*2.2. Curvature*

We present here the notion of curvature of statistical manifolds. We begin by recalling that an *n*-dimensional, C∞ differentiable manifold is defined by a set of points M endowed with coordinate systems CM fulfilling the following two requirements: (1) each element *c* ∈ CM is a one-to-one mapping from M to an open subset of R*<sup>n</sup>*; (2) given any one-to-one mapping *η* : M → R*<sup>n</sup>*, we have that ∀*c* ∈ CM, *η* ∈ CM ⇔ *η* ◦ *c*<sup>−</sup><sup>1</sup> is a C∞ diffeomorphism.

In this paper, we focus on Riemannian manifolds (M, *g*) where the points of M are probability distribution functions. It is worth noting that the manifold structure of M is insufficient to specify in a unique manner the Riemannian metric *g*. On a formal level, an infinite number of Riemannian metrics can be defined on the manifold M. In the context of information geometry however, the selection of the Fisher–Rao information metric (see Equation (7)) as the metric underlying the Riemannian geometry of probability distributions [2,14,15] serves as a primary working assumption. The characterization theorem attributed to Cencov [16] gives significant support for this particular choice of metric. In this characterization theorem, Cencov demonstrates that, up to any arbitrary constant scale factor, the information metric is the only Riemannian metric that is invariant under congruen<sup>t</sup> embeddings (that is, under a family of probabilistically meaningful mappings) of the Markov morphism [16,17].

Upon introducing the Fisher–Rao information metric *<sup>g</sup>μν*(*θ*) in Equation (7), standard differential geometric techniques can be used on the space of probability distributions to describe the geometry of the statistical manifold M*<sup>s</sup>*. The Ricci scalar curvature RM*s* is one example of such a geometric property, where RM*s* is defined as [18]

$$\mathcal{R}\_{\mathcal{M}\_s} \stackrel{\text{def}}{=} \mathcal{g}^{\mu\nu} \mathcal{R}\_{\mu\nu} \tag{13}$$

where *<sup>g</sup>μνgνρ* = *δμρ* and *gμν* = *<sup>g</sup>μν*<sup>−</sup>1. The Ricci tensor <sup>R</sup>*μν* appearing in Equation (13) is given as [18]

$$\mathcal{R}\_{\mu\nu} \stackrel{\text{def}}{=} \partial\_{\boldsymbol{\gamma}} \Gamma^{\boldsymbol{\gamma}}\_{\mu\boldsymbol{\nu}} - \partial\_{\boldsymbol{\nu}} \Gamma^{\boldsymbol{\lambda}}\_{\mu\boldsymbol{\lambda}} + \Gamma^{\boldsymbol{\gamma}}\_{\mu\boldsymbol{\nu}} \Gamma^{\boldsymbol{\eta}}\_{\gamma\boldsymbol{\eta}} - \Gamma^{\boldsymbol{\eta}}\_{\mu\boldsymbol{\gamma}} \Gamma^{\boldsymbol{\gamma}}\_{\nu\boldsymbol{\eta}}.\tag{14}$$

The Christoffel connection coefficients Γ*ρμν* of the second kind that specify the Ricci tensor in Equation (14) are [18]

$$
\Gamma^{\rho}\_{\mu\nu} \stackrel{\text{def}}{=} \frac{1}{2} \mathcal{g}^{\rho\sigma} \left( \partial\_{\mu} \mathcal{g}\_{\sigma\nu} + \partial\_{\nu} \mathcal{g}\_{\mu\sigma} - \partial\_{\sigma} \mathcal{g}\_{\mu\nu} \right). \tag{15}
$$

Next we consider geodesic curves on statistical manifolds**.** A geodesic on an *n*dimensional statistical manifold M*s* can be interpreted as the maximum probability trajectory explored by a complex system during its change from an initial *θ*initial to final macrostates *θ*final, respectively. Each point along a geodesic path corresponds to a macrostate specified by the macroscopic variables *θ* = *θ*1,. . . , *<sup>θ</sup><sup>n</sup>*. In the context of ED, each component *θj* with *j* = 1,. . . , *n* is a solution of the geodesic equation [6],

$$\frac{d^2\theta^k}{d\xi^2} + \Gamma^k\_{lm} \frac{d\theta^l}{d\xi} \frac{d\theta^m}{d\xi} = 0. \tag{16}$$

