The book begins with a modern treatment of general topology, reinterpreting Kuratowski’s early ideas [1] in a contemporary framework. Readers are introduced to essential topological concepts—such as open sets, continuity, convergence, and compactness—establishing the language and tools necessary for later discussions.

Building on the groundwork of topology, the text moves into the realm of manifolds. Chapter 2 explains topological manifolds—spaces that locally resemble Euclidean space—while also presenting modelled manifolds based on S. Lang’s influential works [2, 3]. For additional reference, the text suggests consulting [4] to cover aspects not fully developed in this book. This chapter serves as a bridge, transitioning from abstract topological spaces to concrete geometric structures.

Differentiability is explored next with a focus on Gateaux derivatives. This notion is particularly useful in infinite-dimensional settings, especially in probability theory where a norm may not be available. The text carefully contrasts Gateaux differentiability with the stronger concept of Fréchet differentiability, ensuring students understand the subtleties and applications of both approaches.

Fiber bundles are introduced as indispensable tools in differential geometry. This chapter outlines the local-to-global perspective that fiber bundles offer, demonstrating how complex geometric structures can be understood by piecing together simpler, locally trivial components.

Delving deeper into the geometric framework, Chapter 5 covers connections, parallel transport, and covariant derivatives—key concepts for understanding how geometric data evolves along a manifold. Classical references such as [5] and [6] provide further reading on these fundamental topics. Additionally, an introduction to sheafs is provided, offering a gentle entry point into their role in modern geometry, with [7] recommended for a complete reference. This material lays the groundwork for applying geometric techniques within information geometry later in the book.

The second part shifts focus to probability theory and statistics. It introduces standard definitions, theorems, and methodologies, employing familiar examples—such as coin flipping—to illustrate key concepts. The chapter also discusses the Radon–Nikodym derivative, linking measure theory with probabilistic reasoning. References such as [8, 9] for measure theory and [10] for statistics are suggested for further reading.

These chapters are particularly innovative, blending philosophy with mathematics. Beginning with a discussion inspired by Klein’s geometry and Plato’s ideas, the text presents a novel perspective on how categorical structures naturally arise in the study of probability distributions. By considering manifolds of probability measures and the role of Markov kernels, the reader is guided through a modern generalization of classical geometry into a probabilistic and categorical setting. Key references for this section include [11, 12, 13, 14, 15].

The final part of the book centers on Frobenius manifolds—geometric structures that elegantly encapsulate the interplay between algebra and geometry, coming from 2D Topological Field Theory. Here, the text explains how Frobenius structures emerge naturally within the framework of information geometry. For foundational references on Frobenius manifolds and related topics, the book cites [16, 17].

Chapter 11 also integrates cutting-edge research from 2020 onward [18, 19], demonstrating the latest developments in the field. A notable highlight is the discussion of learning methods pioneered by researchers such as D. Ackley, G. Hinton, and T. Sejnowski [20], showing how deep learning techniques relate to the broader mathematical framework of information geometry.

Conclusion This textbook provides a concise, accessible, and modern overview of several core areas of mathematics—topology, geometry, probability, and category theory—culminating in the study of information geometry. Through careful exposition and a judicious selection of topics, it equips students with the necessary tools to explore this young and rapidly evolving branch of mathematics, bridging classical theory with modern research and applications.

We then refer to this as a topology generated by \( \mathfrak{B}\) .

An equivalent definition can be given. A basis \( \mathfrak{B}\) for a topology \( \mathcal{T}\) is a family of elements of \( \mathcal{T}\) such that for each \( x\in X\) and \( \mathcal{U}\in \mathcal{T}\) , with \( x\in \mathcal{U}\) , there exists \( \mathcal{B}\in \mathfrak{B}\) such that \( x\in \mathcal{B}\) and \( \mathcal{B}\subset \mathcal{U}\) .

An example of a topology widely used in practice is the metric topology.

\( \bullet\) The coarsest topology is the trivial topology.

\( \bullet\) The finest topology is the discrete topology.

In other words, we take as open sets of \( A\) the intersections of open sets of \( X\) with \( A\) .

We will recall some definitions and theorems related to various properties of subsets of a topological space \( X\) .

Complete metric spaces, as well as locally compact Hausdorff spaces, provide natural examples of Baire space.

The Heine-Borel theorem concerns finite-dimensional spaces, it is not necessarily true for an arbitrary topological space. In particular, a closed bounded set with a non-empty interior of an infinite-dimensional normed vector space is never compact, for the norm topology.

\( \star\) Warning! A space can be connected without being locally connected.

A universal covering space is the "simplest" simply connected space that covers a given space. Rigorously, it is defined as follows.

\( \star\) Warning! The converse statement is false.

Note: This definition of the disjoint sum allows to take into account the case where the \( E_{i}\cap E_{j}\ne \emptyset\) . Indeed, the elements of the disjoint sum are ordered pairs \( (x,i)\) . Here \( i\in I\) serves as an auxiliary index that indicates from which \( E_i\) the element \( x\) came from. Each of the \( E_i\) sets are canonically isomorphic to the set \( \tilde E_i=\{(x,i): x\in E_i\}\) . So, if we define \( \varphi_j:E_j \to \coprod_{i\in I}E_i\) by \( \varphi_j(x)=(x,j), \) where \( j\in I\) then \( \varphi_{j}\) is a bijection from \( E_j\) onto \( \tilde E_j=\{(x,i)\mid x\in E_j\}\) and the \( \tilde E_j\) are disjoint, even if the \( E_i\) are not.

Usually to state that the union is a disjoint union we replace \( \bigcup\) by \( \bigsqcup\) .

\( \star\) In what follows, we impose the Hausdorff condition on all manifolds under consideration. Furthermore, any construction performed subsequently—such as products, tangent bundles, or fibered structures—will be required to yield spaces that remain Hausdorff. This ensures a well-behaved topological framework in which geometric operations preserve separation properties.

In the formulation given above, particularly in condition \( \mathcal{M} 2\) , no global assumption was imposed on the structure of the topological vector spaces \( \mathcal{E}_i\) used as local models. In particular, we did not require that all \( \mathcal{E}_i\) be the same, nor that there exist continuous linear isomorphisms between them.

However, one of the most natural and frequently encountered cases in practice is that in which there exists a fixed topological vector space \( \mathcal{E}\) such that each \( \mathcal{E}_i\) is isomorphic to \( \mathcal{E}\) , allowing a uniform model for the local structure of \( \mathcal{M}\) . In such a setting, one can regard \( \mathcal{M}\) as being locally modeled on a single space \( \mathcal{E}\) , which provides a more rigid but also more manageable framework for various constructions.

This remark leads us to the notion of modeled manifold:

If the transition maps (i.e. maps of the type \(\varphi_j \circ \varphi_i^{-1}\)) satisfy additional compatibility conditions (such as differentiability, smoothness, or analyticity), then the \(\mathcal{E}\)-manifold is endowed with some extra structure.

In the following, we primarily focus on modeled manifolds whose model space is a vector space that admits a well-defined differentiable structure. This is the case for manifolds modeled on Banach spaces or Hilbert spaces, where a differentiable structure can be naturally defined. However, we are also interested in a generalization of normed spaces, namely locally convex topological vector spaces.

Let us recall that a locally convex space \( X\) is a vector space over \( \mathbb{K}\) , a (sub)field of the complex numbers (it can be \( \mathbb{C}\) itself or \( \mathbb{R}\) for instance).

A locally convex space is defined either in terms of convex sets or equivalently in terms of seminorms. In fact, a topological vector space \( X\) is said to be locally convex if it verifies one of the following two equivalent properties. We start with the convex set definition.

In any topological vector space, every neighborhood of the origin is absorbent.

A second possible viewpoint on this notion can be achieved using seminorms. A seminorm on \( X\) is a function

such that:

1. Non-negativity: \( p\) is nonnegative or positive semidefinite i.e.

   \[ p(x)\geq 0,~ \text{for all}~ x\in X. \]

2. Scaling property: \( p\) is positive homogeneous or positive scalable:

   \[ p(sx)=|s|\cdot p(x),~ \text{for every scalar} ~ s \text{and} x\in X. \]

   So, in particular, \( p(0)=0\) ,

3. Subadditivity: \( p\) is subadditive and it satisfies the triangle inequality:

   \[ p(x+y)\leq p(x)+p(y), ~ \text{for all}~ x,y\in X. \]

As mentioned earlier, both definitions are equivalent. We outline a sketch of the proof showing this equivalence.

\( \star\) The equivalence of those two definitions follows from a construction known as the Minkowski functional or Minkowski gauge. The key feature of seminorms which ensures the convexity of their \( \varepsilon\) -balls is the triangle inequality.

For an absorbing set \( C\) such that: if \( x\in C\) then \( t\cdot x\in C\) , whenever \( 0\leq t\leq 1\) , let us define the Minkowski functional of \( C\) to be

From this definition, it follows that \( \mu _{C}\) is a seminorm if \( C\) is balanced and convex (it is also absorbent by assumption).

Conversely, given a family of seminorms, the sets

form a base of convex absorbent balanced sets.

Let us mention some interesting properties.

• If \( p\) is positive definite (which states that if \( p(x)=0\) then \( x=0\) ) it implies that \( p\) is a norm.

  \( \star\) While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms and separatedness, defined below.
• If \( X\) is a vector space and \( \mathcal{P}\) is a family of seminorms on \( X\) then a subset \( \mathcal{Q} \subset \mathcal{P}\) is called a base of seminorms for \( \mathcal{P}\) if for all \( p\in \mathcal{P}\) there exists a \( q\in \mathcal{Q} \) and a real number \( r>0\) such that \( p\leq rq\) .
• If \( X\) is a real vector space and \( C\) a convex subspace, if \( x\in C\) , then \( C\) contains the line segment between \( x\) and \( -x\) .
• If \( X\) is a complex vector space and \( C\) a convex subspace, for any \( x\in C\) convex space, contains the disk with \( x\) on its boundary, centered at the origin, in the one-dimensional complex subspace generated by \( x\) .

A locally convex topological vector space is a topological vector space in which every neighborhood of \(0\) contains an open neighborhood \(U\) of \(0\) such that, for all \(x, y \in U\) and \(0 \leq t \leq 1\),

This property turns out to be essential in the context of the implicit function theorem, for Banach spaces.

The latter result, known as the Banach Space Implicit Function Theorem, can be stated as follows:

Let \(X\), \(Y\) and \(Z\) be Banach spaces and let \(\mathcal{U}\) be an open subset of \(X \times Y\). Suppose that \(f : \mathcal{U} \to Z\) is a continuously differentiable (\(C^1\)) mapping such that \(f(a, b) = 0\) for some \((a, b) \in \mathcal{U}\) and the partial derivative \(D_y f(a, b)\) is a linear isomorphism from \(Y\) onto \(Z\).

