curvatures and generalized normalized -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.
Casorati curvatures; statistical manifolds; Sasaki-like statistical space forms; η-Einstein
Information geometry provides a deeper understanding and a geometric approach to families of statistical models. In general, it is related to the study of the differential geometry of statistical manifolds. Information geometry has had a large scope of applications (e.g., physics, chemistry, biology and finance). It has also enabled a joint approach to many problems in the field of differential geometry. The purpose of information geometry is to use tools from Riemannian geometry to extract information from the underlying statistical models. The idea has been successfully used in different areas, including statistical inferences and manifold learning. Amari  showed that there are statistical relationships between families of probability densities in terms of the geometric properties of Riemannian manifolds. It is the study of the intrinsic properties of manifolds of probability distributions. In 1989, the notion of statistical submanifolds was introduced and studied by Vos . However, to-date it has made very little progress due to the difficulty in finding classical differential geometric approaches for the study of statistical submanifolds. Furuhata  studied hypersurfaces in statistical manifolds and provided some examples as well. In 2006, Takano  introduced and studied the statistical structure on Sasakian manifolds, called Sasaki-like statistical manifolds. He also studied Sasaki-like statistical submersions. In 2017, Furuhata et al.  gave another notion for the statistical structures on Sasakian manifolds, called Sasakian statistical manifolds, and obtained several results. Recently, some results have been published for statistical manifolds and submanifolds by different geometers [6,7,8,9].
In order to provide an answer to an open question raised by S. S. Chern concerning the existence of minimal immersions into Euclidean spaces of arbitrary dimension, in the early 1990s Prof. B. Y. Chen introduced new types of Riemannian invariants known as Chen invariants or -invariants and established general inequalities involving the new intrinsic invariants and the main extrinsic invariant for arbitrary Riemannian manifold. Such invariants and inequalities have many nice applications in several areas of Mathematics. In 1999, Casorati  introduced a new extrinsic invariant known as the Casorati curvature. Afterwards, various geometers discussed the geometrical importance of the Casorati curvature [11,12,13]. Due to its geometric importance, a number of results have been obtained in terms of the Casorati curvatures [14,15,16,17,18,19,20,21].
Recently, Lee et al.  derived extremities for normalized -Casorati curvature for statistical submanifolds in statistical manifold with constant curvature. The purpose of this article is to show that normalized scalar curvature is bounded above by Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds of constant -sectional curvature. Further, we find the condition under which a Sasaki-like statistical manifold becomes -Einstein, and vice-versa. We also derive the condition which shows that the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.
2. Sasaki-Like Statistical Manifolds
Let be a Riemannian manifold and and be torsion-free affine connections on . Then, the Riemannian manifold is said to be statistical if
Here, we remark the following:
The connections and are called conjugate connections.
If is a statistical structure on , then is also a statistical structure on .
For the dual connections and , we have
where is the Levi-Civita connection for .
The curvature tensor fields and of and , respectively, satisfy
Let be a -dimensional manifold and let be a n-dimensional submanifold of . Then, the Gauss formulae are :
where and are symmetric, bilinear, imbedding curvature tensors of in for and , respectively. Let us denote the normal bundle of by . The linear transformations and are defined by
for any and . The corresponding Weingarten formulas are :
where , , and and are Riemannian dual connections with respect to the induced metric on .
The corresponding Gauss equations are given by :
where and are Riemannian curvature tensors with respect to ∇ and , respectively.
Let a tensor be of type , a vector field , a 1-form on an odd dimensional manifold satisfying the conditions
for any , then we say has an almost contact structure .
Let be a contact metric manifold. Then, is said to be an η-Einstein if its Ricci tensor S has the following form:
for any smooth function μ and ν on . Moreover, if the function in Equation (13), then the manifold becomes Einstein, and if both the functions μ and ν vanish in Equation (13), then the manifolds are known as Ricci-flat manifolds.
Let and be the tangent orthonormal frame and normal orthonormal frame, respectively, on . The mean curvature vector fields are given by
We also set
Let denote the sectional curvature of a Riemannian manifold of the plane section at a point . Then,
where is the scalar curvature. The normalized scalar curvature is defined as
We also put
. The squared norms of the second fundamental form and are denoted by and , respectively, and are given as
called Casorati curvatures of the submanifold [19,23].
Let be an r-dimensional subspace of , , and is an orthonormal basis of . Then
is called the scalar curvature of the r-plane section. The Casorati curvatures and of that r-plane section are [19,23]:
The normalized -Casorati curvatures and are defined as [20,23]:
Similarly, the dual normalized -Casorati curvatures and are defined as [14,19]:
For a positive real number , put
then the generalized normalized -Casorati curvatures and are given as [20,23]:
if , and
Further, the dual generalized normalized -Casorati curvatures and are given as [14,19]:
be a hyperplane of , and a quadratic form given by
Then f has a global solution,
3. Normalized -Casorati Curvature
In this section, we mainly show that the normalized scalar curvature is bounded above by the normalized -Casorati curvatures for statistical submanifolds of Sasaki-like statistical manifold with constant -sectional curvature.
