The -Hessian Quotient for Riemannian Metrics

axioms

Article The χ-Hessian Quotient for Riemannian Metrics

Pi¸scoranLaurian-Ioan 1,* , Akram Ali 2 , Barbu C˘at˘alin 3 and Ali H. Alkhaldi 2

1 North University Center of Baia Mare, Department of Mathematics and Computer Science, Technical University of Cluj Napoca, Victoriei 76, 430122 Baia Mare, Romania 2 Department of Mathematics, King Khalid University, Abha 9004, Saudi Arabia; [email protected] (A.A.); [email protected] (A.H.A.) 3 “Vasile Alecsandri” National College, str. Vasile Alecsandri nr. 37, 600011 Bac˘au,Romania; [email protected] * Correspondence: [email protected]

Abstract: Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One of the main goals of this paper will be to find a link between these two research fields. In this respect, in the present paper, we will introduce and analyze two important quantities in pseudo-Riemannian geometry, namely the H-distorsion and, respectively, the Hessian χ-quotient. This second quantity will be investigated using the Frobenius (Hilbert–Schmidt) norm. Some important examples will be also given, which will prove the validity of the developed theory along the paper.

Keywords: pseudo Riemannian manifold; Frobenius norm

MSC: Primary 53B20; Secondary 15B40, 15A69

Citation: Laurian-Ioan, P.; Ali, A.; 1. Introduction C˘at˘alin,B.: Alkhaldi, A. H. The The Hessian structural geometry is a fascinating emerging area of research. It is, in χ-Hessian Quotient for Riemannian particular, related to Kaehlerian geometry, and also with many important pure mathemati- Metrics. Axioms 2021, 10, 69. https:// cal fields of research, such as: affine differential geometry, cohomology, and homogeneous doi.org/10.3390/axioms10020069 spaces. A strong relationship can also be established with the geometry of information in applied mathematics. This systematic introduction to the subject initially develops the Academic Editor: Anna Maria Fino foundations of Hessian structures on the basis of a certain pair of a flat connection and a

Received: 20 March 2021 Riemannian metric, and then describes these related fields as theoretical applications. Accepted: 13 April 2021 In Finsler geometry, respectively, in Riemannian geometry are few known invariants. Published: 19 April 2021 Accordingly, one of the main path of research is to find new invariants and to study their impact in some concrete examples. n Publisher’s Note: MDPI stays neutral As we know, Bx represents, in Finsler geometry, the unit ball in a Finsler space centered with regard to jurisdictional claims in at p ∈ M, where (M, F) is a Finsler manifold, i.e., published maps and institutional affil- n = ∈ n| = ( ) < iations. Bx {y R |y| F x, y 1}

and Bn represents the unit ball in the Euclidean space centered at origin:

Bn = {x ∈ Rn||x| < 1} Copyright: © 2021 by the authors. q Licensee MDPI, Basel, Switzerland. i j where |x| = δijx x . This article is an open access article distributed under the terms and Additionally, in Finsler geometry, we know that σF(x) is given by: conditions of the Creative Commons Attribution (CC BY) license (https:// volume(Bn) ( ) = creativecommons.org/licenses/by/ σF x n . volume(Bx ) 4.0/).

Axioms 2021, 10, 69. https://doi.org/10.3390/axioms10020069 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 69 2 of 7

Let now recall one classical deﬁnition

Deﬁnition 1 ([1]). For a Finsler space, the distorsion τ = τ(x, y) is given by:  q  det(gij) τ(x, y) = ln . σF(x)

q i j Additionally, according to Shen ([1]), when F = gij(x)y y , is Riemannian, then

q σF(x) = det(gij(x)).

Deﬁnition 2 ([2]). A pseudo-Riemannian metric of metric signature (p, q) on a smooth manifold M of dimension n = p + q, is a smooth symmetric differentiable two-form g on M, such that, at each point, x ∈ M, gx is non-degenerate on Tx M with the signature (p, q). We call (M, g) a pseudo-Riemannian manifold.