Then, there exists:

• an open set \(\mathcal{W} \subset X\) with \(a \in \mathcal{W}\);
• an open set \(\mathcal{V} \subset \mathcal{U} \subset X \times Y\) with \((a, b) \in \mathcal{V}\);
• a continuously differentiable (\(C^1\)) mapping \(g : \mathcal{W} \to Y\),

such that

In other words, the implicit equation \( f(x,y)=0\) has for \( x\in W\) a solution \( y=g(x)\) of class \( C^1\) such that \( (x,y)\in \mathcal{V}\) .

This solution is unique in an open set \( \mathcal{W}'\subset \mathcal{W}\) .

\( \star{263C}\) Warning! It is important to note that not every locally convex topological vector space admits a differentiable structure in a meaningful way. Nevertheless, a particular class of locally convex topological vector spaces, known as Fréchet spaces, plays a fundamental role in various areas of mathematics, such as statistics and the theory of partial differential equations.

A Fréchet space \( X\) is defined as a locally convex, metrizable, and complete topological vector space, meaning that every Cauchy sequence in \( X\) converges to a point in \( X\) .

For general normed vector space \( X,Y\) , the Fr
specialChar{39}echet directional derivative of a function \( f:\mathcal{U}\to Y\) , where \( \mathcal{U}\) is an open subset of \( X\) , exists at \( x\in \mathcal{U}\) if there exists a bounded linear operator \( A:X\to Y\) such that

The Fréchet derivative in finite-dimensional spaces is the usual derivative. it is represented in coordinates by the Jacobian matrix.

Suppose that \( X\) and \( Y\) are locally convex topological vector spaces. Assume \( U \subset X\) is open and that we have the map \( F: X \to Y\) . The Gateaux differential \( dF(x;\varphi )\) of \( F\) at \( x \in U\) in the direction \( \varphi\) in \( X\) is defined as

If this limit exists for all \( \varphi\) , then \( F\) is said to be Gateaux differentiable in \( x\) .

The limit is taken relatively to the topology of \( Y\) .

Indeed:

1. if \( X\) and \( Y\) are real topological vector spaces, then the limit is taken for real \( t\) .
2. If \( X\) and \( Y\) are complex topological vector spaces, then the limit above in (2) is usually taken as \( t \to 0 \) in the complex plane, as usually in the definition of complex differentiability.
3. In some cases, a weak limit is taken instead of a strong limit, which leads to the notion of a weak Gateaux derivative.

This short introduction to Gateaux’s derivative guides us towards the more standard terminology of differentiable manifolds.

We start with a definition on differentiable manifolds.

Moreover, we have the following compatibility criterion.

Remark that this notion of compatibility is in fact an equivalence relation.

It is enough to have an admissible atlas to define the structure of a manifold.

Notice, that these definitions do not depend on the chart.

A particularly interesting class of Riemannian manifolds is the manifold endowed with the model space \( \mathcal{E} = \mathbb{R}^{n} \). The usual topology and differentiability structure on \( \mathbb{R}^{n} \) induce a differentiable structure on \( \mathcal{M} \) , making it a real \( n \) -dimensional differentiable manifold.

In the following, by an \( n\) -dimensional (real) manifold \( \mathcal{M} \) , we mean a Hausdorff topological space in which every point has a neighborhood homeomorphic to \( \mathbb{R}^n\) .

A natural way to describe the structure of an \( n\) -dimensional differentiable manifold is through local coordinate systems. These coordinate systems are given by charts, which map open subsets of the manifold to open subsets of \( R ^n\) , allowing us to analyze the manifold locally as if it were a Euclidean space.

Local coordinate systems provide a way to describe differentiable manifolds in terms of open subsets \( \mathbb{R}^{n}\) , allowing us to define smooth functions and analyze local geometric properties. In particular, this enables the construction of tangent spaces, which capture the local linear structure of the manifold at each point. The dual spaces to these, known as cotangent spaces, naturally arise when considering differential forms and gradients of functions, playing a fundamental role in differential geometry and analysis on manifolds.

The curve is said to be smooth if it is a \( C^\infty\) map.

More generally:

2.3.4.1.1 Cotangent vectors & Differential 1-forms

2.3.4.1.2 Finite dimensional real manifold

Let \( \mathcal{M}\) a \( n-\) dimensional real manifold. A chart \( (\mathcal{U},\varphi)\) with a local coordinate system \( \varphi:{\scriptstyle M}\to \varphi({\scriptstyle M})=x=(x^1,\dots,x^n)\in \mathbb{R}^n\) at \( M \in \mathcal{M}\) being given, the local coordinate basis on the tangent space is given by

and its dual basis is

Had we chosen a different chart, say \( (\mathcal{U}',\varphi')\) with \( \varphi': {\scriptstyle M}\to x'=({x'}^1,\dots,{x'}^n)\) at \( {\scriptstyle M}\in \mathcal{M}\) , a local coordinate basis can be given by \( \{e'_{i}=\partial/\partial {x'}^{1}\}_{i=1}^{n}\) as well as its dual basis \( \{{\epsilon'}^{i}=d{x'}^{i}\}_{i=1}^{n}\) . We can certainly express one local coordinate basis in terms of the other on the open set \( \varphi(\mathcal{U})\cap\varphi'(\mathcal{U}')\) .

Let us break this statement down. Under a given change of a coordinates, the local coordinates of a tangent vector are described as follows:

So, the vector transformation law is given by:

Similarly, for the covector we have

Having explored the implications of a changing the chart for vectors and covectors, a logical next step is to establish a rigorous definition of a function’s differential.

In the natural basis

More generally, let \( \{e_{i}\}_{i=1}^{n}\) be a basis in \( \mathcal{T}_{\scriptstyle M}(\mathcal{M})\) and let \( \{\epsilon_{i}\}_{i=1}^{n}\) be its dual basis. The value of \( df|_{\scriptstyle M}\in \mathcal{T}^{\star}_{_{M}}(\mathcal{M})\) at \( X_{\scriptstyle M}\in\mathcal{T}_{\scriptstyle M}(\mathcal{M})\) is given by

Hence,

Having previously established the fundamental geometric notions of vectors, differential forms (linear maps on vectors), along with differentials of functions (a type of 1-form), we naturally arrive to the notion of tensors, which encode multilinear maps between these objects.

The following denominations are often used in the literature.

• A (covariant) tensor of order \( r\) at \( \scriptstyle M\) is a tensor on \( \mathcal{T}_{\scriptstyle M}(\mathcal{M})^{\otimes r}\) . It is of type \( (r,0)\)
• A contravariant tensor of order \( s\) at \( \scriptstyle M\) , is a tensor on \( {\mathcal{T}^\star _{\scriptstyle M}(\mathcal{M})}^{\otimes s}\) . It is of type \( (0,s)\) .

Admitting this definition, an \( (r,s)\) -tensor \( T\) can be seen as the tensorial product of \( r\) covariant vectors and \( s\) tangent vectors at \( \scriptstyle M \in \mathcal{M}\) as we can see:

where the \( T_{(i)}\) are covectors and the \( T^{(k)}\) are contravectors.

BY abuse of notation, we have omitted the index \( \scriptscriptstyle M\) to have a more convenient formula. However, it is important to remember that the definition is local.

To conclude, the space of tensors of type \( (r,s)\) can be identified with the tensorial product space:

Let us choose a basis of \( T_{_{M}}^{\star}(\mathcal{M})^{\otimes r} \otimes T_{_{M}}(\mathcal{M})^{\otimes s }\) given by the \( n^{(r+s)}\) vectors:

where \( \{\epsilon^{i}\}_{i=1}^{n}\) is the dual basis of \( \{e_{j}\}_{j=1}^{s}\) .

If we work in the framework of local coordinate basis the associate basis of the \( (r,s)\) -tensor space is

then the \( (r,s)\) -tensor \( T\) is given by:

To enhance the clarity, it is advantageous to adopt the Einstein summation convention, where repeated upper and lower indices are implicitly summed over. In the following formula, we illustrate this convention.

where the components are given by,

Under a change of coordinates, an \( (r,s)\) -tensor transforms via \( r\) applications of the Jacobian (for covariant indices) and \( s\) applications of its inverse (for contravariant indices):

We leverage this framework to study symmetries of tensors. The study of tensor symmetries within this framework reveals fundamental insights into their invariant properties and multilinear algebraic behaviour.

Let us first consider a (covariant) tensor of order \( r\) . Let \( S_{r} \) be the group of permutations of the \( r\) integers \( \{1,2,\dots,r\}\) . By definition, it acts on the tangent space at \( \scriptstyle M \in \mathcal{M}\) in the following way:

or in local coordinates

so that \( T\) has the symmetry defined by \( \sigma\) .

One can define a symmetrization operator \( S\) as well as an antisymmetrization operator \( A\) on the (covariant) tensor \( T\) . This is done below:

which in local coordinates is defined by

where we have used the Einstein summation rule and

is the Kronecker tensor.

The symmetry properties of a contravariant tensor is defined similarly.

In the following we restrict ourselves to the case where the topological space \( \mathcal{F}_{\scriptstyle M}= \pi^{-1}({\scriptstyle M}), {\scriptstyle M}\in \mathcal{M}\) are homeomorphic to a space \( F\) , called typical fibre.

A vector bundle is a \( G\) -bundle with \( G\) the linear group of \( \mathcal{F}\) .

The notion of tangent spaces can be upgraded to the structure of a tangent bundle. We describe this in the next section below.

In the case of a \( n\) -dimensional manifold, the tangent-bundle \( (\mathcal{T}(\mathcal{M}), \mathcal{M}, \pi,G)\) can be identified to the natural bundle defined by

• a fiber at \( {\scriptstyle M}\in \mathcal{M}\) , \( \mathcal{F}_{{\scriptstyle M}}= \mathcal{T}_{\scriptstyle M}(\mathcal{M})\) and typical fiber \( \mathbb{R}^{n}\) ,
• a projection \( \pi:({\scriptstyle M},X_{{\scriptstyle M}}) \mapsto {\scriptstyle M}\) ,
• the structural group \( G=GL(n,\mathbb{R})\) of linear automorphisms on \( \mathbb{R}^{n}\) .

Let \( \mathcal{M}\) a \( n\) -dimensional manifold. A frame \( \phi_{\scriptscriptstyle M}\) associated with the tangent bundle \( \mathcal{T}_{\scriptstyle M}(\mathcal{M})\) of \( \mathcal{M}\) is a set of \( n\) linearly independent vectors \( \{f_{1},\dots,f_{n}\}\) which can be expressed as a linear combination of a particular basis \( \{e_{1},\dots,e_{n}\}\) of \( \mathcal{T}_{\scriptstyle M}(\mathcal{M})\) , such that

This implies that there exists a bijection between the set of all frames in \( \mathcal{T}_{\scriptstyle M}(\mathcal{M})\) and the group \( GL(n,\mathbb{R})\) .

Furthermore, a frame bundle associated with a vector bundle forms a principal \( GL(n,\mathbb{K})\) -bundle, where \( \mathbb{K}\) is a field of characteristic 0 such as \( \mathbb{R}\) or \( \mathbb{C}\) . The general linear group acts freely and transitively on the frames via basis changes.