At this juncture, we reiterate the fact that each macrostate *θ* is in one-to-one correspondence with the probability distribution *p*(*x*|*θ*), with the latter characterizing a distribution of the microstates *x*. It is useful to recognize that the scalar curvature RM*s* can be readily recast as the sum of sectional curvatures K *<sup>e</sup>ρ*, *<sup>e</sup>σ* of all tangent space planes *Tp*M*s* with *p* ∈ M*s* spanned by pairs of orthonormal basis vectors .*eρ* = *∂θρ*(*p*)/,

$$\mathcal{R}\_{\mathcal{M}\_s} \stackrel{\text{def}}{=} \mathcal{R}\_a^a \stackrel{\text{def}}{=} \sum\_{\rho \ne \sigma} \mathcal{K} \left( \mathfrak{e}\_{\rho \prime} \mathfrak{e}\_{\sigma} \right), \tag{17}$$

where K(*<sup>a</sup>*, *b*) is given by [18]

$$\mathcal{K}(a,b) \stackrel{\text{def}}{=} \frac{\mathcal{R}\_{\mu\nu\rho\sigma}a^{\mu}b^{\nu}a^{\rho}b^{\sigma}}{(\mathcal{g}\_{\mu\sigma}\mathcal{g}\_{\nu\rho} - \mathcal{g}\_{\mu\rho}\mathcal{g}\_{\nu\sigma})a^{\mu}b^{\nu}a^{\rho}b^{\sigma}},\tag{18}$$

with

$$a \stackrel{\text{def}}{=} \sum\_{\rho} \langle a, \, \varepsilon^{\rho} \rangle \varepsilon\_{\rho \prime} \, b \stackrel{\text{def}}{=} \sum\_{\rho} \langle b, \, \varepsilon^{\rho} \rangle \varepsilon\_{\rho \prime} \text{ and } \langle \varepsilon\_{\rho \prime} \, \varepsilon^{\sigma} \rangle \stackrel{\text{def}}{=} \delta\_{\rho}^{\sigma}. \tag{19}$$

We observe that the Riemann curvature tensor <sup>R</sup>*αβρσ* [18] is fully determined by the sectional curvatures K *<sup>e</sup>ρ*, *<sup>e</sup>σ* where

$$\mathcal{R}^{\mathfrak{a}}{}\_{\beta\mathfrak{p}\sigma} \stackrel{\text{def}}{=} \mathcal{g}^{\mathfrak{a}\gamma} \mathcal{R}\_{\gamma\beta\mathfrak{p}\sigma} \stackrel{\text{def}}{=} \partial\_{\sigma} \Gamma^{\mathfrak{a}}{}\_{\beta\mathfrak{p}} - \partial\_{\rho} \Gamma^{\mathfrak{a}}{}\_{\beta\sigma} + \Gamma^{\mathfrak{a}}{}\_{\lambda\sigma} \Gamma^{\lambda}{}\_{\beta\mathfrak{p}} - \Gamma^{\mathfrak{a}}{}\_{\lambda\rho} \Gamma^{\lambda}{}\_{\beta\sigma} . \tag{20}$$

The negativity of the Ricci scalar curvature RM*s* is a strong (i.e., a sufficient but not necessary) criterion of local dynamical instability. Moreover, the compactness of the manifold M*s* is required to specify genuine chaotic (that is, temporally complex) dynamical systems. In particular, it is evident from Equation (17) that the negativity of RM*s* implies that negative principal curvatures (i.e.**,** extrema of sectional curvatures) are more dominant than positive ones. For this reason**,** the negativity of RM*s* is a sufficient but not necessary requirement for local instability of geodesic flows on statistical manifolds. It is worth mentioning the possible circumstance of scenarios in which negative sectional curvatures are present, but the positive curvatures dominate in the sum of Equation (17) such that R M*s* is a non-negative quantity despite flow instability in those directions. For additional mathematical considerations related to the concept of curvature in differential geometry, we refer to [19].

### *2.3. Jacobi Fields*

We introduce here the concept of the Jacobi vector field. It is worth noting that the analysis of stability/instability arising in natural (geodesic) evolutions is readily accomplished by means of the Jacobi–Levi–Civita (JLC) equation for geodesic deviation. This equation is familiar in both theoretical physics (for example, in the case of General Relativity) as well as in Riemannian geometry. The JLC equation describes in a covariant manner, the degree to which neighboring geodesics locally scatter. In particular, the JLC equation effectively connects the curvature properties of an underlying manifold to the stability/instability of the geodesic flow induced thereupon. Indeed, the JLC equation provides a window into a diverse and mostly unexplored field of study concerning the connections among topology, geometry and geodesic instability, and thus to complexity and chaoticity. The use the JLC equation in the setting of information geometry originally appeared in [20].