Also a well known results from [2], is the following:

Theorem 1 ([2]). Given a pseudo-Riemannian manifold (M, g), there exists a unique linear connection ∇g on M, called the Levi–Civita connection of g, such that:

(a) ∇gg = 0,

(b) ∇g is torsion free, i.e., T = 0.

1 n k For a coordinate chart (U, x , ··· , x ), the Christoffel symbols Γij of the Levi–Civita connection are related to the components of the metric g in the following way:

1 ∂g ∂gjl ∂gij Γk = gkl li + − . ij 2 ∂xj ∂xi ∂xl

If f : M → R is a smooth function, then the second covariant derivative of the function f is given by: ∂2 f ∂ f ∇2 f = − Γk dxi ⊗ dxj g ∂xi∂xj ij ∂xk is called the Hessian of the function f . In the paper [3], the authors used the following notations, which we will also use in this paper:

∂ f f = , ,i ∂xi 2 ∂ f k f ij = − Γ f,m , ∂xi∂xj ij ∂ f,ij f = − Γl f − Γl f . ,ijk ∂xk ki ,lj kj ,li Some important results regarding the “size” of a matrix were recently established in a series of papers. In this respect, please see [4,5]. As we know, the two norm for a matrix A is given by

kAxk kAk = max kAxk = max . 2 kxk=1 x6=0 kxk Axioms 2021, 10, 69 3 of 7

The Frobenius (or the Hilbert–Schmidt norm) of a matrix A = (Aij) is defined, as follows: v u n n u 2 kAkHS = t∑ ∑ Aij. i=1 j=1

The operator norm of a matrix A = (Aij) is given by

kAkop = maxkxk=1kAxk.

Next we will recall some properties of these norms. We will be focused on Hilbert– Schmidt norm k·kHS, because we will use it to establish some new main results of this paper:

kA · BkHS ≤ kAkHS · kBkHS (1)

kA · Bkop ≤ kAkop · kBkop (2) √ kAkHS ≤ nkAkop. (3)

In the previous inequality, the equality take place when A = In. Additionally,

n |det(A)| ≤ kAkHS (4) √ √ r kAkHS ≤ rkAk2 = . (5) σmin(A) Here, A denotes any positive deﬁnite symmetric matrix, r is the rank of A, and σmin(A) denotes the minimum singular value of A. Some interesting results regarding the Minkowski norm on Finsler geometry are presented in [6].

2. Main Results Now, using the distortion deﬁnition, we will introduce the following deﬁnition:

Deﬁnition 3. For a pseudo-Riemannian manifold, we will denote, by v u ( ) σF(x) u det gij σF,∇F = = t (6) σ∇F(x) 2 det ∇g f

2 the H-distortion, if and only if σF,∇F(x) =constant. Example 1. We will consider a pseudo-Riemannian manifold, (x1, x2) | x1 6= −1 ⊂ R2, en- dowed with the following metric:

(1 + x1)2 x1 + 2 g = . (7) ij x1 1

After tedious computations, we obtain the following Christoffel symbols for this metric: 1 2 1 2 1 1 2 2 1 Γ11 = −1, Γ11 = x + 1, Γ22 = Γ12 = Γ21 = Γ21 = Γ12 = Γ22 = 0. Now, we get: Axioms 2021, 10, 69 4 of 7

∂2 f f = − Γ1 f + Γ2 f ,11 ∂x1∂x1 11 ,1 11 ,2 ∂2 f f = − Γ1 f + Γ2 f ,12 ∂x1∂x2 12 ,1 12 ,2 ∂2 f f = − Γ1 f + Γ2 f ,21 ∂x2∂x1 21 ,1 21 ,2 ∂2 f f = − Γ1 f + Γ2 f . ,22 ∂x2∂x2 21 ,1 22 ,2 With the above Christoffel symbols, after replacing, we get the following Hessian matrix:

2 2 ! ∂ f + ∂ f − (x1 + 1) ∂ f ∂ f ∇2 f = ∂x1∂x1 ∂x1 ∂x2 ∂x1∂x2 g ∂2 f ∂2 f . ∂x2∂x1 ∂x2∂x2