The topology on the fiber bundle associated to a vector bundle \( E\) is constructed using local trivializations of \( E\) . Each trivialization induces a bijection between the fiber over an open set \( U_i\) and \( U_i\times GL(n,\mathbb{K})\) , with the final topology ensuring compatibility across overlapping regions. The fibers of the frame bundle are \( GL(n,\mathbb{K})\) -torsors. A torsor (or principal homogeneous space) for a Lie group \( G\) is a homogeneous space for \( G\) in which the stabilizer subgroup of every point is trivial. A principal homogeneous space for a group \( G\) is a non-empty set on which \( G\) acts freely and transitively.

We remark the following fact, relating the manifold structure and the frame bundle.

A natural manifold structure can be given if we notice that the open sets of the typical fiber \( GL(n,\mathbb{R})\) are in bijection with the open sets of \( \mathbb{R}^{n^{2}}\) , and that the structural group of diffeomorphisms of the typical fiber onto itself is simply \( GL(n,\mathbb{R})\) .

Let us introduce the space \( \mathfrak{T}(\mathcal{M})\) of all \( \mathcal{C}^{\infty}\) vectors fields on \( \mathcal{M}\) , that is the space of all \( X(f)\) which are differentiable for all \( \mathcal{C}^{\infty}\) functions \( f\) on \( \mathcal{M}\) . Under the addition and the multiplication operation:

this forms a module on the ring \( C^{\infty}(\mathcal{M})\) , but cannot be an algebra for the product of vector fields.

Note A ring is a set \( \mathfrak{R}\) with two internal laws, addition and multiplication. For the addition \( \mathfrak{R}\) is an abelian group, the multiplication is associative and distributive with respect to addition.

A module \( \mathfrak{T}\) over the ring \( \mathfrak{R}\) is an abelian group together with an external operation, called scalar multiplication such that

The product \( XY\) of two vector fields defined by \( (XY)(f)=X(Y(f))\) does not satisfy the Leibnitz rule. Indeed,

Therefore \( XY\) does not form a vector field.

However, notice that the Lie bracket \( [\cdot,\cdot]\) defined by

does form a vector field. The multiplication, defined by the lie bracket is :

• distributive with respect to the addition,
• anti-commutative,

• not associative but satisfies the Jacobi identity:

  \[ \begin{equation} [[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0,~ X,Y,Z\in \mathfrak{T}, \end{equation} \]

  (33)

Notice that a moving frame may not exist globally on \( T(\mathcal{M})\) .

To end this section, let us mention that vector fields can be used for instance in Index Theory. The Poincaré–Hopf theorem relates zeros of vector fields on \( \mathcal{M}\) to the Euler characteristic of \( \mathcal{M}\) .

In the case of a n-dimensional manifold, the cotangent-bundle \( (\mathcal{T}^{\star}(\mathcal{M}), \mathcal{M}, \pi,G)\) can be identified to the natural bundle

• fiber at \( {\scriptstyle M} \in \mathcal{M}\) , \( \mathcal{F}_{\scriptstyle M}= \mathcal{T}^{\star}_{\scriptstyle M}(\mathcal{M})\) and typical fiber \( \mathcal{F} =\mathbb{R}^n\) ,
• projection \( \pi : ({\scriptstyle M},\omega_{\scriptstyle M}) \mapsto {\scriptstyle M} \) ,
• The structural group \( G=GL(n,\mathbb{R})\) , of linear automorphisms on \( \mathbb{R}^n\) .

A covariant vector field \( \omega\) associates to each \( {\scriptstyle M}\in \mathcal{M}\) a covariant vector \( \omega_{\scriptstyle M} \in X\in \mathcal{T}^{\star}_{\scriptstyle M}(\mathcal{M})\) by the mapping

The covector field \( \omega \in \mathcal{T}^{\star}(\mathcal{M})\) acts on the vector field \( X\in T(\mathcal{M})\) by

If we denote by \( \{dx^{i}\}_{i=1}^{n}\) the dual basis of the natural basis \( \{\partial_{i}=\partial/\partial x^{i}\}_{i=1}^{n}\) ,

and

where the index \( {\scriptstyle M}\in \mathcal{M}\) .

From the equation (20), we can define the differential 1-form field by:

In the natural basis

and in an arbitrary local basis \( \{e_{i}\}_{i=1}^{n}\) in \( \mathcal{T}_{\scriptstyle M}(\mathcal{M})\) , where \( \{\varepsilon_{i}\}_{i=1}^{n}\) is the dual basis, it implies that we have:

and finally

Assume \( \mathcal{M}\) and \( \mathcal{M}'\) are two manifolds. Let:

be a differentiable mapping between \( \mathcal{M}\) and \( \mathcal{M}'\) , such that \( {\scriptstyle M}\in \mathcal{M}\) is mapped to \( {\scriptstyle M'}=f({\scriptstyle M})\in \mathcal{M}'\) .

The map \( f\) induces a linear (Jacobian) mapping  the Jacobian mapping is sometimes denoted \( f'\) . denoted \( f_{\star}\) , between the tangent space \( T_{\scriptstyle M}(\mathcal{M})\) at \( {\scriptstyle M}\) and the tangent space \( T_{\scriptstyle M'}(\mathcal{M}')\) at \( {\scriptstyle M'}=f({\scriptstyle M})\) ,

defined in the following way.

Let \( g\) a differentiable function in the neighborhood of \( f({\scriptstyle M})\) . Then, we obtain:

Let \( \omega_x\in T^{\star}_x(\mathcal{M})\) and \( \omega'_{\scriptstyle M'}\) be such that

The pull-back (reciprocal image) \( f^{\star}: \mathcal{T}^{\star}_{\scriptstyle M'}(\mathcal{M}') \to \mathcal{T}^{\star}_{\scriptstyle M} (\mathcal{M})\) of a covariant vector \( \omega'_{\scriptstyle M'}\) under the differentiable mapping \( f\) is defined by the equality:

Therefore, the pull-back \( f^\star: T^{\star}(\mathcal{M}') \to T^{\star}(\mathcal{M})\) of the 1-form \( \omega'\) under a differentiable mapping \( f\) is defined by

We generalise the discussion in the previous chapter about tensors. This notion also can evolve using the concept of fiber bundles.

In particular, a fiber bundle where the base space is the manifold \( \mathcal{M}\) and the fiber is identified with \( \otimes^{p}T^{\star}_{\scriptstyle M}(\mathcal{M})\otimes^{q}T_{\scriptstyle M}(\mathcal{M})\) at all \( {\scriptstyle M}\in \mathcal{M}\) is called a \( (p,q)\) -tensor bundle.

Notice that a tangent bundle is a \( (0,1)\) -tensor bundle and that the cotangent bundle is a \( (1,0)\) -tensor bundle.

Operations defined in section 2.3.5 for tensors at a point, are carried over fiber-wise allowing to define similar operations on tensor fields.

Under these operations, the set of all \( (p,q)\) -tensor fields of class \( \mathcal{C}^{r}\) is a module on the ring \( \mathcal{C}^{r}(\mathcal{M})\) .

Let \( X\) and \( Y\) be two vector fields. By (50), they can be expressed as

which can be identified to the derivative (4) in the direction of \( Y\) . This leads us to the formulation of a definition of the covariant derivative.

The covariant derivative \( \nabla_{_{Y}}X\) is linear in \( Y\)

where

In the frame \( \{e_{i}\}_{i=1}^{n}\) , we obtain:

and thus

where \( \nabla_{e_{i}}\) denote the covariant derivative in the direction \( e_i\) .

The answer to this question is yes. The notion of a covariant derivative in the direction of \( X\in \mathcal{T}_{_{M}}(\mathcal{M})\) can be extended—in a neightborhood of \( M\in \mathcal{M}\) — to an arbitrary type of tensor field, under the following assumptions:

Therefore, if \( t\) is a \( (p,q)\) -tensor its covariant derivative forms a \( (q+1,p)\) -tensor:

where

We will study the above definition on the example of a 1-form. By the fourth condition above, we have that

which implies that

If we substitute \( X\) by \( e_{i}\) this gives

and thus

which means in particular that

This allows us to deduce that

We now consider a (2,0)-tensor and explore the notion of covariant derivative of a (2,0)-tensor. Let \( g=g_{\mu\nu}\varepsilon^\mu\otimes \varepsilon^{\nu}\) be a \( (2,0)\) -tensor.

The covariant derivative of \( g\) is obtained from the properties (3) and (4) outlined in the list of conditions above.

Therefore,

or

We continue with the next remark.

If \( (\varphi,U)\) is a local chart, the component of a point \( \gamma(t)\) on the curve \( \gamma\) are

and the components of \( u\) are given by \( u^{i}=dx^{i}/dt\)

We shortly digress on the notion of geodesics, as they intimately related to the topic discussed in this section.

The concept of geodesics generalizes the notion of straight line in Euclidean space.

There are some easy computations that can be done given \( \gamma:I\to M,   I=[a,b]\subset \mathbb{R}\) , the length of the smooth curve is given by

In particular, this notions starts to be interesting as soon as we want to understant better the intrinsic distant between two points on a manifold. Assume \( \mathcal{M}\subset \mathbb{R}^n\) is an \( m\) -dim smooth manifold. Take a pair of distinct points \( p,q\in \mathcal{M}\) . Their Euclidean distance is given by \( |p-q|\) in the ambient Euclidean space. However, this type of distance does not tell us much, from the viewpoint of the manifold \( \mathcal{M}\) . We therefore introduce the notion of an intrinsic distance in \( \mathcal{M}\) .

Let us now write the equation of geodesics in local chart \( (\varphi,U)\) the component of a point \( \gamma(t)\) on the curve \( \gamma\) are \( x^{i} (t)=\varphi\circ\gamma(t)\) and the component of \( u\) are \( u^{i}=dx^{i}/dt\)

Let \( (\mathcal{M}, g)\) be a Riemannian manifold, that is a smooth manifold equipped with a Riemannian metric. The Riemannian curvature tensor is defined as a map

characterized by the formula:

where \( [X,Y]\) is a Lie bracket of vector fields and where \( \Gamma(\mathcal{T}\mathcal{M})\) are tangent bundle sections.

In the scope of using a more modern language, we can consider the notion of Riemannian curvature in a more categorical framework, which relies on the notion of sheaf. To give a rough idea, a sheaf assigns some local data to open sets, and patches these local data together into a global object.

This definition means that to each open set \( U\subset X\) , we assign an object \( \mathcal{F}(U\) ) in the category \( \mathscr{C}\) . The category \( \mathscr{C}\) can be the category of sets, abelian groups, rings etc.

To each inclusion of open sets \( V\subset U\) , we assign a restriction morphism \( \rho_{U,V}:\mathcal{F}(U)\to \mathcal{F}(V)\) , satisfying:

• Identity: \( \rho_{U,U}\) is the identity;
• Transitivity: \( \rho_{V,W}\circ\rho_{U,V}=\rho_{U,W}\) for \( W\subset V\subset U\) .