In what follows, we take into consideration two neighboring geodesic paths *θα*(*ξ*) and *θα*(*ξ*) + *δθα*(*ξ*), where the quantity *ξ* denotes an affine parameter satisfying the geodesic equations,

$$\frac{d^2\theta^\alpha}{d\xi^2} + \Gamma^a\_{\beta\gamma}(\theta) \frac{d\theta^\beta}{d\xi} \frac{d\theta^\gamma}{d\xi} = 0,\tag{21}$$

and

$$\frac{d^2[\theta^a + \delta\theta^a]}{d\xi^2} + \Gamma^a\_{\beta\gamma}(\theta + \delta\theta) \frac{d\left[\theta^{\beta} + \delta\theta^{\beta}\right]}{d\xi} \frac{d[\theta^{\gamma} + \delta\theta^{\gamma}]}{d\xi} = 0,\tag{22}$$

respectively. Noting that to first order in *δθα*,

$$
\Gamma^{a}\_{\beta\gamma}(\theta + \delta\theta) \approx \Gamma^{a}\_{\beta\gamma}(\theta) + \partial\_{\eta}\Gamma^{a}\_{\beta\gamma}\delta\theta^{\eta},\tag{23}
$$

after some algebraic calculations, to first order in *δθα*, Equation (22) becomes

$$\frac{d^2\theta^\kappa}{d\xi^2} + \frac{d^2(\delta\theta^\kappa)}{d\xi^2} + \Gamma^\ll\_{\beta\gamma}(\theta) \frac{d\theta^\delta}{d\xi} \frac{d\theta^\gamma}{d\xi} + 2\Gamma^\ll\_{\beta\gamma}(\theta) \frac{d\theta^\delta}{d\xi} \frac{d(\delta\theta^\gamma)}{d\xi} + \partial\_\eta\Gamma^\ll\_{\beta\gamma}(\theta)\delta\theta^\eta \frac{d\theta^\delta}{d\xi} \frac{d\theta^\gamma}{d\xi} = 0. \tag{24}$$

The equation of geodesic deviation can be found by subtracting Equation (21) from Equation (24),

$$\frac{d^2(\delta\theta^a)}{d\xi^2} + 2\Gamma^a\_{\beta\gamma}(\theta)\frac{d\theta^\beta}{d\xi}\frac{d(\delta\theta^\gamma)}{d\xi} + \partial\_\eta\Gamma^a\_{\beta\gamma}(\theta)\delta\theta^\eta\frac{d\theta^\beta}{d\xi}\frac{d\theta^\gamma}{d\xi} = 0. \tag{25}$$

Equation (25) can be conveniently recast via the covariant derivatives (see [21], for instance) along the curve *θα*(*ξ*),

$$\frac{D^2(\delta\theta^a)}{D\xi^2} = \frac{d^2(\delta\theta^a)}{d\xi^2} + \partial\_\beta \Gamma^a\_{\rho\sigma} \frac{d\theta^\beta}{d\xi} \delta\theta^\rho \frac{d\theta^\sigma}{d\xi} + 2\Gamma^a\_{\rho\sigma} \frac{d(\delta\theta^\rho)}{d\xi} \frac{d\theta^\sigma}{d\xi} + $$

$$-\Gamma^a\_{\rho\sigma} \Gamma^\sigma\_{\kappa\lambda} \delta\theta^\rho \frac{d\theta^\kappa}{d\xi} \frac{d\theta^\lambda}{d\xi} + \Gamma^a\_{\rho\sigma} \Gamma^\rho\_{\kappa\lambda} \delta\theta^\kappa \frac{d\theta^\lambda}{d\xi} \frac{d\theta^\sigma}{d\xi} \tag{26}$$

The covariant derivative is defined as *<sup>D</sup>ξδθα* def = *∂ξδθα* + Γ*αξκδθκ* with *<sup>D</sup>ξ* def = *D*/*Dξ* and *∂ξ* def = *∂*/*∂ξ*, respectively. By combining Equations (25) and (26), and performing some tensor algebra calculations, we obtain

$$\frac{D^2(\delta\theta^a)}{D\xi^2} = \left(\partial\_\rho \Gamma^a\_{\eta\sigma} - \partial\_\eta \Gamma^a\_{\rho\sigma} + \Gamma^a\_{\lambda\sigma} \Gamma^\lambda\_{\eta\rho} - \Gamma^a\_{\eta\lambda} \Gamma^\lambda\_{\rho\sigma}\right) \delta\theta^\eta \frac{d\theta^\rho}{d\xi} \frac{d\theta^\sigma}{d\xi}.\tag{27}$$

