axioms Article The c-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 c-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 c-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 2 M, where (M, F) is a Finsler manifold, i.e., published maps and institutional affil- n = 2 nj = ( ) < iations. Bx fy R jyj F x, y 1g and Bn represents the unit ball in the Euclidean space centered at origin: Bn = fx 2 Rnjjxj < 1g Copyright: © 2021 by the authors. q Licensee MDPI, Basel, Switzerland. i j where jxj = dijx x . This article is an open access article distributed under the terms and Additionally, in Finsler geometry, we know that sF(x) is given by: conditions of the Creative Commons Attribution (CC BY) license (https:// volume(Bn) ( ) = creativecommons.org/licenses/by/ sF 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 t = t(x, y) is given by: 0 q 1 det(gij) t(x, y) = ln@ A. sF(x) q i j Additionally, according to Shen ([1]), when F = gij(x)y y , is Riemannian, then q sF(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 2 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 rg on M, called the Levi–Civita connection of g, such that: (a) rgg = 0, (b) rg is torsion free, i.e., T = 0. 1 n k For a coordinate chart (U, x , ··· , x ), the Christoffel symbols Gij of the Levi–Civita connection are related to the components of the metric g in the following way: 1 ¶g ¶gjl ¶gij Gk = 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 r2 f = − Gk 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 = − G f,m , ¶xi¶xj ij ¶ f,ij f = − Gl f − Gl 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 fol- lows: 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) p kAkHS ≤ nkAkop. (3) In the previous inequality, the equality take place when A = In. Additionally, n jdet(A)j ≤ kAkHS (4) p p r kAkHS ≤ rkAk2 = . (5) smin(A) Here, A denotes any positive definite symmetric matrix, r is the rank of A, and smin(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 ( ) sF(x) u det gij sF,rF = = t (6) srF(x) 2 det rg f 2 the H-distortion, if and only if sF,rF(x) =constant. Example 1. We will consider a pseudo-Riemannian manifold, (x1, x2) j 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 G11 = −1, G11 = x + 1, G22 = G12 = G21 = G21 = G12 = G22 = 0. Now, we get: Axioms 2021, 10, 69 4 of 7 ¶2 f f = − G1 f + G2 f ,11 ¶x1¶x1 11 ,1 11 ,2 ¶2 f f = − G1 f + G2 f ,12 ¶x1¶x2 12 ,1 12 ,2 ¶2 f f = − G1 f + G2 f ,21 ¶x2¶x1 21 ,1 21 ,2 ¶2 f f = − G1 f + G2 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 r2 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 r2 f = H f (x) = . g k 0 2 2 Subsequently, the determinant of the Hessian will be det(rg f ) = −k = constant. Finally, let us conclude that in this case, the H-distorsion will be: det(gij) −1 2 = = = sF,rF 2 2 constant. det(rg 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 cH = , (8) kH f1(x1)kHS the Hessian c-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 cH must be also a constant. Next, we will investigate some of cH 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 p 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 cH quotient for two smooth functions f , f1 : M ! R, which are considered for the same pseudo-Riemannian manifold (M, g), the following inequality holds: p Q r cH ≤ , (9) smin(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 smin H f1 (x) is −1 the minimum singular value of the Hessian H f1 (x).