A sheaf is a presheaf with an extra gluing condition. Namely, for any open covering \( U=\bigcup\limits_{\alpha} U_\alpha\) and a family of local sections that agree on overlaps (i.e. compatible local sections), there exists a unique global section that restricts to them. In particular, \( \mathcal{F}\) is a sheaf if, for any open cover \( U=\bigcup\limits_{\alpha} U_\alpha\) , the sequence is exact:

Let us breakdown this construction:

• The first map sends a global section \( s\in \mathcal{F}(U)\) to its restriction \( s|{_{U_\alpha}}\) .
• The second pair of maps send \( (s_{\alpha})\) to their restrictions on overlaps \( s_{\alpha}|_{{{U_\alpha}\cap U_{\beta}}}\) and \( s_{\beta}|_{{{U_\alpha}\cap U_{\beta}}}\) .

Going back to the definition of curvature, the main differences, between the previous language and the new formalism are depicted below:

• the tangent bundle sections are viewed as a sheaf \( \mathcal{E}\) of \( \mathcal{O}_\mathcal{M}\) -modules.
• The covariant derivative is interpreted as a morphism of sheaves rather than an operation on vector fields.
• The curvature tensor is introduced as a functorial transformation of the module structure.
• The bracket operation is associated with the intrinsic structure of the Lie algebroid given by the sheaf \( \mathcal{E}\) .

Let \( (\mathcal{M}, g)\) be a Riemannian manifold, where \( \mathcal{M}\) is a smooth manifold equipped with a sheaf \( \mathcal{O}_{\mathcal{M}}\) of smooth functions and a sheaf \( \mathcal{E} = \mathcal{T}_{\mathcal{M}}\) of sections of the tangent bundle, endowed with a Riemannian metric \( g\) . The connection \( \nabla\) is then a morphism of sheaves of \( \mathcal{O}_{\mathcal{M}}\) -modules:

where \( \Omega^1_{\mathcal{M}}\) is the sheaf of Kähler differentials on \( \mathcal{M}\) .

The Riemannian curvature tensor is then a natural transformation of the \( \mathcal{O}_{\mathcal{M}}\) -module functor given by

defined by the relation

where \( [X,Y]\) is the Lie bracket of vector fields, arising from the structure of the Lie algebroid associated with \( \mathcal{E}\) . The operator \( [\nabla_X, \nabla_Y]\) is a commutator of differential operators.

Since the right hand side only depends on the values of \( X, Y, Z\) at a given point, \( R\) is a tensorial object, i.e., a section of the sheaf: \( R \in \Gamma(\mathcal{M}, \operatorname{End}(\mathcal{E}) \otimes \Omega^2_{\mathcal{M}})\) .

This tensor encodes the non-commutativity of the covariant derivative and measures the curvature of the category of \( \mathcal{O}_{\mathcal{M}}\) -modules endowed with \( \nabla\) .

Let \( \mathcal{M}\subset \mathbb{R}^n\) be a smooth \( m\) -dimensional submanifold. Let \( p\in \mathcal{M}\) and let \( E\subset \mathcal{T}\mathcal{M}\) be a 2-dimensional linear subspace of the tangent space. The sectional curvature of \( \mathcal{M}\) at \( (p,E)\) is the number

where \( u,v\in E\) are linearly independent and \( R_p\) is the Riemannian curvature tensor.

The space of symmetric positive definite matrices has many interesting applications.

• Optimisation (convex programming).
• Information geometry and harmonic analysis (via Wishart laws). These are important in random matrix theory and statistics.
• Medicine (diffusion tensor Imaging) as a model of the anisotropic diffusion of water in the tissues.
• Machine learning, one is interested in finding the correlation between given properties (for example one could take weight / volume or weight and height). One models the relationship with a covariance matrix.

Let \( M\) be a smooth manifold, and let \( \nabla\) be an affine connection on \( M\) . The presence of torsion in \( \nabla\) quantifies the extent to which infinitesimal parallel transport fails to be symmetric.

In local coordinates, \( \{x^i\}\) , the torsion tensor can be expressed in terms of the connection coefficients \( \Gamma_{ij}^k\) :

where \( \boldsymbol{T}^i_{jk}\) are the components of the torsion tensor, and \( \Gamma_{jk}^i\) are the Christoffel symbols of the connection.

• The torsion measures the failure of parallel transport to preserve the commutativity of vector fields.
• Let us mention that the torsion affects the holonomy of a connection, which describes how vectors rotate when parallel transported around closed loops.

In Riemannian geometry, the Levi-Civita connection is the unique torsion-free connection that is compatible with the metric. This connection is central to the study of curvature and geodesics.

In the presence of torsion, the equation of geodesic deviation (which describes how nearby geodesics spread apart or converge) includes additional terms involving the torsion tensor.

Intuitively, imagine two nearby geodesics on a surface. If the surface is flat (such as a plane), the geodesics are straight lines, and the distance between them remains constant. However, if the surface is curved (for instance, a sphere), the geodesics may converge or diverge over time. Geodesic deviation quantifies this behavior.

Let \( \mathcal{M}\) be a smooth manifold equipped with a sheaf \( \mathcal{O}_{\mathcal{M}}\) of smooth functions and a sheaf \( \mathcal{E} = \mathcal{T}_{\mathcal{M}}\) of sections of the tangent bundle. A Riemannian structure (or a semi-Riemannian structure in the case of Lorentzian manifolds) is given by a metric \( g: \mathcal{E} \otimes_{\mathcal{O}_{\mathcal{M}}} \mathcal{E} \to \mathcal{O}_{\mathcal{M}}\) .

A geodesic on \( \mathcal{M}\) is a path satisfying the equation

where \( \nabla\) is the Levi-Civita connection associated with \( g\) .

Now, consider a one-parameter family of geodesics, represented by the morphism

where \( I\) is an interval in \( \mathbb{R}\) parametrizing proper time or arc length, and the second parameter represents a perturbation of the initial geodesic. The geodesic deviation vector field is given by

which defines a section of \( \mathcal{E}\) along the central geodesic \( \gamma(s=0)\) . This vector field satisfies the Jacobi equation, which in sheaf-theoretic terms can be written as

Here, \( R: \mathcal{E} \times_{\mathcal{O}_{\mathcal{M}}} \mathcal{E} \to \operatorname{End}(\mathcal{E})\) is the Riemann curvature tensor, viewed as a section of the sheaf \( \operatorname{End}(\mathcal{E}) \otimes_{\mathcal{O}_{\mathcal{M}}} \Omega^2_{\mathcal{M}}\) .

2.5.5.3.1 Flat vs. Curved Geometry:

The torsion tensor is skew-symmetric in its lower indices:

2.5.5.4.1 Vanishing Torsion

The torsion tensor and the curvature tensor \( R\) of a connection are related via the Bianchi identity:

The First Bianchi Identity for a connection with torsion can be written as:

In components, this can be rewritten as:

For a more categorical approach, let \( \mathcal{E}\) be a sheaf of sections of the tangent bundle (or a vector bundle over a space), and let \( \nabla\) be a connection on \( \mathcal{E}\) . The curvature \( R^{\nabla}\) and torsion \( \boldsymbol{T}\) appear as components of a functorial construction on the Atiyah algebroid (the sheaf of first-order differential operators preserving a given structure).

In particular, using the Lie-algebroid formalism, the first Bianchi identity corresponds to the failure of the Spencer differential \( d_\nabla\) acting on torsion to be zero, that is:

where

• \( d_\nabla\) is the Spencer differential associated with the connection.
• \( \wedge_{\nabla}\) denotes the wedge operation compatible with the connection.

Specifically, if you parallel transport a vector \( Y\) along a curve tangent to \( X\) ,

The basic notion in probability theory is the one of random experiment. Such an experiment is mathematically described by a probability space characterized by the triplet \( (\Omega,\mathbf{S}, P)\) where:

1. \( \Omega\) is a sample space i.e. the space of all possible outcomes (samples) \( \omega\) of the random experiment.
2. \( {\mathbf{S}}\) is the family of events. An event \( A\in {\mathbf{S}}\) , is identified with a subset \( A\) of \( \Omega\) .
3. The probability associates to each event a real number between 0 and 1. A probability is defined by a map from the space of events \( {\mathbf{S}}\) to the set \( [0,1]\) .

• The triplet \( (\Omega,{\mathbf{S}},P)\) is called a probability space.
• An event \( A \in {\mathbf{S}}\) such that \( P[A] = 0\) is called a \( P\) -null set.
• An event \( A \in {\mathbf{S}}\) such that \( P[A] = 1\) is said to be realized almost surely (usually denote: “a.s.”).

From this definition, we can obtain have useful relations:

• \( P[A^c] = 1-P[A],   A\in {\mathbf{S}}\)
• \( P[A\setminus B] = P[A] -P[B],   B\subseteq A \in {\mathbf{S}}\) , in particular \( P[B] \leq P[A]\)
• \( P[A\cup B] =P[A] +P[B]-P[A\cap B],  A B \in {\mathbf{S}}\) .

Now, consider the following situation:

To end our discussion on the example of the flipping coin, we observe the following.

More generally, the flipping coin example can be re-contextualized as follows.

Let us first consider a random experiment described by the probability space \( (\Omega,\mathbf{S},P)\) . A real random variable is an application from the sample space \( \Omega\) to a subspace \( E\in \mathbb{R}\) which preserves the algebra of events.

• A real random variable on the measurable space \( (\Omega,\mathbf{S})\) is a measurable function \( X\) with value in \( E \subseteq\mathbb{R}\) (or \( \overline{\mathbb{R}}=\mathbb{R}\cup\{-\infty,+\infty\}\) ) and \( \mathcal{B}\) is the \( \sigma\) -algebra generated by the topology on \( E\) .
• If \( E\) is finitely or infinitely many countable we speak about finite or discrete random variables.
• If \( E\) is a vector space we speak about random vectors.

Let Let \( (\Omega, \mathbf{S},P)\) a probability space, the integral over \( \Omega\) , if there exist, of a random variable \( X\) is called the expectation (or mean) of \( X\) , and denoted

That is \( \mathbb{E}_P[X]\) exist if \( X\in L(\Omega,\mathbf{S},P)\) .

Let \( u\in L^{2}(\Omega,\mathbf{S}, P_{\theta})\) be a tangent vector to the manifold \( \mathsf{S}\) (of dimension \( n\) ) of probability distributions (we assume of exponential type) at the point \( P_{\theta}\) .

where \( \mathbb{E}_{P_{\theta}}[u]\) is the expectation value, with respect to the probability \( {P_{\theta}}\) .

The tangent space to the considered manifold \( \mathsf{S}\) is (locally) isomorphic to the \( n\) -dimensional linear space generated by the family of (centered) random variables (called score vector), \( ( \partial_{i}\ell_\theta),\{i=1,\dots,n\}\) where \( \ell_{\theta} = \ln \rho_{\theta}.\)

In the basis \( \partial_i\ell_{\theta}\) the (Fisher–Rao) metric is the covariance matrix of the score vector:

with

where \( \{a^{i}\}\) form a dual basis to \( \{\partial_j\ell_{\theta}\}\) :

and

Suppose that we perturb infinitesimally \( \theta \) such that \( \theta'=\theta+d\theta\) . Consider the linear map

(this map depends on \( d\theta\) and reduces to the identity map as \( d\theta\) tends to 0).