Finally, the geodesic deviation equation expressed in component form becomes

$$\frac{D^2 l^\alpha}{D\xi^2} + \mathcal{R}^\alpha\_{\rho\eta\sigma} \frac{d\theta^\rho}{d\xi} l^\eta \frac{d\theta^\sigma}{d\xi^\nu} = 0,\tag{28}$$

where *Jα* def = *δθα* is the *α*-component of the Jacobi vector field [18]. Equation (28) is known formally as the JLC equation. We observe from the JLC equation in Equation (28) that neighboring geodesics accelerate relative to each other at a rate measured in a direct manner by the Riemannian curvature tensor *<sup>R</sup>αβγδ*. The quantity *Jα* is defined as,

$$J^{a} = \delta\theta^{a} \stackrel{\text{def}}{=} \delta\_{\phi}\theta^{a} = \left(\frac{\partial\theta^{a}\left(\xi;\phi\right)}{\partial\phi}\right)\_{\tau=\text{constant}}\delta\phi,\tag{29}$$

where {*θμ*(*ξ*; *φ*)} denotes the one-parameter *φ* family of geodesics whose evolution is specified by means of the affine parameter *ξ*. The Jacobi vector field intensity *J*<sup>M</sup>*s* on the manifold M*s* is given by

$$J\_{\mathcal{M}\_s} \stackrel{\text{def}}{=} \left( J^a \mathcal{g}\_{a\beta} J^\beta \right)^{1/2}. \tag{30}$$

In general, the JLC equation is intractable even at low dimensions. However, in the case of isotropic manifolds, it reduces to

$$\frac{D^2 J^\mu}{D\xi^2} + \mathcal{K}l^\mu = 0.\tag{31}$$

The sectional curvature K in Equation (31) assumes a constant value throughout the manifold. In particular, when K < 0, unstable solutions of Equation (31) become

$$J^{\mu}(\xi) = \frac{\omega\_0^{\mu}}{\sqrt{-\mathcal{K}}} \sinh \left(\sqrt{-\mathcal{K}}\xi\right),\tag{32}$$

with initial conditions *Jμ*(0) = 0 and *d Jμ*(0) *dξ* = *ω<sup>μ</sup>*(0) = *ωμ*0 = 0, respectively, for any 1 ≤ *μ* ≤ *n* with *n* being the dimensionality of the underlying manifold. For additional remarks concerning the JLC equation, we refer to [18,21,22].

We point out that it would be intriguing to understand the behavior of the Jacobi vector fields within the geometry of the Kaniadakis statistical mechanics emerging from a one deformation parameter *κ* [13]. We leave this fascinating line of study to future scientific inquiry. For a schematic description of the behavior of the IGE and the Jacobi field for two-dimensional surfaces with distinct (Gaussian) curvatures, we refer to Table 1 and Figure 1.

In the next section, making use of the complexity quantifiers introduced in Equations (3), (17), and (30), we present numerous illustrative examples within the IGAC framework.

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**Table 1.** Schematic description of the behavior of the IGE and the Jacobi field for different types of two-dimensional surfaces characterized by distinct constant values of their Gaussian curvature. For such surfaces, the sectional and the scalar curvatures coincide, while the Gaussian curvature is simply one-half of the scalar curvature. In particular, positive curvature causes geodesics to converge while negative curvature causes geodesics to spread out. More specifically, in flat, positively, and negatively curved manifolds, the geodesic deviation equation yields deviations of nearby geodesics that exhibit linear, oscillatory, and exponential behaviors, respectively. Moreover, the volumes of the manifolds regions explored during the entropic motion tend to increase while transitioning from positively to negatively curved manifolds. Correspondingly, the IGE exhibits its maximum growth (that is, linear growth) in the presence of exponential instability on negatively curved manifolds.

**Figure 1.** Graphical depictions of the links among curvature, Jacobi fields, and IGE. In (**a**), we depict the constant scalar curvature of a positively curved manifold (solid line), a flat manifold (dashed line), and a negatively curved manifold (dotted line). In (**b**), we illustrate the behavior of the normal components of the Jacobi fields quantifying how nearby geodesics are changing in the normal direction (that is, the direction that is orthogonal to the unit tangent vector of the geodesic) as we move along the geodesics. In the positive, flat, and negative curvature cases, we observe oscillatory behavior (solid line), linear behavior (dashed line), and exponential behavior (dotted line), respectively. Finally, in (**c**), we plot the temporal behavior of the IGE in the positive (sublogarithmic behavior, solid line), flat (logarithmic behavior, dashed line), and negative (linear behavior, dotted line) curvature cases.