We will search a function that respects the condition that the determinant of the Hessian will be a constant. In this respect, let us consider the following function f : R2 → R, f (x1, x2) = kx1x2 + 1, where k is a non-positive real constant. For this function, at a critical point x, the above Hessian matrix became:

kx2 + (x1 + 1)kx1 k ∇2 f = H f (x) = . g k 0

2 2 Subsequently, the determinant of the Hessian will be det(∇g f ) = −k = constant. Finally, let us conclude that in this case, the H-distorsion will be:

det(gij) −1 2 = = = σF,∇F 2 2 constant. det(∇g f ) k

Now, using the theory of Frobenius norms, we will introduce the following quantity:

Deﬁnition 4. For a pseudo-Riemannian manifold (M, g), we will denote, by

kH f (x)kHS χH = , (8) kH f1(x1)kHS

the Hessian χ-quotient for two smooth function f , f1 : M → R. Here, k·kHS represent the Frobenius (Hilbert–Schmidt) norm of a Hessian matrix that is attached to the pseudo-Riemannian manifold. Here, the point x represents the critical point for the ﬁrst Hessian of the smooth function f and the point x1 represents the critical point of the Hessian of the second function f1.

Remark 1. Because kH f (x)kHS and kH f1(x1)kHS are two constants, then we can conclude that χH must be also a constant. Next, we will investigate some of χH properties.

First, let us recall the following well known properties regarding Frobenius norms: for two matrices A = (aij), respectively, B = (bij), as we know, the following inequalities hold:

kA · BkHS ≤ kAkHS · kBkHS

respectively √ B−1 ≤ r B−1 . HS HS Hence, now we can formulate the following: Axioms 2021, 10, 69 5 of 7

Theorem 2. For the Hessian χH quotient for two smooth functions f , f1 : M → R, which are considered for the same pseudo-Riemannian manifold (M, g), the following inequality holds: √ Q r χH ≤ , (9) σmin(H f1(x1))

where Q = kH f (x)kHS, x is a critical point of the Hessian H f (x), r represents the rank of the −1 second Hessian H f1(x1), x1 is the critical point of the Hessian H f1(x1), and σmin H f1 (x) is −1 the minimum singular value of the Hessian H f1 (x).

Proof. Starting with the inequality

kA · BkHS ≤ kAkHS · kBkHS,

−1 for A = H f (x) and B = (H f1(x1)) , one obtains:

−1 −1 χH = H f (x) · H f (x1) ≤ kH f (x)k · H f (x1) . 1 HS HS 1 HS Using the hypothesis of the theorem, we get, from here, the following inequality:

−1 χH ≤ Q · H f (x1) . 1 HS Additionally, we can remark the following fact: r −1 −1 −1 H f (x1) = tr H f (x1) × H f (x1) 1 HS 1 1 s n 1 n 1 = ≤ = −1( ) ∑ 2 ∑ tr H f1 x1 . i=1 λi i=1 λi

However, using the inequalities between Frobenius norm and of the two-norm, we obtain: √ −1 −1 H f (x1) ≤ r · H f (x1) 1 1 2

−1( ) = 1 and, since H f1 x1 −1 , we ﬁnd the desired result immediately. 2 σmin(H f1 (x1))

Remark 2. Let us observe that, because

−1 −1 H f (x1) ≥ H f (x1) , 1 HS 1 op

−1 where k·kop is the operator norm of the Hessian H f1 (x1), we obtain: −1 H f (x) · H f (x1) ≤ χH. 1 op

Additionally, because the general property √ kAk2 ≤ kAkHS ≤ rkAk2

holds, we can now give the following:

Corollary 1. For the Hessian χH quotient for two smooth functions f , f1 : M → R, considered for the same pseudo-Riemannian manifold (M, g), the following inequality holds: √ −1 −1 H f (x)H f (x1) ≤ χH ≤ r H f (x)H f (x1) , (10) 1 2 1 2 Axioms 2021, 10, 69 6 of 7

where x is a critical point of the Hessian H f (x), r represents the rank of the second Hessian −1 H f1(x1), x1 is the critical point of the Hessian H f1(x1), and σmin H f1 (x) is the minimum −1 singular value of the Hessian H f1 (x).