The given vector

is mapped to \( (u^k+d\theta^i\Gamma_{ij}^ku^j)\partial_k\in\mathcal{T}_{\theta}\) .

This construction allows to establish a correspondence between the vectors in \( T_{\theta}\) and \( T_{\theta'}\) .

The function \( \Gamma_{ij}^k(\theta)\) are the coefficients of the affine connection.

Let \( \pi:E\to \mathsf{S}\) be a vector bundle. A covariant derivative (also knows as a connection) is an \( \mathbb{R}\) -bilinear map

Consider the intrinsic change in the \( j\) -th basis vector \( \partial_j (\theta)\) as \( \theta\) deforms into \( \theta'\) , in the direction of \( \partial_i\) . We obtain the following vector field:

This is a covariant derivative of the vector field \( \partial_j\) along \( \partial_i\) . It is determined from the coefficients of the affine connection (namely \( \Gamma_{ij}^k(\theta)\) ).

There exists a skewness tensor. It is a fully symmetric covariant tensor of rank 3:

given by

so that we have:

The skewness tensor was introduced to formalize the notion of statistical curvature, via the affine connections \( \{\nabla^{\alpha}\}_{\alpha\in \mathbb{R}}\) . We have the following:

for any couple of vector fields \( X, Y\) over \( \mathsf{S}\) ; \( \nabla^{0}\) is the Levi–Civita connection and \( (\cdot)\) is the “contraction” (of two tensors).

The coefficients of the affine connection are:

This called the \( \alpha\) -connection.

Recall that the third order Amari–Chentsov tensor is defined to be:

We use this tensor to simplify calculations of the \( \alpha\) -connection i.e.:

The \( \alpha\) -connection has a meaning of its own, depending on \( \alpha\) . It plays, for example, an important role in statistical inference.

Let us begin by considering a broad and fundamental question: what is, in general, the aim of geometry?

One can state succinctly that geometry is primarily concerned with:

• the study of spaces.
• The transformations of these spaces, namely, their motions or the actions of groups.
• Entire classes of geometric objects.

Section 3.2.1.1.1

For us, however, we will think of those (Klein) figures as figures in the sense of Plato. Plato’s world of geometric figures refers to his philosophical concept of the World of Forms, where perfect and unchanging geometric shapes exist, independently of the physical world.

In Plato’s philosophy, as described in dialogues such as the Republic and the Phaedo, mathematical objects: circles, triangles, and other ideal forms do not exist merely as physical approximations but as pure, abstract entities in a higher, non-material realm. In The Republic, Plato’s “Allegory of the Cave” illustrates how ordinary humans perceive mere shadows (imperfect physical objects), while true philosophers seek to understand the Forms, including perfect geometric structures. A drawn triangle or sphere in the physical world is an imperfect shadow of the true ideal triangle or sphere that exists in the intelligible realm.

It is precisely this concept of Plato’s world of figures/forms which is used in Klein’s perception and which we will use also to describe geometric structures in probability and measure theory.

In the present work, we extend this viewpoint beyond its original domain by considering a formalism in which Klein’s approach is adapted to the realm of probability theory and statistics. The motivation for such a generalization arises naturally: if one seeks to construct a statistical geometry, it is necessary to first identify the appropriate notion of geometric figures and the corresponding transformations that preserve the relevant structures.

Section 3.2.1.3.1

Section 3.2.1.3.2

Section 3.2.1.3.3

The most elementary class of geometric objects consists of subsets of a given space, referred to by Felix Klein as figures.

• Two figures \( A\) and \( B\) are said to be congruent if there exists a motion mapping \( A\) onto \( B\) , along with an inverse motion mapping \( B\) onto \( A\) . The geometer’s principal concern is to investigate the properties of figures that remain invariant under such motions–those properties that are identical for all congruent figures.
• The central problem thus becomes the determination of invariants of figures, namely, real-valued functions that take equal values on congruent figures. A complete system of such invariants allows for a full classification of geometric configurations.

Since any figure can be regarded as a set of points, one is naturally led to a generalization of this concept of geometry–extending beyond the study of figures in the classical sense. Alongside figures in Klein’s sense, which are simply sets, one may consider parametrized sets, thereby introducing a new geometric framework to examine their intrinsic properties.

Let \( \Theta\) be an index set parametrizing such a collection, and let us construct an epimorphism from this parameter set onto the space under consideration. That is, for each parameter value \( \theta \in \Theta\) , a corresponding point of the considered figure is assigned. In general, a single point of the figure may be associated with multiple parameter values, reflecting a richer underlying structure.

Following Klein’s approach to geometry, we may consider parametrized sets as Klein figures, where the parametrization is defined in a specific way. Such examples include:

• finite ordered sequences of points,
• parametrized smooth curves or surfaces,
• smooth manifolds parametrized by locally smooth coordinates–these can be treated both as points or as smooth surfaces.

Furthermore, in the scope of generalizing objects and structures, we can recall what a category is, as we will need it later.

A category is a mathematical structure that captures relationships between objects in an abstract way. At its core, a category consists of objects and morphisms (arrows) between them, which satisfy rules of composition and identity. One should think of a category not just as a formalism but a lens through which mathematics reveals its hidden symmetries and harmonies.

To be more precise, A category \( \mathscr{C}\) consists of the following data:

• A collection of objects, denoted by \( \mathrm{ Ob}(\mathscr{C})\)
• For each pair of objects \( X, Y \in \mathrm{ Ob}(\mathscr{C})\) , a set of morphisms from \( X\) to \( Y\) .

These morphisms must satisfy the following axioms:

(1) Composition

For any three objects \( X,Y,Z\) there exists a composition map \( \circ\) which assigns to each pair of morphisms \( f:X\to Y\) and \( g:Y\to Z\) a composite morphism

(2) Identity Morphisms

For each object \( X\) , there exists an identity morphism \( id\) such that for all \( f:X\to Y\) and \( g:Y\to X\) we have

(3) Associativity

For any morphisms \( f:X\to Y\) , \( g:Y\to Z\) , and \( h:Z\to W\) one requires:

One extends Klein’s notion of figures, beyond these classical cases, namely in the framework of probabilities. The following examples provide examples of generalized Klein figures.

• Sets of probability distributions of a given type;
• Dominated families of probability distributions, namely, all finite families of probability distributions on finite \( \sigma\) -algebras.

(Indeed, any finite or countable family \( \{P_k\}\) of probability measures on a finite \( \sigma\) -algebra is necessarily dominated, as it admits a dominating measure.)

Using the Klein figure formalism, an essential ingredient in constructing a geometric framework for probabilistic or statistical quantities is the study of transformations acting on generalized Klein figures. We now introduce such transformations.

This transition probability distribution describes a Markovian random transition from the measurable space \( (\Omega, \mathbf{S})\) to \( (\Omega', \mathbf{\Sigma})\) . The following theorem establishes an extension property for such transformations.

Let us consider now, classical example of a statistical problems.

We illustrate the congruences, mentioned in the first section of this chapter on a concrete example, which is presented below.

The following theorem provides a condition under which two families of probability distributions are considered congruent, meaning that they share an underlying structural equivalence in terms of how their probability measures are defined. This gives rise to a certain notion of universal property, since it is reminiscent of certain universal properties in algebraic geometry.

Theorem 16 (Universal property)

Let us consider two families of probability distributions, denoted as: \( w_1\) and \( w_2\) which are defined as

\[ w_i=\{P_{\theta}^{{i}}(d\omega^{(i)}), ~ \theta\in \Theta\}, \]

where these families are probability distributions indexed by a parameter \( \theta\) , meaning that for all \( \theta\in \Theta\) we have a corresponding probability measure on the spaces \( (\Omega^{1},\bf{ S}^{(1)})\) and \( (\Omega^{2},\bf{ S}^{(2)})\) , respectively.

The two families \( w_1\) and \( w_2\) are said to be congruent if there exists:

1. A measurable mapping: \( \epsilon=\varphi_{i}(\omega)\) of the measure spaces \( (\Omega^{i},\bf{ S}^{(i)})\) , on which they are defined, onto a finite space

   \[ (\mathscr{E},{\bf S}(\mathscr{E})). \]

2. A common family of probability distributions:

   \[ R_{\theta}(d\epsilon) \]

   on the space \( \mathscr{E}\) , which serves as an intermediate representation of the distributions in both families.

3. A family of transition distributions:

   \[ Q^{(i)}(\epsilon;d\omega^{(i)}) \]

   that describes how the probability measure on \( \mathscr{E}\) is "transferred" back to the original space \( \Omega^{(i)}\) . This relation is formalized as:

   \[ P^{(i)}_{\theta}\{\cdot\}=\int_{\mathscr{E}}R_{\theta}\{d\epsilon\}Q^{(i)}(\epsilon,\cdot), \]

   where the key consistency condition is such that

   \[ Q(\epsilon,\varphi^{-1}(\epsilon))=1. \]

This ensures that each outcome \( \omega^{(i)}_{j}\) belongs to the preimage of its corresponding \( \epsilon\) , meaning that correctly classifies elements into equivalence classes in \( \mathscr{E}\) .

In summary, our construction shows that the property of being dominated is invariant under Markov morphisms, and that dominated families and the families of mutually absolutely continuous distributions naturally form complete subcategories of the category of measurable spaces with Markov morphisms.

In local coordinates, the geodesic equation takes the form:

Since parametrization is essential, any other canonical parametrization \( s = s(t)\) of \( \gamma(t)\) must be an affine function of \( t\) , that is, \( s(t) = at + b\) .

The linear connection operator \( \nabla\) is associated with two multi-linear operators \( T(X,Y)\) and \( R(X,Y)\) :

• The torsion operator:

  \[ T(X,Y) = \nabla_{X}(Y) - \nabla_{Y}(X) - [X,Y]. \]

• The curvature operator:

  \[ R(X,Y)Z = \nabla_{X}(\nabla_{Y}Z) - \nabla_Y(\nabla_{X}Z) - \nabla_{[X,Y]}Z. \]

where \( [X,Y] = XY - YX\) is the commutator.

To better understand the implications of an absolutely equivariant connection, let us recall the fundamental properties of geodesics in canonical coordinates.

• Given two points on a geodesic, one can uniquely define the midpoint of the geodesic segment connecting them.
• This midpoint is determined by a symmetric function of the two endpoints.
• If the connection \( \nabla\) is absolutely equivariant in the category \( \mathcal{C}\) , then the midpoint function is necessarily absolutely covariant with respect to pairs of points.

Thus, the notion of geodesic symmetry emerges naturally in the categorical framework: an equivariant connection ensures that midpoint operations behave consistently across all objects and morphisms of the category.