Let be a statistical submanifold in a Sasaki-like statistical manifold such that is tangent to the structure vector field ξ of . Then, the normalized δ-Casorati curvatures and satisfy
for real t, , where , and . The equality case holds in Equation (30) if and only if the component of ζ satisfies
Applying summation and using Equations (14)–(19) in Equation (33), we obtain
Indeed, from Equation (2), . Then, from Equations (22) and (34) we see that
Now we write a quadratic polynomial as
where is the hyperplane of . Without loss of generality, let us assume that is spanned by , then Equation (36) yields
Now, we consider the quadratic forms such that
We start with the problem
where is a real constant. By comparing Equation (37) and Lemma 1, we get that
Hence, a critical point of the problem has the following form:
Thus, we get
From this, it follows that
which is the required inequality, and the equality in Equation (30) holds if and only if we have the equality in the all the previous inequalities. Thus, the equality holds in Equation (30) if and only if the relations in Equation (31) are true. ☐
A similar result can also be obtained for normalized δ-Casorati curvatures and .
4. Generalized Normalized -Casorati Curvature
In this section, we mainly show that the normalized scalar curvature is bounded above by the generalized normalized -Casorati curvatures for statistical submanifolds of Sasaki-like statistical manifold with constant -sectional curvature. We mainly prove the following result.
Let be a statistical submanifold in a Sasaki-like statistical manifold such that is tangent to the structure vector field ξ of . Then, the generalized normalized δ-Casorati curvatures and satisfy
for real t, , where , and . The equality case holds in Equation (41) if and only if the component of ζ satisfies
Keeping in mind the scalar curvature in Equation (35), we may assume a quadratic polynomial as
where is the hyperplane of . Without loss of generality, let us assume that is spanned by , then from Equation (43) it follows that
Now, we consider the quadratic forms such that
We start with the problem
where is a real constant.
By comparing Equation (44) and Lemma 1, it is easy to see that
Hence, a critical point of the problem has the following form:
Thus, we get
From this it follows that
which is the required inequality. The equality in Equation (41) holds if and only if we have the equality in the all the previous inequalities. Thus, the equality holds in Equation (41) if and only if the relations in Equation (42) are true. ☐
A similar result can also be obtained for generalized normalized δ-Casorati curvatures and .
5. -Einstein Sasaki-Like Statistical Manifolds
In 1962, Okumura  introduced and studied the -Einstein manifold. In 1965, Sasaki  named it -Einstein. Since then a number of papers have been published on this topic due to its application to the physics or in particular to the theory of relativity.
In this section, we obtain the following results.
A Sasaki-like statistical manifold is η-Einstein if and only if . Moreover, μ and ν are constants and are equal to
Taking the inner product of Equation (12) with , we find
Substituting in Equation (48) and taking summation , we get
With , the above equation takes the following form:
Hence, is -Einstein with and . A straight-forward computation proves the converse part. ☐
The Ricci curvature tensor of η-Einstein Sasaki-like statistical manifold in the direction of ξ is
Setting in Equation (51), we obtain the required result. ☐
If the Ricci curvature tensor of an η-Einstein Sasaki-like statistical manifold in the direction of ξ, then
Solving Equations (57) and (58) for and , we have the required result. ☐
The scalar curvature of an η-Einstein Sasaki-like statistical manifold is constant and equal to
Putting in Equation (51) and taking summation , we have our assertion. ☐
We give an example of Theorem 3, which is the following:
We recall from Example 2.2 of Reference  that is a Sasaki-like statistical manifold with and the structure tensors are defined by
We see that trace. Thus, by Theorem 3, we conclude that is an η-Einstein manifold.
Theorem 3 yields the following corollary.
The η-Einstein Sasaki-like statistical manifold becomes Einstein if . Moreover, in that case .
One can easily obtain the result by just substituting in Equation (51). ☐
We have the following conclusions from this work:
By using a different approach, we obtained a relationship between a new extrinsic invariant called the Casorati curvature and an intrinsic invariant called the normalized scalar curvature of statistical manifolds with any co-dimension of Sasaki-like statistical space forms. The derived relations can motivate other researchers to obtain similar relationships for many kinds of invariants of similar nature, for statistical submanifolds in different ambient spaces, such as Kaehler-like statistical manifolds, Kenmotsu-like statistical manifolds, cosymplectic-like statistical manifolds, and statistical warped product manifolds.
An Einstein Sasaki-like statistical manifold can not be Ricci-flat.
The metric of the Sasaki-like statistical manifolds with constant curvature is a solution of the vacuum Einstein field equation if the manifold is -Einstein with constant curvature . In fact, the Einstein field equations consist of 10 equations in Einstein’s general theory of relativity. This theory tells us the fundamental interaction of gravitation. Actually, the Einstein field equations are used to obtain the spacetime geometry which are the the outcome of the presence of linear momentum and mass-energy. Therefore, it is of great interest to see what type of solution we can obtain for Einstein field equations in the case of the metric of Sasaki-like statistical manifold.
All authors have contributed equally to the study and preparation of the article. All authors have read and approved the final manuscript. Conceptualization, all authors; Methodology, all authors; Validation, all authors; Investigation, all authors; Writing—Original Draft Preparation, M.A.; Writing—Review & Editing, all authors; Visualization, M.A.; Supervision, M.H.S.; Funding Acquisition, A.H.A.
The authors wish to thank the referees for their many valuable and helpful suggestions in order to improve this manuscript. The first author (Ali H. Alkhaldi) would like to express his gratitude to King Khalid University, Saudi Arabia for providing administrative and technical supports.
Conflicts of Interest
The authors declare no conflict of interest.
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