Let now recall, from the ﬁrst example where we get the Hessian for the function f : R2 → R, f (x1, x2) = kx1x2 + 1:

(1 + x1)kx1 + kx2 k H f (x) = ∇2 f (x) = , g k 0

where x = (x1, x2) is one critical point for the Hessian H f (x).

For the same metric as in Example1,

(1 + x1)2 x1 + 2 g = , ij x1 1

2 2 we will construct another function f1 : R → R, such that, in a critical point x1 ∈ R , we could be able to compute the Hessian χH quotient. First, let us observe that, for the 1 2 1 2 metric (gij) from the ﬁrst example, the Hessian of the function f (x , x ) = kx x + 1 has the critical point given as a solution of the below equation system: ( ∂ f = kx2 + = ∂x1 1 0 ∂ f = kx1 + = ∂x1 1 0.

1 1 Accordingly, the critical point for this function will be − k , − k . Next, the Hessian that is associated with the metric gij for this critical point will be:

1 1 − 1 k H f (x) = ∇2 f − , − = k . g k k k 0

Accordingly, the Frobenius (Hilbert–Schmidt) norm in this case will be easily computed, as follows: s 2 2 1 1 2 1 kH f (x)HSk = ∇g f − , − = 2k + . k k HS k Let us now consider the following example as a continuation of the above results:

2 1 2 1 1 2 2 Example 2. We will consider the following function f1 : R → R, f (x , x ) = 2x − (x ) x + 2 x for the same matrix as in the ﬁrst example. The critical points for this function f1(x1) are given as a solution of the following equations system ( ∂ f1 = − x1x2 = ∂x1 2 2 0 ∂ f1 = −(x1)2 + = ∂x2 1 0.

Hence, we obtain, for the Hessian of this function f1, two critical points (1, 1), respectively (−1, −1). We will compute the Hessian matrix for this two critical points, as follows:

−2x2 + 2 − 2x1x2 − (x1 + 1)(1 − x1)2 −2x1 H f (x ) = ∇2 f = 1 1 g 1 −2x1 0 2 2 H f (−1, −1) =∇2 f (−1, −1) = . 1 g 1 2 0 √ √ 2 2 2 Accordigly, we derive kH f1(−1, −1)|kHS = 2 + 2 + 2 = 2 3. Axioms 2021, 10, 69 7 of 7

In the same way, we point-out, for the another critical point (1, 1), the following result for the Hessian matrix: −2 −2 H f (1, 1) = ∇2 f (1, 1) = . 1 g 1 −2 0 √ √ 2 2 2 Hence, we find kH f1(1, 1)|kHS = 2 + 2 + 2 = 2 3, the same value as for the prevoius 2 critical point. In conclusion, the Hessian χ-quotient for the above mentioned functions f , f1 : R → R is given by: r 2 2 1 2k + k kH f (x)kHS χH = = √ = constant. kH f1(x1)kHS 2 3 Accordingly, we have find an good example with respect to the definition that we have introduced earlier regarding the χH quotient of some functions f , f1.

3. Conclusions In this paper, we have investigated some interesting properties of the two quantities that we have introduced, namely, the H-distorsion and, respectively, the χH quotient. Finally, we have given some concludent examples regarding this theory.

Author Contributions: Conceptualization, P.L.-I. and A.A.; methodology, formal analysis, B.C.; investigation, A.H.A.; writing—original draft preparation, P.L.-I.; writing—review and editing, A.A.; visualization, B.C.; supervision, A.H.A.; project administration, A.H.A.; funding acquisition, A.A. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors thank the referees for their valuable and constructive comments for modifying the presentation of this work. The authors extend their appreciation to the deanship of scientific research at King Khalid University for funding this work through research groups program under grant number R.G.P.1/50/42. Conflicts of Interest: The authors declare no conflict of interest.

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