Consider the category \( \mathsf{Caph}F\) of collections \( \mathsf{\mathsf{Caph}}(\Omega,\mathbf{S},\mathbf{Z})\) with finite quotient algebras \( \mathbf{S}/ \mathbf{Z}\) . These collections are finite-dimensional manifolds. Any \( \mathbf{Z}\) -dominated measure on \( (\Omega, \mathbf{S})\) , where \( \mathbf{S}/ \mathbf{Z}\) is finite, is completely determined by the vector

Introduce equivalence classes under scaling:

where p is defined up to a positive multiplicative constant. This ensures permutation invariance.

Another type of coordinates, which can be used to solve the problem of chart invariance, is to introduce the rectilinear coordinates. Let \( f_j:\Omega\to \mathbb{R}\) , \( (j=0,…,m-1)\) be \( \bf{ S}\) -measurable functions with \( f_0(\omega)=I(\omega)\) . Assume the matrix \( \bigg(f_j(\omega_i)\bigg)\) has full rank \( m\) . Define coordinates:

where \( \mathbf{M}_Pf(\omega)\) is an expectation of a random variable \( f(\omega)\) , defined as a following integral:

where the function \( f\) must be integrable with respect to the measure \( P\) (or quasi-integrable).

To analyze the structure of statistical manifolds, we introduce tangent vectors and vector fields. Let \( \{P_t\}\) be a family of probability distributions parameterized by a smooth curve \( \mathbf{x}(t)\) . The functional \( (Y)_{P}\) of smooth functions \( f(\mathbf{x})\) is defined as:

which represents a tangent vector at the point \( P=P_{\theta}\) .

Exercise 37, correction.

Proof The proof is divided into three parts.

Firstly, since any measure on \( \mathbf{\Sigma}\) can be extended to a measure on \( \mathbf{\Sigma^*}\) , where it coincides with the induced inner and outer measures, it remains to show that for any fixed \( \mathcal{A} \in \mathbf{\Sigma^*}\) , the transition probability function \( \omega \mapsto \Pi^*(\omega, \mathcal{A})\) is \( \mathbf{S^*}\) -measurable.

Having established the first part of the proof, we now proceed to the second step. Our aim is to show that the real-valued function

is \( \mathbf{S^*}\) -measurable for any fixed \( \mathcal{A} \in \mathbf{\Sigma^*}\) . To do so, it suffices to verify that for any probability measure \( P\) on \( \mathbf{S}\) and any real number \( z \in [0,1]\) , the outer and inner \( P\) -measures of the preimage

of the half-open interval \( [0, z)\) coincide, i.e.,

which ensures that \( U\) is indeed \( \mathbf{S^*}\) -measurable.

Consider probability measures \( P\) and \( Q\) . For any \( \mathcal{A} \in \mathbf{\Sigma^*}\) , there exist two sets \( \mathcal{G}, \mathcal{F} \in \mathbf{\Sigma}\) satisfying:

such that for every \( \omega\) , the following equalities hold:

Since probability measures are monotone, we obtain the inequalities:

Therefore, the functions \( \Pi(\omega; \mathcal{G})\) and \( \Pi(\omega; \mathcal{F})\) are \( \mathbf{S}\) -measurable and equal almost everywhere, since

except possibly on a \( \mathbf{S}\) -measurable \( P\) -null set \( N\) where \( P\{N\} = 0\) .

From this, it follows that:

Moreover, we also have:

Thus, we conclude:

which proves the second part.

Finally, we proceed to the third and last step: proving that the measures \( (P\Pi)^*\) and \( P^*\Pi^*\) , both defined on \( (\Omega',\mathbf{\Sigma^*})\) , coincide. That is, we claim that for any \( \mathcal{A} \in \mathbf{\Sigma^*}\) , we have:

Let us choose sets \( \mathcal{F}\) and \( \mathcal{G}\) satisfying for a given \( \mathcal{A}\) :

Then, as before, we have:

where \( Q = P \Pi\) . Given that the function \( \Pi_{\mathcal{A}}^*(\omega) = \Pi^*(\omega; \mathcal{A})\) is \( \mathbf{\Sigma^*}\) -measurable, we obtain:

By monotonicity, we conclude:

Thus, we deduce:

Since this holds for all \( \mathcal{A} \in \mathbf{\Sigma^*}\) , we obtain the desired equality:

This completes the proof.

Exercise 38.

The measure \( P_0\) is dominating since for each \( P_k\) we have the bound

and the inclusion of null sets:

If we can show that \( P_0\) is a constructive probability measure, the desired conclusion follows from Lemma 3.

Consider a measurable mapping \( \omega = f_k(x)\) , where \( f_k : \mathbf{E} \to \Omega\) determines the constructive distribution

Define \( f_0\) piecewise as follows: for \( 2^{-k} < x < 2^{-k+1}\) , let

and set \( f_0(0) = f_1(0)\) .

This function is measurable, as it is composed of countably many measurable functions, each defined on a distinct measurable subset. Consequently, the \( f_0\) -preimage of any \( \mathbf{S}\) -set is a countable union of disjoint \( g_k\) -preimages.

Furthermore, we have the measure transformation:

since the \( g_k\) -preimage is the \( f_k\) -preimage shifted by \( 2^{-k}\) and contracted by a factor of \( 2^k\) . This completes the proof.

In the 1990s, the concept of Frobenius manifolds gained traction among both algebraic and analytic geometers due to the discovery of three significant classes, all stemming from the profound interplay between mathematics and physics. This development led to the creation of advanced mathematical tools and generated an abundance of challenging and intriguing problems. Some of these arose naturally from attempts to provide a rigorous mathematical interpretation of mirror symmetry (e.g., Homological Mirror Symmetry). Others emerged from the ambition to axiomatize and deepen the mathematical understanding of topological quantum field theory.

The first class of Frobenius manifolds originated in the study of singularities. Specifically, Saito spaces—predating the explicit definition of Frobenius manifolds—were associated with the unfolding spaces of singularities. A particularly straightforward example is the space of complex polynomials of a fixed degree \( d\) with distinct roots, which can also be identified with the configuration space of \( d\) marked points on the complex plane.

Subsequent classes discovered in the 1990s have a formal nature. Examples include the (formal) moduli spaces of solutions to Maurer–Cartan equations modulo gauge equivalence and the formal completions of cohomology spaces of smooth projective (or compact symplectic) manifolds. The latter is commonly referred to as quantum cohomology.

More recently, a new class of Frobenius manifolds has emerged, demonstrating the versatility of this structure. It has been shown that under certain conditions, the manifold of probability distributions can generate a Frobenius manifold (this is the Combe–Manin construction). This discovery inspires many open questions, particularly regarding the complete classification of Frobenius manifolds, the identification of novel classes, and the elucidation of the intricate relationships between the existing ones.

In the first section of this chapter, we provide a concise survey of Frobenius manifolds. We introduce the definitions of Frobenius, pre-Frobenius, and potential pre-Frobenius manifolds and briefly outline the WDVV equations. We also touch upon the hidden algebraic structures intrinsic to Frobenius manifolds. This section concludes with a brief summary.

In the second section, we delve more deeply into the theoretical framework of Frobenius manifolds.

In the third section, we present a collection of exercises and open problems for the reader to explore.

Let \( \mathcal{M}\) be a smooth manifold of dimension \( \mathcal{M}\) . Affine flat structures can be defined in multiple equivalent ways.

Section 4.1.2.1.1

Section 4.1.2.1.2

Section 4.1.2.1.3

A group \( \Lambda\) is called an \( n\) -crystallographic group if it contains a normal, torsion-free, maximal abelian subgroup of rank \( n\) and finite index. Crystallographic groups fit into the short exact sequence:

where \( V\) is a complex vector space, and \( P \leq GL(n, \mathbb{Z}) \cong \mathrm{Aut}(V)\) is a finite group acting faithfully on \( V\) .

A complex crystallographic group is a discrete subgroup \( \Lambda \subset \mathrm{Iso}(\mathbb{C}^n)\) such that \( \mathbb{C}^n / \Lambda\) is compact, where \( \mathrm{Iso}(\mathbb{C}^n)\) denotes the group of biholomorphisms preserving the standard Hermitian metric.

We note the following property:

Among the crystallographic groups, one encounters the Bieberbach groups, which are torsion-free crystallographic groups.

This affine structure leads to interesting algebraic properties on the tangent bundle/tangent sheaf. Let \( \Gamma(TM)\) denote the space of vector fields on a given manifold \( \mathcal{M}\) .

According to the previous chapters, a flat and torsionless connection satisfied the following. An affine connection \( \nabla\) is torsion-free (or torsionless) if

The connection is called flat if

Such a connection defines a covariant differentiation for vector fields \( X, Y \in \Gamma(TM)\) :

Let us consider two well-known structures in topology: the torus and the Klein bottle. These topological objects are the only compact \( 2\) -dimensional manifolds that admit Euclidean structures.

Assume \( \mathcal{M}\) is a closed manifold different from the torus or Klein bottle. Then there exists no affine structure on it. This is a direct consequence of the following result:

When dealing with manifolds of dimension greater than two, there is no definitive criterion for determining the existence of an affine structure.

In particular, according to Smillie’s theorem, a closed manifold does not admit an affine structure if its fundamental group is built up out of finite groups by taking free products, direct products, and finite extensions. Specifically, a connected sum of closed manifolds with finite fundamental groups admits no affine structure. This result provides a profound insight into the interplay between algebraic and geometric properties in the study of such manifolds.

Certain Seifert fiber spaces also do not possess affine structures. This is clarified by the following statement:

We discuss the complex case. Due to works of Kobayashi [22], a classification has been established.

However, it is interesting to note that there is no other compact complex surface admitting even holomorphic affine connections.

Let us consider an affine flat structure on a manifold \( \mathcal{M}\) . To define such a structure, several ingredients are required:

• An atlas with transition functions that are affine and linear (in the affine case).
• A metric \( g\) that is compatible with the affine flat structure.

• A symmetric tensor of rank \( 3\) , denoted as:

  \[ A: S^3(\mathcal{T}_\mathcal{M}) \longrightarrow \mathbb{R}. \]

We define a multiplication operation \( \circ\) on the tangent sheaf \( \mathcal{T}_\mathcal{M}\) .

Define a bilinear symmetric multiplication \( \circ = \circ_{A, g}\) on the tangent sheaf \( \mathcal{T}_\mathcal{M}\) as follows:

such that:

where the prime denotes partial dualization.

A compatibility relation between the rank-\( 3\) tensor \( A\) , the rank-\( 2\) tensor \( g\) , and the multiplication operation \( \circ\) is given by:

This invariance of the metric with respect to multiplication ensures that the structure is well-defined.

Certain additional requirements on the algebraic structure of the tangent sheaf \( (\mathcal{T}_\mathcal{M}, \circ)\) lead to having a Frobenius manifold.

There are however two important axioms to keep under consideration and that we shall express below.

An important axiom to have is the one of potentiality. Namely, this axiom requires the existence of a family of local potentials \( \Phi\) , such that:

This axiom is particularly important regarding the relations to the Monge–Ampère equations.

If \( \mathscr{D}\) is a strictly convex bounded subset of \( \mathbb{R}^n\) then for any nonnegative function \( f\) on \( \mathscr{D}\) and continuous \( \tilde{g}:\partial \mathscr{D} \to \mathbb{R}^n\) there is a unique convex smooth function \( \Phi\in C^{\infty}( \mathscr{D})\) such that

in \( D\) and \( \Phi=\tilde{g}\) on \( \partial \mathscr{D}\) .

An elliptic Monge–Ampère equation domain refers to the geometric data generated by \( (\mathscr{D}, \Phi)\) , where

• \( \mathscr{D}\) is a strictly convex domain
• \( \Phi\) a real convex smooth function (with arbitrary and smooth boundary values of \( \Phi\) )

such that Eq. (74) is satisfied.

We state the following result:

An associative pre-Frobenius manifold is a pre-Frobenius manifold such that there exists an associativity property

where \( X,Y,Z\) are vector fields. We will discuss this axiom fully in the context of Frobenius manifolds, below. A Frobenius manifold is a pre-Frobenius manifold where the axioms of potentiality and associativity both hold.

To derive the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation, let us rewrite the associativity of the multiplication \( \circ\) :

This yields a non-linear system of associativity equations, which are partial differential equations for the potential \( \Phi\) . For all \( a, b, c, d\) , these equations are written as:

These equations are highly non-linear and of third order.

Let \( \mathcal{M}\) be a manifold. A Frobenius algebra \( (\mathcal{A}, \circ)\) over a field \( \mathbb{K}\) is a commutative, associative, and unital algebra with a multiplication operation \( \circ\) equipped with a symmetric bilinear form \( \langle -, - \rangle\) satisfying:

for all \( x, y, z \in \mathcal{A}\) .

A manifold \( \mathcal{M}\) admits the structure of a Frobenius manifold if:

• At any point of \( \mathcal{M}\) , the tangent space has the structure of a Frobenius algebra \( \mathcal{A}\) .
• The invariant inner product \( \langle -, - \rangle\) defines a flat metric on \( \mathcal{M}\) .
• The unity vector field satisfies \( \nabla e = 0\) .
• The tensor of rank 4, \( (\nabla_W A)(X, Y, Z)\) , is fully symmetric, where \( X,Y,Z,W\) are vector fields.
• A vector field \( E\) , called the Euler field, exists on \( \mathcal{M}\) such that \( \nabla(\nabla E) = 0\) .

The Euler field \( E\) belongs to the class of affine vector fields. Its existence is inherently tied to the affine structure on \( \mathcal{M}\) . This can be observed through the equivalence of the following statements:

• (1) \( E\) is an affine vector field.
• (2) \( \nabla(\nabla E) = 0\) .
• (3)

  For all vector fields \( Y, Z\) on \( \mathcal{M}\) :

  \[ \nabla_Y(\nabla_Z E) = \nabla_{\nabla_Y Z} E. \]
• (4)

  The coefficients of \( E\) are affine functions. Writing \( E = \sum_m E^m \partial_m\) , we have:

  \[ E^m = a^m_j x^j + b^m, \]

  where \( a^m_j\) and \( b^m\) are constants in \( \mathbb{R}\) .

Thus, we propose a more concise definition of Frobenius manifolds based on Frobenius bundles. This new approach offers a practical and geometrical perspective.

Consider a pre-Frobenius manifold given by a triple \( (\mathcal{M}, g, A)\) . Define the following geometrical objects:

• A connection:

  \[ \nabla_0: \mathcal{T}_\mathcal{M} \longrightarrow \Omega_M^1 \otimes \mathcal{T}_\mathcal{M}, \]

  where \( \nabla_0\) is determined by the condition that flat fields are \( \nabla_0\) -horizontal.

• A pencil of connections depending on a parameter \( \lambda\) :

  \[ \nabla_{\lambda, X}(Y) := \nabla_{0, X}(Y) + \lambda (X \circ Y). \]

  This is called the structure connection of \( (\mathcal{M}, g, A)\) .

We sumarize the key elements existing for pre-Frobenius manifolds, in the example of statistical manifolds. This is as follows:

• An affine flat structure. This structure exists for statistical manifolds of exponential type (see Exercise \( 12.3\) ).
• A triple \( (\mathsf{S}, g, T)\) , where \( \mathsf{S}\) is a finite-dimensional statistical manifold:
• \( g\) is a Riemannian (Fisher) metric,
• \( T\) is the rank-3 symmetric Amari–Chentsov tensor,

• A multiplication defined by:

  \[ u \circ \nu = \bar{T}(u, \nu), \]

  where locally,

  \[ \bar{T}_{ij}^k = \sum g^{km} T_{ijm}, \]

• Metric invariance:

  \[ g(u, \nu \circ w) = g(u \circ \nu, w), \text{for flat vector fields } u,v,w. \]
• The flatness condition, which requires that the affine space of connections \( \nabla_{\lambda}\) is flat.

Let \( (\mathcal{M}, g, A)\) be a triple, where \( \mathcal{M}\) is an associative pre-Frobenius manifold of dimension \( n\) . We introduce the following definition, which will play a fundamental role in the structure theory of such manifolds:

Let \( (e_1, e_2, \dots, e_n)\) be a local basis of \( \mathcal{T}_\mathcal{M}\) . In this local basis, the multiplication takes the form:

In the simplest case, this reduces to

Thus, the basis \( (e_i)\) provides a well-defined (up to renumbering) family of idempotents. If \( \mathcal{M}\) is semisimple, there exists an unramified covering of \( \mathcal{M}\) (of degree at most \( n!\) ) on which the induced pre-Frobenius structure becomes a splitting structure.

Let \( (e_i)\) be a local coordinate basis and let \( (\epsilon^i)\) be its dual basis i.e. \( 1\) -forms. The structure tensor \( A\) , encoding the pre-Frobenius structure, is given by the condition:

where \( g\) is the flat metric and \( \circ\) denotes the associative product on \( \mathcal{T}_\mathcal{M}\) . Since the basis vectors satisfy \( e_i \circ e_j = \delta_{ij} e_i\) , it follows, by a direct application of the above identity, that

We introduced the notation \( \eta_i\) , for the diagonal components of the metric, so that \( \eta_i=g(e_i, e_i)\) .

With this notation, the three-tensor \( A\) takes the form:

exhibiting its diagonalizability in a chosen coordinate system.

Finally, considering the identity element \( e = \sum_{i=1}^n e_i\) in \( (\mathcal{T}_\mathcal{M}, \circ)\) , we obtain the corresponding co-identity:

As a consequence of the above discussion, we are able to reformulate our initial definition differently.

We are now in a position to articulate a fundamental characterization of Frobenius structures within this formalism. This characterization, which encapsulates the essential interplay between the multiplication, the metric, and their compatibility conditions, will serve as a guiding principle in the subsequent development of the theory.

The function \( \eta\) is referred to as the potential metric of the structure. This metric corresponds to a Hessian metric of the form:

where \( \Phi\) is the potential. Note that the canonical coordinates \( \{u^i\}\) are defined up to renumbering and constant shifts.

We now turn to an analysis of the geometric properties of the pencil of connections \( \nabla_\lambda\) , particularly in relation to the structures of a pre-Frobenius and Frobenius manifold.

Let \( \nabla_\lambda\) denote the structure connection associated with the pre-Frobenius manifold \( (\mathcal{M}, g, A)\) . The curvature of this connection satisfies a quadratic relation in the parameter \( \lambda\) , taking the form:

where \( R_1, R_2, R_3\) are curvature terms determined by the pre-Frobenius structure. A key observation is that the term \( R_3 \) coincide with \( \lambda_0^2\) which satisfies \( \lambda_0^2 = 0\) , leading to the simplification:

This relation gives rise to the following fundamental theorem, characterizing the Frobenius condition in terms of the vanishing of specific curvature terms.

4.1.6.6.1 Webs

4.1.6.6.2 Some results

Let us consider an object \( X_{d}\) , a \( d\) -dimensional surface residing in an \( n\) -dimensional projective space \( \mathbb{P}^{n}\) , where the constraint \( d \leq n\) holds.

This embodies a duality, intrinsic to projective geometry. Notably, in the case where \( d = n\) , the hyperplane \( P_{I}\) identifies to the point \( p\) , and \( P_{II}\) becomes the \( (n-1)\) -dimensional surface devoid of the point \( p\) . This situation reflects the classical notion of duality in projective spaces, leading to the identification of \( X_{n}\) with the projective space \( \mathbb{P}^{n}\) .

More precisely, an m-pair is a pair consisting of:

1. An m-plane: A linear subspace of dimension \( m \) in an \( n \) -dimensional space.
2. An (n-m-1)-plane: A linear subspace of dimension \( n-m-1\) in the same \( n\) -dimensional space.

These two planes are typically considered in the context of projective geometry or linear algebra, where they may satisfy certain geometric or algebraic relationships.

• 0-pair: A point (0-plane) and a plane (2-plane). This can be identified with the projective space \( \mathbb{P}^3\)
• 1-pair: A line (1-plane) and another line (1-plane). Two lines in 3D space can either intersect at a point, be parallel (not intersecting but lying on a common plane), or be skew (not intersecting and not parallel).

• 1-pair: A line (1-plane) and a plane (2-plane). In 4D space, a line and a plane can intersect at a point, not intersect at all, or intersect along a line if the line lies entirely within the plane.
• 2-pair: Two planes (2-planes). In 4D space, two planes can intersect at a point, along a line, or not intersect at all.

In projective geometry, an \( m \) -pair can be used to describe configurations of points, lines, and planes at infinity.

The concept of \( m \) -pairs is closely related to Grassmannians, which are spaces that parameterize all \( m \) -dimensional subspaces of an \( n\) -dimensional space. The study of \( m \) -pairs can be seen as a way to explore the relationships between different Grassmannians.

In computer vision, \( m\) -pairs can be used to model the relationship between different views of a scene. For example, in structure from motion, the relationship between 2D image planes (2-planes) and 3D space (3-planes) can be analyzed using the concept of \( m \) -pairs.

The examples above illustrate how \( m \) -pairs can be applied in various contexts, from projective geometry to computer vision.

Normalized surfaces associated with an \( m\) -pair space possesses the following properties:

A paracomplex number is an element of the form:

where \( x, y \in \mathbb{R} \) are real numbers, and \( \epsilon \) is the paracomplex unit satisfying:

The set of all paracomplex numbers is denoted by \( \mathbb{P}\) .

The algebra of paracomplex numbers \( \mathfrak{P} \) is a two-dimensional commutative algebra over the real numbers \( \mathbb{R}\) . It is isomorphic to the direct sum \( \mathbb{R} \oplus \mathbb{R} \) , and its multiplication rule is given by:

• Idempotent Basis: Paracomplex numbers can be expressed in terms of idempotent elements. Define:

  \[ e_+ = \frac{1 + \epsilon }{2}, ~ e_- = \frac{1 - \epsilon }{2}. \]

  These elements satisfy \( e_+^2 = e_+ \) , \( e_-^2 = e_- \) , and \( e_+ e_- = 0 \) . Any paracomplex number \( z = x + \epsilon y \) can be written as:

  \[ z = (x + y)e_+ + (x - y)e_-. \]

• Conjugation: The paracomplex conjugate of \( z = x + \epsilon y \) is defined as:

  \[ \overline{z} = x - \epsilon y. \]

• Norm: The norm of a paracomplex number \( z = x + \epsilon y \) is given by:

  \[ \|z\| = z \overline{z} = x^2 - y^2. \]

  Note that this norm is not positive definite, as it can take negative values.

Let \( \mathfrak{P}\) be the algebra of paracomplex numbers. A module over \( \mathfrak{P}\) is an abelian group M together with a scalar multiplication:

satisfying the following properties for all \( z, z_1, z_2 \in \mathfrak{P}\) and \( m, m_1, m_2 \in M \) :

1. \( z \cdot (m_1 + m_2) = z \cdot m_1 + z \cdot m_2 \) ,
2. \( (z_1 + z_2) \cdot m = z_1 \cdot m + z_2 \cdot m \) ,
3. \( (z_1 z_2) \cdot m = z_1 \cdot (z_2 \cdot m) \) ,
4. \( 1 \cdot m = m \) .

• Paracomplex Vector Space: The simplest example of a module over \( \mathfrak{P}\) is \( \mathfrak{P}^n\) , the set of n -tuples of paracomplex numbers. Scalar multiplication is defined component-wise:

  \[ z \cdot (z_1, z_2, \dots, z_n) = (z z_1, z z_2, \dots, z z_n). \]

• Decomposition into Real Submodules: Using the idempotent basis \( e_+ \) and \( e_- \) , any module \( M\) over \( \mathfrak{P} \) can be decomposed into two real submodules:

  \[ M = e_+ M \oplus e_- M. \]

  Here, \( e_+ M \) and \( e_- M \) are real vector spaces, and the action of \( \mathfrak{P}\) on \( M \) is determined by the actions of \( e_+ \) and \( e_- \) .

Modules over paracomplex algebras appear in various areas of mathematics and physics, including:

• Geometry: Paracomplex structures are used in the study of para-Hermitian and para-Kähler manifolds.
• Physics: Paracomplex numbers and modules are used in the study of supersymmetry and integrable systems.

Gromov–Witten invariants are fundamental numerical invariants in symplectic geometry and algebraic geometry, capturing intersection properties of (pseudo-)holomorphic curves in a given space.

Given a compact symplectic manifold \( (\mathcal{M},\varpi)\) , the Gromov–Witten invariant counts the number of (pseudo-)holomorphic maps

from a compact Riemann surface \( \mathscr{S}\) (with complex structure \( j\) ) into \( \mathcal{M}\) , satisfying certain constraints on their homology class and intersection conditions with given cycles.

• This notion appears in Enumerative Geometry: Gromov–Witten invariants count the number of holomorphic curves passing through prescribed points or satisfying intersection constraints.
• Quantum Cohomology: They define a deformation of classical cohomology, giving rise to a quantum product that encodes curve counts in a ring structure.
• Holomorphic Curve Moduli Space: The counts arise from integration over the moduli space of stable maps. We refer to [17] for more information on this topic.

The point of view that we adopt, here, is inspired from quantum cohomology, which is a formal Frobenius manifold.

Let us consider the (formal) Frobenius manifold \( (H,g)\) . We denote by \( k\) a field of characteristic 0 (such as \( \mathbb{C}\) or \( \mathbb{R}\) ). Let \( H\) be a \( k\) -module of finite rank and

an even symmetric pairing (which is non degenerate). We denote \( H^*\) the dual to \( H\) .

An important part of the Frobenius manifold structure is encoded in the existence of a potential function

which governs the multiplication structure on the manifold. In local coordinates, under suitable conditions, this function can be expressed as:

where \( Y_n\in (H^*)^{\otimes n}\) is a symmetric multilinear map

This system of multilinear forms defines a system of abstract correlation functions on the pair \( (H,g)\) , where \( H\) is a vector space equipped with a non-degenerate pairing \( g\) . These functions are symmetric and (in the context of Gromov–Witten theory) correspond to intersection numbers on the moduli space of stable maps. The Gromov–Witten invariants are generated from those multi-linear maps.

In the context of Gromov–Witten invariants, the potential function \( \Phi\) serves as the generating function for the intersection numbers of moduli spaces of holomorphic curves. The symmetric multilinear maps \( Y_n\) correspond to the correlation functions computed via topological field theory techniques.

The function \( \Phi\) satisfies the WDVV equations (associativity conditions on quantum cohomology), which govern the Frobenius manifold structure.

The maps \( Y_n\) define the higher-order correlation functions, whose values give the Gromov–Witten invariants.

This formulation provides a bridge between Frobenius manifolds, quantum cohomology, and Gromov–Witten theory, showing how the potential function encodes geometric intersection theory in an algebraic and formal power series framework. The abstract correlation functions \( Y_n\) serve as the structural foundation from which the Gromov–Witten invariants emerge, linking the geometry of moduli spaces with the algebraic structure of Frobenius manifolds.

In the framework of statistical geometry, we return to the study of statistical manifolds, emphasizing the discrete case of the exponential family. The fundamental relation governing this structure is given by the expansion:

where:

• The parameter \( \beta\) , given by \( \beta=(\beta_0,....,\beta_n)\in \mathbb{R}^{n+1}\) , provides an affine canonical parametrisation.
• The objects \( X_j(\omega)\) are directional co-vectors, forming a finite set of necessary and sufficient statistics, denoted by \( \mathcal{X}_n\) .
• The co-vectors, \( X_1(\omega),\dots ,X_{n}(\omega)\) are linearly independent co-vectors and we impose the normalization \( X_0(\omega)\equiv 1\) .

The family in (83) defines an analytic \( n\) -dimensional hypersurface within the statistical manifold, which can be uniquely determined by \( n+1\) points in general position.

We introduce the notion of Gromov–Witten invariants for statistical manifolds (GWS) as follows:

Equivalently, these invariants may be written in terms of the generating expansion:

These invariants naturally emerge as part of the potential function \( \tilde\bf{ \Phi}\) , which is identified with the Kullback–Liebler entropy function of the statistical system. The entropy function itself is expressed in the form:

This formulation suggests a deeper geometric and categorical interpretation of statistical learning, where the intersection theory of statistical structures plays a fundamental role. Within this perspective, the entropy function \( \tilde\bf{ \Phi}\) governs the geometry of statistical families, much like the potential function in Frobenius manifolds or Gromov–Witten theory encodes intersection numbers in moduli spaces of holomorphic curves.

Therefore, we state the following:

More precisely, \( \tilde\bf{ \Phi}\) arises as a generating function whose coefficients encode the multilinear maps \( \tilde{Y_n}\) , which define the GWS structure. These invariants characterize the underlying statistical geometry by capturing the intersection properties of statistical hypersurfaces.

The tangent fiber bundle is denoted by the quintuple \( (T\mathsf{S},\mathsf{S},\pi,G,F)\) , where:

• \( T\mathsf{S}\) is the total space of the tangent bundle, consisting of all possible tangent vectors to points in \( \mathsf{S}\) .
• \( \pi: T\mathsf{S}\to \mathsf{S}\) is a continuous surjective map that projects each tangent vector to its base point in \( \mathsf{S}\) .
• \( F\) is the fiber, representing the space of allowable tangent vectors at each point of \( \mathsf{S}\) .
• \( G\) is a Lie group that acts on the fibers, encoding the parallel transport structure within the statistical manifold.

For any point \( \rho\in \mathsf{S}\) , the tangent space at \( \rho\) , denoted \( T_{\rho}\mathsf{S}\) , is given by:

This identification follows from the fact that infinitesimal perturbations of a probability distribution \( \rho\) can be described by signed measures that respect the probabilistic constraints imposed by \( \mathsf{S}\) . The ideal \( I\) of the \( \sigma\) -algebra corresponds to the subspace of measures that do not contribute to the variations in probability distributions, ensuring consistency with the underlying measure-theoretic structure.

The Lie group \( G\) acts freely and transitively on each fiber of \( T\mathsf{S}\) , meaning that every element of the fiber can be transformed into any other through the group action. The action is given by:

where:

• \( f\) is an element of the total space \( T\mathsf{S}\) i.e., a tangent vector at some \( \rho\in \mathsf{S}\) .
• \( h\) is a parallel transport within the statistical manifold, representing an infinitesimal displacement in the space of probability distributions.

This affine structure on the fibers implies that the action of \( G\) acts as a translation group, ensuring a well-defined parallel transport mechanism in the space of probability measures. Such a structure is crucial for describing information geometry, as it encodes how probability distributions evolve under statistical transformations.

We define a learning process in terms of the Ackley–Hinton–Sejnowski method [20], which is based on minimizing the Kullback–Leibler divergence as a measure of distance between probability distributions. This process can be interpreted in a geometric framework as follows:

More precisely, let \( \mathsf{S}\) be a statistical manifold equipped with a fiber bundle structure \( (T\mathsf{S}, \mathsf{S}, \pi, G, F) \) . Consider a geodesic path \( \gamma: I \to \mathsf{S} \) , parametrized by an interval \( I \subset \mathbb{R}\) . The fiber over \( \gamma\) , denoted by \( F_{\gamma}\) , is assumed to decompose into two connected components:

Each component \( \gamma^+ \) and \( \gamma^- \) is contained in a totally geodesic submanifold of \( TS \) , denoted by \( E^+ \) and \( E^- \) , respectively:

In particular, the learning process is considered successful whenever the distance between the geodesic \( \gamma^+ \) and its orthogonal projection into \( E^+ \) decreases towards zero. That is, the process converges if:

This formulation provides a rigorous geometric criterion for assessing the success of learning, leveraging the underlying differential geometry of the statistical manifold.

In other words:

More formally, as was depicted in [25] (sec. 3) let us denote by \( \Upsilon\) the set of (centered) random variables over \( (\Omega,\mathcal{F},P_{\theta})\) which admit an expansion in terms of the scores under the following form:

By direct calculation, one finds that the log-likelihood \( \ell= ln\rho\) of the usual (parametric) families of probability distributions belongs to \( \Upsilon_p\) as well as the difference \( \ell -\ell^*\) of log-likelihood of two probabilities of the same family.

Being given a family of probability distributions such that \( \ell \in \Upsilon_P\) for any \( P\) , let \( \mathcal{U}_P\) , let us denote \( P^*\) the set such that \( \ell-\ell^*\in \Upsilon_p\) . Then, for any \( P^*\in \mathcal{U}_p\) , we define \( K(P,P^*)=\mathbb{E}_P[\ell - \ell^*]\) .

Similarly as in the classical (GW) case, the (GWS) count intersection numbers of the para-holomorphic curves generated by \( \gamma^+\) and \( \gamma^-\) . In fact, we have the following statement: