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Article

Advanced Manifold–Metric Pairs

Independent Researcher, 13 Allee Turcat Mery, 13008 Marseille, France
Mathematics 2025, 13(15), 2510; https://doi.org/10.3390/math13152510
Submission received: 1 July 2025 / Revised: 29 July 2025 / Accepted: 1 August 2025 / Published: 4 August 2025

Abstract

This article presents a novel mathematical formalism for advanced manifold–metric pairs, enhancing the frameworks of geometry and topology. We construct various D-dimensional manifolds and their associated metric spaces using functional methods, with a focus on integrating concepts from mathematical physics, field theory, topology, algebra, probability, and statistics. Our methodology employs rigorous mathematical construction proofs and logical foundations to develop generalized manifold–metric pairs, including homogeneous and isotropic expanding manifolds, as well as probabilistic and entropic variants. Key results include the establishment of metrizability for topological manifolds via the Urysohn Metrization Theorem, the formulation of higher-rank tensor metrics, and the exploration of complex and quaternionic codomains with applications to cosmological models like the expanding spacetime. By combining spacetime generalized sets with information-theoretic and probabilistic approaches, we achieve a unified framework that advances the understanding of manifold–metric interactions and their physical implications.

1. Introduction

Currently, natural systems, such as those in physics and cosmology, are often modelled using algebraic and topological frameworks, including subfields like differential geometry. Recognizing that a space is a mathematical structure of arbitrary finite dimension, where objects have relative positions, we seek to explore its abstract properties, such as generalized metrics or probabilistic structures. In differential geometry, a manifold M is a topological space that is locally Euclidean and equipped with properties like smoothness, which we generalize by associating with non-traditional metrics, m, such as higher-rank tensors.
A manifold is essentially a space that resembles locally a Euclidean space in the sense that M is covered by some arbitrary coordinate patches. These kinds of structures allow differentiation to be defined, but they do not distinguish intrinsically between different coordinate systems. In turn, the only concepts defined by the manifold structure are those that are independent of the choice of a coordinate system. This framework is rooted in differential geometry, algebraic geometry, and topology, where manifolds are equipped with a differentiable structure, enabling the definition of tangent spaces, tensor fields, and metrics [1,2]. Furthermore, analysis on manifolds provides tools to study smooth functions and their properties, such as differentiability and integrability, which are essential for our generalization [3,4]. To build such novel objects, we need tensors, and/or advanced generalized tensors [5]. Advanced generalized tensors are novel generalized tensors that are defined using not only objects from standard calculus, but also objects from fractional calculus [6]. In this manuscript, we will focus on the tensors that have already been explored. Note that manifolds can also be viewed from the perspective of category theory [5]. A formal formulation can be found in Hawking and Ellis [7]. However, in this study, the former formulation is sufficient.
The motivation of our studies lies in the fact that, as a mathematics and physics community, we are interested in building higher and more sophisticated mathematical and theoretical physics structures than the ones we currently have, with the goal of understanding our universe at a deeper level. Therefore, this study enriches recent developments of manifold–metric pair constructions [8].
Furthermore, a plethora of metrics exists, which we can construct beyond the standard ones, starting from the simplest ones classified by Bianchi [9], known as the Bianchi classification. These metrics are typically defined on Riemannian or pseudo-Riemannian manifolds, where properties such as symmetry and positive-definiteness (for Riemannian metrics) are well-established [10]. Interestingly, several studies are currently exploring the possibility that one of these new metrics possesses the property of extra dimensions. Notably, Polyakov [11], Deser and Zumino [12] have considered theories that include the ones discussed in a string theory framework. An easy way to express this argument is a set of different types of dimensions; there is the extra dimension set, as well as a spectrum of dimensions, which can be represented in a continuous manner, such as a probabilistic set of dimensions.
In this context, the manifold dimension refers to the number of independent coordinates defining M , typically a fixed integer D [13]. Extra dimensions extend this concept to higher-dimensional manifolds, such as the 11-dimensional manifolds considered in cosmological models. Spectrum dimensions arise from spectral properties of operators on M , such as eigenvalues of the metric tensor, which may characterize geometric or physical properties. Probabilistic sets of dimension incorporate stochastic structures, where dimensions are defined via probabilistic measures or entropic quantities, as explored in our generalized metric framework with higher-rank tensors, such as the ( 0 , 3 ) -tensor. These concepts are interconnected, as extra dimensions expand the manifold’s structure, spectrum dimensions provide analytical tools, and probabilistic dimensions introduce stochastic generalizations, collectively enabling novel manifold–metric pairs for applications in cosmology and beyond.
We designate a functional tensor as a generalized metric because it extends the role of standard metrics in measuring geometric properties on the manifold M , enabling the modeling of higher-order interactions, such as those in multiple geometric spaces, and topological spaces, probabilistic or entropic spacetimes, which are critical for applications in cosmology and theoretical physics. Like standard pseudo-Riemannian metrics, such tensors are symmetric and define geometric structures, but their trilinear or multilinear forms allow for richer interactions beyond the bilinear framework of traditional metrics [10,13,14].
In fact, even probabilistic concepts in gravitational theories have not been thoroughly studied in science, yet there is significant interest in the subject [15,16,17,18]. In 1996, Cho et al. [15] studied several properties of probabilistic metric spaces, while in 2018, Bachir [18] introduced and studied the concepts of probabilistic 1-Lipschitz maps. The latter authors used the space of all probabilistic 1-Lipschitz maps to present a new method for constructing a probabilistic metric completion. In particular, they expanded the concepts of probabilistic invariant metric group completion. Proving that the space of all probabilistic 1-Lipschitz maps is defined on a probabilistic invariant metric group, then the latter group can be endowed with a semigroup structure. This probabilistic invariant completes Menger groups, which were characterized by the space of all probabilistic 1-Lipschitz maps and which map functionals in the spirit of the classical Banach–Stone theorem [18]. Fast forward in time, the mathematical structure of manifold–metric pairs is still studied under different Riemannian and non-Riemannian geometries [19,20].
It is worth noting that we have some mild, yet important, evidence for probabilistic dimensions, as we can observe by the limits on the number of dimensions D of the spacetime continuum, which has been measured to be D 4 ± 0.1 from gravitational wave estimates [21]. These results suggest not only that our standard paradigm, which is described with ( 1 , 3 ) -dimensional manifolds, is valid, but also that there is further room for improvements and generalization of these concepts. These findings result in an inaugural interest in concepts of probabilistic spaces in the literature, which motivates us to explore further these probabilistic concepts and possibly apply them to D-spaces and D-dimensional manifolds. Note that these theories will be tested using current and future cosmological experiments with novel methods such as those found by Philcox and Slepian [22]. These advanced manifold–metric pairs can potentially be studied in the framework of functors of action theories [23,24] and in the framework of analytical dynamical analyses, as carried out in previous studies [25,26,27]. Note that as in the Ntelis [5] companion paper, we present some applications. In this study, we go beyond the concepts of generalization of functional metrics studied by Jleli and Samet [28].

2. Advanced Manifold–Metric Space Pairs

To interpret experimental data across scales, from astronomical to cosmological, physical systems are typically described using differential equations on (1,3)-dimensional manifolds (i.e., one temporal and three spatial dimensions) with corresponding metrics. These manifolds, often equipped with pseudo-Riemannian metrics of the Lorentzian signature, model phenomena in general relativity and cosmology. In this study, we generalize the concepts of manifold and metric spaces to accommodate higher-rank tensors, functional dependencies, and extended codomains, enabling broader applications in theoretical physics. We formalize this generalization through a theorem and supporting propositions, systematically constructing a generalized manifold–metric pair from the standard framework.

Generalized Manifold–Metric Pair Theorem

Theorem 1 (Generalized Manifold–Metric Pair). 
Let M D be a D-dimensional differentiable manifold. There exists a generalized manifold–metric pair ( M G , F ) , where M G is a manifold supporting a ( U , L ) -rank functional tensor F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U Q , with C α i being D-dimensional coordinate spaces, c C , f ( c ) : C R a smooth function, and Q the quaternion field. The corresponding line element is as follows:
d s U + L = F μ 1 μ L ν 1 ν U [ c , f ( c ) ] d c μ 1 d c μ L d c ν 1 d c ν U ,
where d c ν i = g ν i λ d c λ for a symmetric, positive-definite (0,2)-rank tensor g μ ν . The tensor F transforms as a ( U , L ) -tensor under coordinate changes and may be symmetric in its covariant and contravariant indices separately, ensuring a well-defined distance function.
Proof. 
We construct the generalized manifold–metric pair ( M G , F ) from the standard pair ( M , m ) through a series of propositions, each addressing a key step: extending the tensor rank, functionalizing the tensor, and generalizing the codomain.
Proposition 1 (Extension to (U,L)-Rank Tensor). 
A D-dimensional differentiable manifold M with a standard manifold–metric pair ( M , m μ ν : C C R ) , where m μ ν is a symmetric (0,2)-tensor, can be generalized to a proposition pair ( M , m μ 1 μ L ν 1 ν U : C μ 1 C ν U R ) , where m μ 1 μ L ν 1 ν U is a ( U , L ) -rank tensor.
Proof. 
Consider a standard manifold–metric pair:
( M , m ) ( M , m μ ν : C C R ) ,
where M is a D-dimensional differentiable manifold, C is the D-dimensional coordinate space with coordinates c μ , and m μ ν is a symmetric (0,2)-tensor, defining the line element as follows:
d s 2 = m μ ν ( c ) d c μ d c ν .
The metric m μ ν is symmetric ( m μ ν = m ν μ ) and transforms under coordinate changes c μ c μ = f μ ( c ) as follows:
m μ ν ( c ) = c ρ c μ c σ c ν m ρ σ ( c ) ,
ensuring invariance of d s 2 [2].
Define coordinate spaces C α i , each a copy of C, for α i { μ 1 , , μ L , ν 1 , , ν U } . Construct the tensor product spaces as follows:
C L : = C μ 1 C μ 2 C μ L ,
C U : = C ν 1 C ν 2 C ν U ,
forming C L C U = C μ 1 C ν U . Define a ( U , L ) -rank tensor m μ 1 μ L ν 1 ν U : C L C U R , which transforms as follows:
m μ 1 μ L ν 1 ν U = i = 1 U c ν i c ρ i j = 1 L c σ j c μ j m σ 1 σ L ρ 1 ρ U .
This tensor preserves the differentiable structure of M [1]. The manifold-metric pair becomes the following:
( M , m ) M , m μ 1 μ L ν 1 ν U : C μ 1 C ν U R .
Symmetry in indices (e.g., m μ 1 μ L ν 1 ν U = m μ π ( 1 ) μ π ( L ) ν 1 ν U for permutations π ) may be imposed as needed. □
Proposition 2 (Functional Tensor Generalization). 
The ( U , L ) -rank tensor m μ 1 μ L ν 1 ν U can be promoted to a functional tensor F μ 1 μ L ν 1 ν U [ c , f ( c ) ] , where f ( c ) : C R is a smooth function, mapping C μ 1 C ν U R .
Proof. 
Start with the pair ( M , m μ 1 μ L ν 1 ν U : C μ 1 C ν U R ) . Introduce a smooth function f ( c ) : C R , where c C . Define the functional tensor F μ 1 μ L ν 1 ν U [ c , f ( c ) ] , which depends on both coordinates c and the function f ( c ) . The tensor F inherits the transformation properties of m μ 1 μ L ν 1 ν U :
F μ 1 μ L ν 1 ν U = i = 1 U c ν i c ρ i j = 1 L c σ j c μ j F σ 1 σ L ρ 1 ρ U ,
adjusted for the functional dependence on f ( c ) , which transforms appropriately under coordinate changes [4]. The smoothness of f ( c ) ensures that F is differentiable, supporting a well-defined geometric structure on M . □
Proposition 3 (Codomain Generalization to Quaternions). 
The codomain of the functional tensor F μ 1 μ L ν 1 ν U [ c , f ( c ) ] can be extended from R to the quaternion field Q , defining a generalized manifold–metric pair ( M G , F ) .
Proof. 
The functional tensor F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U R can be generalized to map to Q , the quaternion field, which includes real and complex numbers as subsets. Define the generalized manifold M G , which may include additional structure (e.g., a bundle or algebraic structure) to support quaternionic outputs [29]. The pair is as follows:
( M G , F ) M G , F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U Q .
The line element is the following:
d s U + L = F μ 1 μ L ν 1 ν U [ c , f ( c ) ] d c μ 1 d c μ L d c ν 1 d c ν U ,
where d c ν i = g ν i λ d c λ for a symmetric, positive-definite (0,2)-rank tensor g μ ν . The quaternionic codomain accommodates non-positive-definite metrics, suitable for generalized physical theories. The transformation properties of F ensure invariance of the line element under coordinate changes. □
The generalized manifold–metric pair is constructed by combining Propositions 1–3. Start with the standard pair ( M , m μ ν ) . Proposition 1 extends the tensor to a ( U , L ) -rank tensor m μ 1 μ L ν 1 ν U . Proposition 2 functionalizes it to F μ 1 μ L ν 1 ν U [ c , f ( c ) ] . Proposition 3 extends the codomain to Q , yielding the following:
( M G , F ) M G , F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U Q .
The line element
d s U + L = F μ 1 μ L ν 1 ν U [ c , f ( c ) ] d c μ 1 d c μ L d c ν 1 d c ν U
is well-defined, with F transforming as a ( U , L ) -tensor, and g μ ν ensuring proper handling of covariant differentials. Symmetry in indices and differentiability are preserved, completing the construction. □
Remark 1 (Restriction to Real Numbers). 
Current physical theories are tested using real numbers, as observables are typically real-valued. To ensure testability, the codomain can be restricted to R + , simplifying the pair to a measurable metric space. However, complex numbers and quaternions have applications in electronics, wave mechanics, and field theories, suggesting future observables may involve C or Q .
Remark 2 (Beyond Other Generalizations). 
This generalization extends beyond families of manifolds such as Euclidean spaces R n , n-spheres S n , n-tori T n , real projective spaces R P n , complex projective spaces C P n , quaternionic projective spaces H P m , flag manifolds, Grassmann manifolds, Stiefel manifolds, or Finsler manifolds [30]. Unlike Finsler manifolds, which use a Minkowski functional with specific properties, our functional tensor F allows arbitrary inputs (e.g., quaternions and functions) and outputs to Q , accommodating diverse manifold structures.

3. Metrizability Conditions

A D-dimensional manifold M is metrizable if there exists a metric d : M × M R 0 that induces the manifold’s topology. Not all manifolds are metrizable, but for topological manifolds (Hausdorff, second-countable, and locally Euclidean), metrizability is guaranteed under certain conditions [31].

3.1. Metrizability Conditions in Standard Manifold–Metric Pairs

A topological space M is metrizable if and only if it satisfies the following conditions:
  • Hausdorff: For any two distinct points x , y M , there exist disjoint open neighborhoods U x and U y such that x U x and y U y .
    x , y M , x y , U x , U y open , U x U y = , x U x , y U y .
  • Second-Countable: The topology of M has a countable basis, i.e., there exists a countable collection B = { B 1 , B 2 , } of open sets such that every open set in M can be expressed as a union of elements of B .
    B = { B n } n N such that U open in M , U = B i U B i .
  • Locally Euclidean: For a D-dimensional manifold, every point x M has a neighborhood homeomorphic to an open subset of R D .
    x M , U open , x U , and ϕ : U V R D homeomorphism .
  • Paracompact: Every open cover of M has a locally finite open refinement. That is, for any open cover { U α } α A , there exists a locally finite open cover { V i } i I such that each V i U α for some α .
    { U α } α A , { V i } i I locally finite , i I , α A , V i U α .
According to the Urysohn Metrization Theorem, a second-countable, Hausdorff, and regular space is metrizable. Since topological manifolds are locally Euclidean, Hausdorff, and second-countable, they are metrizable [31]. Additionally, smooth manifolds admit a Riemannian metric, ensuring metrizability.

3.2. Standard Distance Function for (0,2)-Rank Tensor

For a smooth manifold M satisfying the above conditions, a (0,2)-rank tensor g i j d x i d x j that is symmetric ( g i j = g j i ) and positive-definite ( g i j v i v j > 0 for all non-zero v i T p M ) defines a Riemannian metric. The infinitesimal distance element is as follows:
d s 2 = g i j d x i d x j .
The standard distance function between points p , q M is as follows:
d ( p , q ) = inf γ a b g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t ,
where γ : [ a , b ] M is a smooth curve from γ ( a ) = p to γ ( b ) = q , and g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) = g i j ( γ ( t ) ) γ ˙ i ( t ) γ ˙ j ( t ) . This distance is non-negative, symmetric, satisfies the triangle inequality, and induces the manifold’s topology, as guaranteed by the metrizability conditions and paracompactness, which allows a partition of unity to construct g i j .

4. Metrizability for Generic (U,L)-Rank Tensors

For a generic (U,L)-rank tensor (with U covariant and L contravariant indices), metrizability requires constructing a distance function that induces the manifold’s topology. For example,
  • a (2,0)-rank tensor g i j is the inverse of a (0,2)-rank tensor and can be associated with a Riemannian metric;
  • tensors of rank (1,1), (0,3), etc., require additional structure to define a distance function equivalent to a (0,2)-rank tensor.

4.1. Additional Structure for (U,L)-Rank Tensors

To make a (U,L)-rank tensor function as a metric, it must define a distance function that induces the manifold’s topology. The additional structure required includes the following:
  • A (0,2)-Rank Tensor: A symmetric, positive-definite tensor g i j d x i d x j that defines a Riemannian metric.
  • Tensor Operations: Operations such as contractions or inversions to relate the (U,L)-rank tensor to g i j or its inverse g i j . For example,
    • for a (2,0)-rank tensor g i j , the additional structure is a (0,2)-rank tensor g i j such that g i k g k j = δ j i ;
    • for a (1,1)-rank tensor T j i , a (0,2)-rank tensor g i j such that T j i = g i k h k j for some tensor h i j , or T preserves g i j as an isometry ( T k i g i j T l k = g j l );
    • for a (0,3)-rank tensor T i j k , a vector field v k such that g i j = T i j k v k is symmetric and positive-definite.
  • Symmetry and Positive-Definiteness: The resulting (0,2)-rank tensor must be symmetric ( g i j = g j i ) and positive-definite ( g ( v , v ) > 0 for all non-zero v T p M ).
  • Smoothness: All tensors and auxiliary fields (e.g., v k and ω k ) must vary smoothly over M.
The manifold must still be Hausdorff, second-countable, and paracompact to ensure the existence of a global (0,2)-rank tensor via a partition of unity, satisfying the metrizability conditions.

4.2. Analysis of (0,3)-Rank Tensor Distance Construction

Consider a (0,3)-rank tensor m a b c defining an infinitesimal element as follows:
d s 3 = m a b c d x a d x b d x c .
A proposed distance function is as follows:
d ( p , q ) = inf γ a b m a b c ( γ ( t ) ) γ ˙ a ( t ) γ ˙ b ( t ) γ ˙ c ( t ) 1 / 3 d t ,
where the cube root is taken to yield a length-like quantity, and the absolute value ensures non-negativity. For example, in 2D with d s 2 D 3 = d x 3 + d y 3 , the tensor has components m 111 = 1 , m 222 = 1 , and others zero, so along a curve γ ( t ) = ( x ( t ) , y ( t ) ) ,
d ( p , q ) = inf γ a b x ˙ ( t ) 3 + y ˙ ( t ) 3 1 / 3 d t .
To function as a metric, the following additional structure is required:
  • Symmetry: The tensor m a b c must be symmetric ( m a b c = m b c a = m c a b ) to ensure d ( p , q ) = d ( q , p ) .
  • Positive-Definiteness: The expression m a b c v a v b v c 0 for all non-zero v a , with equality only at v a = 0 . In the 2D example, x ˙ 3 + y ˙ 3 can be negative, so the absolute value is necessary.
  • Topology Equivalence: The distance must induce the manifold’s topology, requiring m a b c to be compatible with local Euclidean charts.
  • Smoothness: The tensor m a b c must be smooth.
The construction can define a metric if these conditions are met, and the manifold is Hausdorff, second-countable, and paracompact.

4.3. Generalized Metrizability for (0,L)-Rank Tensors

For a (0,L)-rank tensor m i 1 i L to define a metric on a D-dimensional manifold M, it must induce a distance function that satisfies the manifold’s topology. The infinitesimal element is as follows:
d s L = m i 1 i L d x i 1 d x i L ,
and the distance function is the following:
d ( p , q ) = inf γ a b m i 1 i L ( γ ( t ) ) γ ˙ i 1 ( t ) γ ˙ i L ( t ) 1 / L d t ,
where γ : [ a , b ] M is a smooth curve from p to q, and the absolute value ensures non-negativity. The additional structure and conditions required are as follows:
  • Symmetry: The tensor m i 1 i L must be symmetric under all index permutations:
    m i σ ( 1 ) i σ ( L ) = m i 1 i L , permutations σ .
  • Positive-Definiteness: For all non-zero vectors v i T p M ,
    m i 1 i L v i 1 v i L 0 ,
    with equality only when v i = 0 .
  • Topology Equivalence: The distance d ( p , q ) must induce the manifold’s topology, requiring m i 1 i L to be compatible with local Euclidean charts.
  • Smoothness: The tensor m i 1 i L must be smooth over M.
  • Auxiliary Structure (Optional): If needed, L 2 vector fields v k 1 , , v k L 2 such that
    g i j = m i j k 1 k L 2 v k 1 v k L 2 ,
    where g i j is a symmetric, positive-definite (0,2)-rank tensor that can be used to define a standard Riemannian metric.
  • Manifold Conditions: The manifold must be on of the following:
    • Hausdorff: To separate points.
    • Second-Countable: To ensure a countable basis.
    • Paracompact: To allow a partition of unity for constructing m i 1 i L .
These conditions generalize the ( 0 , 3 ) -rank case, ensuring the ( 0 , L ) -rank tensor defines a valid metric.

4.4. Generalized Metrization Theorem

Theorem 2 (Generalized Metrization Theorem for Manifolds). 
A D-dimensional topological manifold M is metrizable with a (0,2)-rank tensor (Riemannian metric) if and only if it is Hausdorff, second-countable, and paracompact. The metric tensor g = g i j d x i d x j is symmetric, positive-definite, and induces the manifold’s topology via the associated distance function. For (U,L)-rank tensors, including (0,L)-rank tensors, metrizability requires constructing a distance function, possibly via a (0,2)-rank tensor, using additional structure such as auxiliary fields and tensor operations.

4.5. Proof of the Generalized Theorem

Proof. 
We prove that a D-dimensional topological manifold M is metrizable with a (0,2)-rank tensor if it is Hausdorff, second-countable, and paracompact, and extends to (0,L)-rank tensors.
Suppose M is a D-dimensional topological manifold, i.e., it is Hausdorff, second-countable, and locally Euclidean. By the Urysohn metrization theorem, a secondtyku, second-countable, Hausdorff, and regular space is metrizable with a distance function d : M × M R 0 . To show regularity, note that manifolds are locally Euclidean, so for any point x M and closed set C x , there exists a chart ( U , ϕ ) with x U and ϕ : U R D . Since R D is regular, we can separate ϕ ( x ) and ϕ ( C U ) with disjoint open sets, which pull back to M, ensuring regularity. Thus, M admits a metric d.
For a smooth manifold, we seek a (0,2)-rank tensor (Riemannian metric). Since M is paracompact, there exists a smooth partition of unity { ψ i } subordinate to a locally finite open cover { U i } , where each U i is homeomorphic to an open subset of R D . In each chart ( U i , ϕ i ) , define a local metric g i = j = 1 D d x j d x j (the Euclidean metric). Using the partition of unity, construct a global metric:
g = i ψ i g i .
Since { ψ i } is locally finite and ψ i 0 with i ψ i = 1 , and each g i is positive-definite, g is a smooth, symmetric, positive-definite (0,2)-rank tensor. The distance function induced by g is as follows:
d ( p , q ) = inf γ a b g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t ,
which generates a topology equivalent to the manifold’s topology, as the local charts ensure compatibility with Euclidean distances.
For a (0,L)-rank tensor m i 1 i L , we define the distance as follows:
d ( p , q ) = inf γ a b m i 1 i L ( γ ( t ) ) γ ˙ i 1 ( t ) γ ˙ i L ( t ) 1 / L d t .
This requires m i 1 i L to be symmetric, with m i 1 i L v i 1 v i L 0 for non-zero v i , and smooth. The manifold must be Hausdorff, second-countable, and paracompact. Optionally, m i 1 i L can be related to a (0,2)-rank tensor g i j via g i j = m i j k 1 k L 2 v k 1 v k L 2 , using L 2 smooth vector fields. The construction of m i 1 i L or g i j uses the partition of unity, relying on paracompactness.
Conversely, if M is metrizable with a (0,2)-rank tensor, it must be Hausdorff and second-countable (as these are properties of metric spaces). Paracompactness is necessary for smooth manifolds to ensure the existence of a partition of unity, which is required to construct the global metric g or m i 1 i L . Thus, the conditions are necessary and sufficient.
Hence, a D-dimensional topological manifold is metrizable with a (0,2)-rank tensor if and only if it is Hausdorff, second-countable, and paracompact. For (0,L)-rank tensors, metrizability is achieved by defining a suitable distance function with the above conditions. □

4.6. Generalized Metrizability for (U,L)-Rank Tensors

For a (U,L)-rank tensor F i 1 i L j 1 j U to define a metric on a D-dimensional manifold M, consider the infinitesimal element as follows:
d s U + L = F i 1 i L j 1 j U d x i 1 d x i L d x j 1 d x j U ,
where d x i k are differentials of coordinates, and d x j k are differentials of covector components, interpreted via a (0,2)-rank tensor g i j such that d x j k = g j k k d x k . The distance function is as follows:
d ( p , q ) = inf γ a b F i 1 i L j 1 j U ( γ ( t ) ) γ ˙ i 1 ( t ) γ ˙ i L ( t ) γ ˙ j 1 ( t ) γ ˙ j U ( t ) 1 / ( U + L ) d t ,
where γ : [ a , b ] M is a smooth curve from p to q, and γ ˙ j k = g j k k γ ˙ k . The additional structure and conditions required are as follows:
  • Symmetry: The tensor F i 1 i L j 1 j U must be symmetric under permutations of covariant indices { i 1 , , i L } and contravariant indices { j 1 , , j U } separately:
    F i σ ( 1 ) i σ ( L ) j τ ( 1 ) j τ ( U ) = F i 1 i L j 1 j U ,
    for permutations σ and τ , to ensure d ( p , q ) = d ( q , p ) .
  • Positive-Definiteness: For all non-zero vectors v i T p M and covectors w j T p * M ,
    F i 1 i L j 1 j U v i 1 v i L w j 1 w j U 0 ,
    with equality only when v i = 0 or w j = 0 . If γ ˙ j k = g j k k γ ˙ k , the integrand must be non-negative.
  • Topology Equivalence: The distance d ( p , q ) must induce the manifold’s topology, requiring F i 1 i L j 1 j U to be compatible with local Euclidean charts.
  • Smoothness: The tensor F i 1 i L j 1 j U and any auxiliary fields (e.g., g i j ) must be smooth over M.
  • Auxiliary Structure: A (0,2)-rank tensor g i j , symmetric and positive-definite, to relate covariant and contravariant components: γ ˙ j k = g j k k γ ˙ k . Optionally, U + L 2 fields (e.g., L 1 vector fields v k 1 , , v k L 1 and U 1 covector fields ω l 1 , , ω l U 1 ) such that
    g i j = F i j k 1 k L 1 l 1 l U v k 1 v k L 1 ω l 1 ω l U ,
    where g i j is a Riemannian metric, if needed to simplify the construction.
  • Manifold Conditions: The manifold must be one of the following:
    • Hausdorff: To separate points.
    • Second-Countable: To ensure a countable basis.
    • Paracompact: To allow a partition of unity for constructing F i 1 i L j 1 j U .
These conditions ensure the ( U , L ) -rank tensor defines a valid metric, generalizing the ( 0 , L ) -rank case.

4.7. Generalized Metrization Theorem

Theorem 3 (Generalized Metrization Theorem for Manifolds). 
A D-dimensional topological manifold M is metrizable with a (0,2)-rank tensor (Riemannian metric) if and only if it is Hausdorff, second-countable, and paracompact. The metric tensor g = g i j d x i d x j is symmetric, positive-definite, and induces the manifold’s topology via the associated distance function. For (U,L)-rank tensors, metrizability requires constructing a distance function, possibly via a (0,2)-rank tensor, using additional structure such as auxiliary fields and tensor operations.

4.8. Proof of the Generalized Metrization Theorem

Proof. 
We prove that a D-dimensional topological manifold M is metrizable with a (0,2)-rank tensor if it is Hausdorff, second-countable, and paracompact, and extends to (U,L)-rank tensors.
Suppose M is a D-dimensional topological manifold, i.e., it is Hausdorff, second-countable, and locally Euclidean. By the Urysohn metrization theorem, a second-countable, Hausdorff, and regular space is metrizable with a distance function d : M × M R 0 . To show regularity, note that manifolds are locally Euclidean, so for any point x M and closed set C x , there exists a chart ( U , ϕ ) with x U and ϕ : U R D . Since R D is regular, we can separate ϕ ( x ) and ϕ ( C U ) with disjoint open sets, which pull back to M, ensuring regularity. Thus, M admits a metric d.
For a smooth manifold, we seek a (0,2)-rank tensor (Riemannian metric). Since M is paracompact, there exists a smooth partition of unity { ψ i } subordinate to a locally finite open cover { U i } , where each U i is homeomorphic to an open subset of R D . In each chart ( U i , ϕ i ) , we define a local metric g i = j = 1 D d x j d x j (the Euclidean metric). Using the partition of unity, we construct a global metric:
g = i ψ i g i .
Since { ψ i } is locally finite and ψ i 0 with i ψ i = 1 , and each g i is positive-definite, g is a smooth, symmetric, positive-definite (0,2)-rank tensor. The distance function induced by g:
d ( p , q ) = inf γ a b g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t ,
which generates a topology equivalent to the manifold’s topology, as local charts ensure compatibility with Euclidean distances.
For a (U,L)-rank tensor F i 1 i L j 1 j U , we define the distance as follows:
d ( p , q ) = inf γ a b F i 1 i L j 1 j U ( γ ( t ) ) γ ˙ i 1 ( t ) γ ˙ i L ( t ) γ ˙ j 1 ( t ) γ ˙ j U ( t ) 1 / ( U + L ) d t ,
where γ ˙ j k = g j k k γ ˙ k using a (0,2)-rank tensor g i j . This requires F i 1 i L j 1 j U to be symmetric in its covariant and contravariant indices, with m i 1 i L j 1 j U v i 1 v i L w j 1 w j U 0 , and smooth. The manifold must be Hausdorff, second-countable, and paracompact. Optionally, F i 1 i L j 1 j U can be related to a (0,2)-rank tensor g i j via contractions with L 1 vector fields and U 1 covector fields. The construction of F i 1 i L j 1 j U or g i j uses the partition of unity, relying on paracompactness.
Conversely, if M is metrizable with a (0,2)-rank tensor, it must be Hausdorff and second-countable (as these are properties of metric spaces). Paracompactness is necessary for smooth manifolds to ensure the existence of a partition of unity, which is required to construct the global metric g or F i 1 i L j 1 j U . Thus, the conditions are necessary and sufficient.
Hence, a D-dimensional topological manifold is metrizable with a (0,2)-rank tensor if and only if it is Hausdorff, second-countable, and paracompact. For (U,L)-rank tensors, metrizability is achieved by defining a suitable distance function with the above conditions. □

4.9. Comparison of the Generalized Manifold–Metric Pair with a Hessian Structure

A Hessian structure on a manifold M of dimension D is defined by a smooth potential function ϕ : M R , whose Hessian is a (U,L)-rank tensor provided by the following:
H μ 1 μ L ν 1 ν U = μ 1 μ L ν 1 ν U ϕ ,
where μ i = / c μ i are derivatives with respect to contravariant coordinates, and ν j = / c ν j are derivatives with respect to covariant coordinates. This Hessian is equated to the generalized metric tensor:
H μ 1 μ L ν 1 ν U = F μ 1 μ L ν 1 ν U [ c , f ( c ) ] .
The line element for the Hessian structure is as follows:
d s U + L = H μ 1 μ L ν 1 ν U d c μ 1 d c μ L d c ν 1 d c ν U ,
and the distance function is
d ( p , q ) = inf γ a b H μ 1 μ L ν 1 ν U ( γ ( t ) ) γ ˙ μ 1 ( t ) γ ˙ μ L ( t ) γ ˙ ν 1 ( t ) γ ˙ ν U ( t ) 1 / ( U + L ) d t ,
where γ ˙ ν k = g ν k λ γ ˙ λ using a symmetric, positive-definite (0,2)-rank tensor g μ ν . The manifold typically requires a structure supporting mixed derivatives, such as a flat connection or a bundle structure.
The generalized manifold–metric pair is as follows:
( M G , F ) M G , F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U Q ,
with the same line element and distance function. Since F μ 1 μ L ν 1 ν U = H μ 1 μ L ν 1 ν U , the two structures are equivalent, with the Hessian providing a specific construction of F via the potential ϕ . We can compare the two as follows:
  • Tensor Rank: Both H μ 1 μ L ν 1 ν U and F μ 1 μ L ν 1 ν U are (U,L)-rank tensors, making them structurally identical.
  • Line Element: The line elements are identical, as F = H , both of order U + L .
  • Functional Dependence: The Hessian depends solely on ϕ , while F includes f ( c ) , which may be incorporated into ϕ or represent coordinate parameterization.
  • Symmetry: Both tensors are symmetric in their covariant and contravariant indices due to the commutativity of partial derivatives in H , satisfying the symmetry requirement for F .
  • Positive-Definiteness: Both require F μ 1 μ L ν 1 ν U v μ 1 v μ L w ν 1 w ν U 0 , ensured by the absolute value in the distance function and the auxiliary metric g μ ν .
  • Applications: The Hessian structure is used in information geometry and statistical mechanics, while F supports cosmological models like FLRW, where F μ ν EHI [ t , a ( t ) ] (with U = 0 , L = 2 ) is unlikely to be a Hessian unless a suitable ϕ is constructed.
The equivalence F = H implies the generalized manifold–metric pair is a Hessian structure, with ϕ defining the metric tensor, suitable for applications requiring higher-rank tensors.

5. Generic Advanced Manifold–Metric Pair Applications

This section applies the generalized manifold–metric pair, as defined in Section 2, to physical systems in arbitrary dimensions, including spatial and spacetime manifolds. We formalize these applications through a theorem supported by propositions and examples, demonstrating the construction of higher-rank tensor metrics with real or quaternionic codomains, suitable for diverse physical contexts such as cosmology.

5.1. Applications of Generalized Manifold–Metric Pairs

Theorem 4 (Applications of Generalized Manifold–Metric Pairs). 
Let M D be a D-dimensional differentiable manifold. The generalized manifold–metric pair ( M G , F ) , where F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U Q is a ( U , L ) -rank functional tensor, can be applied to construct metrics for physical systems, including the following:
  • Higher-rank ( 0 , L ) -tensor metrics with real codomain R + on Euclidean manifolds.
  • ( 0 , L ) -tensor metrics with complex or quaternionic codomains for quantum or field-theoretic applications.
  • Functional ( 0 , L ) -tensor metrics with time-dependent scaling for cosmological spacetimes, such as the Friedmann–Lemaître–Robertson–Walker (FLRW) model.
The line element is provided as follows:
d s U + L = F μ 1 μ L ν 1 ν U [ c , f ( c ) ] d c μ 1 d c μ L d c ν 1 d c ν U ,
where d c ν i = g ν i λ d c λ for a symmetric, positive-definite ( 0 , 2 ) -rank tensor g μ ν , and F transforms as a ( U , L ) -tensor, with optional symmetry in indices.
Proof. 
The theorem is established through three propositions, each constructing a specific class of generalized manifold–metric pairs, followed by examples illustrating their application. The propositions address higher-rank tensors, complex/quaternionic codomains, and cosmological applications, ensuring the theorem’s claims are systematically verified. □
Proposition 4 (Higher-Rank Tensor Metrics with Real Codomains). 
On a D-dimensional differentiable manifold M D , a generalized manifold–metric pair ( M , m μ 1 μ L : C μ 1 C μ L R + ) can be constructed with a symmetric ( 0 , L ) -rank tensor m μ 1 μ L , defining a line element d s L = m μ 1 μ L d c μ 1 d c μ L .
Proof. 
Consider a D-dimensional differentiable manifold M D with coordinate space C and coordinates c μ . Define a ( 0 , L ) -rank tensor m μ 1 μ L : C μ 1 C μ L R + , symmetric under permutations of indices, i.e., m μ π ( 1 ) μ π ( L ) = m μ 1 μ L for any permutation π . The tensor transforms under coordinate changes c μ c μ = f μ ( c ) as follows:
m μ 1 μ L = j = 1 L c σ j c μ j m σ 1 σ L .
The line element is as follows:
d s L = m μ 1 μ L d c μ 1 d c μ L ,
which is invariant under coordinate transformations due to the tensor’s transformation properties [1]. The codomain R + ensures a measurable metric, and symmetry guarantees that the line element is well-defined for physical applications. The cubic form d s 3 can take positive, negative, or zero values, which is expected for a non-traditional (0,3)-tensor metric and does not pose an issue as it generalizes the standard pseudo-Riemannian metric, defined as a symmetric (0,2)-tensor [10,13], to capture higher-order interactions, which is consistent with advanced geometric frameworks [14]. □
Example 1 (Two-Dimensional Manifold with a (0,3)-Tensor Metric). 
Consider a two-dimensional Euclidean manifold M 2 D with coordinates c μ = ( x , y ) . Define a ( 0 , 3 ) -tensor metric as follows:
m μ ν ρ = 1 if ( μ , ν , ρ ) = ( 1 , 1 , 1 ) or ( 2 , 2 , 2 ) , 0 otherwise ,
which is symmetric under index permutations. The line element is as follows:
d s 3 = m μ ν ρ d c μ d c ν d c ρ = d x 3 + d y 3 .
This pair ( M 2 D , m μ ν ρ ) is a special case of Proposition 4 with L = 3 , U = 0 , and codomain R , generalizing the Pythagorean theorem.
Proposition 5 (Complex/Quaternionic Codomain Metrics). 
On a D-dimensional differentiable manifold M D , a generalized manifold–metric pair ( M , m μ 1 μ L : C μ 1 C μ L C ) can be constructed with a symmetric ( 0 , L ) -rank tensor m μ 1 μ L , where C Q , suitable for quantum or field-theoretic applications.
Proof. 
Start with a D-dimensional differentiable manifold M D and coordinate space C with coordinates c μ . Define a ( 0 , L ) -rank tensor m μ 1 μ L : C μ 1 C μ L C , symmetric under index permutations. The tensor transforms as follows:
m μ 1 μ L = j = 1 L c σ j c μ j m σ 1 σ L ,
preserving the differentiable structure [1]. The line element is as follows:
d s L = m μ 1 μ L d c μ 1 d c μ L C .
The complex codomain, a subset of Q , supports applications in quantum field theory, where complex-valued metrics model wave-like phenomena [29]. The symmetry and transformation properties ensure a consistent geometric framework. □
Example 2 (Three-Dimensional Manifold with a Complex (0,3)-Tensor Metric). 
Consider a three-dimensional manifold M 3 D with coordinates c μ = ( x , y , z ) . Define a ( 0 , 3 ) -tensor metric as follows:
m μ ν ρ = i if ( μ , ν , ρ ) = ( 1 , 1 , 1 ) , 1 if ( μ , ν , ρ ) = ( 2 , 2 , 2 ) or ( 3 , 3 , 3 ) , 0 otherwise ,
yielding
d s 3 = i d x 3 + d y 3 + d z 3 .
This pair ( M 3 D , m μ ν ρ ) is a special case of Proposition 5 with L = 3 , U = 0 , and codomain C .
Example 3 (Seven-Dimensional Manifold with a Functional (0,7)-Tensor Metric). 
Consider a seven-dimensional manifold M 7 D with coordinates c μ = ( τ , t , x 1 , , x 5 ) . Define a functional ( 0 , 7 ) -tensor metric as follows:
m μ 1 μ 7 ( c ) = a 7 ( τ ) i if μ 1 = = μ 7 = 1 , b 7 ( τ , t , x 1 ) if μ 1 = = μ 7 = 2 , f 7 ( x ; k ) if μ 1 = = μ 7 { 3 , , 7 } , 0 otherwise ,
where a ( τ ) is a scale factor, b ( τ , t , x 1 ) is a temporal scaling function, and f ( x ; k ) models spatial curvature with k { 1 , 0 , 1 } . The line element is as follows:
d s 7 = a 7 ( τ ) i d τ 7 + b 7 ( τ , t , x 1 ) d t 7 + f 7 ( x ; k ) d x 7 C .
This pair exemplifies Proposition 5 with L = 7 , U = 0 , and codomain C , incorporating functional dependence suitable for cosmological applications [19].

An Expanding Homogeneous and Isotropic Manifold and Metric Pair

Proposition 6 (Cosmological Manifold–Metric Pair). 
The generalized manifold–metric pair ( M EHI , F μ ν EHI [ t , a ( t ) ] : C C R + ) can be constructed to describe the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime, with a ( 0 , 2 ) -rank functional tensor F μ ν EHI incorporating a time-dependent scale factor a ( t ) .
Proof. 
Consider a four-dimensional differentiable manifold M EHI representing a homogeneous and isotropic spacetime. Define a ( 0 , 2 ) -rank functional tensor F μ ν EHI [ t , a ( t ) ] : C C R + as follows:
F μ ν EHI [ t , a ( t ) ] = c 2 0 0 0 0 a 2 ( t ) 0 0 0 0 a 2 ( t ) 0 0 0 0 a 2 ( t ) ,
where a ( t ) is the scale factor, c is the speed of light, and the metric has Lorentzian signature ( , + , + , + ) . The line element is as follows:
d s 2 = c 2 d t 2 + a 2 ( t ) d x i d x j δ i j ,
where δ i j is the Kronecker delta. The tensor is symmetric ( F μ ν EHI = F ν μ EHI ) and transforms as a ( 0 , 2 ) -tensor under coordinate changes [2]. The functional dependence on a ( t ) , constrained by cosmological observations [32], preserves the homogeneity and isotropy of the FLRW spacetime, making the pair ( M EHI , F EHI ) a special case of the generalized framework with U = 0 and L = 2 . □
Example 4 (FLRW Spacetime). 
The manifold–metric pair ( M EHI , F EHI ) is defined as follows:
( M EHI , F EHI ) M EHI , F μ ν EHI [ t , a ( t ) ] : C C R + ,
with the metric tensor from Equation (55). This pair, with U = 0 , L = 2 , and codomain R + , represents the FLRW spacetime, consistent with Proposition 6.

5.2. Comparison of a FLRW Manifold–Metric Pair with a Warped Product Structure

This subsection demonstrates that the Friedmann–Lemaître–Robertson–Walker (FLRW) manifold–metric pair, as a special case of the generalized framework in Section 2, can be equivalently represented as a warped product manifold. We formalize this equivalence through a theorem, supported by a proposition and an example, highlighting the structural alignment between the functional tensor F μ ν EHI [ t , a ( t ) ] and the warped product metric.
Theorem 5 (FLRW as a Warped Product Manifold).
The FLRW manifold–metric pair ( M EHI , F μ ν EHI [ t , a ( t ) ] : C C R + ) with a line element, using Equation (56), which is equivalent to a warped product manifold M = B × f F , where B = R is the time coordinate with metric g B = c 2 d t 2 , F = R 3 is the spatial slice with metric g F = δ i j d x i d x j , and f ( t ) = a ( t ) is the warping function. The functional tensor F μ ν EHI [ t , a ( t ) ] captures the time-dependent scaling of the spatial metric, preserving the homogeneity and isotropy of FLRW spacetime.
Proof. 
The proof is established by constructing the FLRW manifold–metric pair as a warped product through a proposition, followed by an example verifying the construction for Euclidean spatial slices and general curvatures. The equivalence is shown by matching the line element and tensor structure of F μ ν EHI to the warped product metric. □
Proposition 7 (Construction of FLRW as a Warped Product). 
The FLRW manifold–metric pair ( M EHI , F μ ν EHI [ t , a ( t ) ] ) with a metric matrix, from Equation (55), can be constructed as a warped product manifold M = B × f F , where B = R has metric g B = c 2 d t 2 , F = R 3 has metric g F = δ i j d x i d x j , and the warping function f ( t ) = a ( t ) is the scale factor, yielding the line element using Equation (56).
Proof. 
Consider a four-dimensional differentiable manifold M EHI representing the FLRW spacetime. A warped product manifold is defined as M = B × f F , where B and F are manifolds with metrics g B and g F , respectively, and f : B R + is a smooth warping function. The warped product metric is as follows:
g = π * g B + f 2 σ * g F ,
where π : B × F B and σ : B × F F are projection maps, and π * g B and σ * g F are the pullbacks of the respective metrics [2]. In local coordinates, with B having coordinates ( x μ ) and metric g B μ ν , and F having coordinates ( y i ) and metric g F i j , the line element is as follows:
d s 2 = g B μ ν d x μ d x ν + f 2 ( x ) g F i j d y i d y j .
For the FLRW spacetime, we define the following:
  • Base manifold B = R , with coordinate t and metric g B = c 2 d t 2 .
  • Fiber manifold F = R 3 , with coordinates ( x 1 , x 2 , x 3 ) and metric g F = δ i j d x i d x j , where δ i j is the Kronecker delta.
  • Warping function f ( t ) = a ( t ) , i.e., the scale factor, depending only on the time coordinate.
The warped product manifold is M EHI = R × R 3 , and the metric is as follows:
g = π * ( c 2 d t 2 ) + a 2 ( t ) σ * ( δ i j d x i d x j ) .
The line element becomes the following:
d s 2 = c 2 d t 2 + a 2 ( t ) d x i d x j δ i j ,
matching the FLRW line element. The functional tensor F μ ν EHI [ t , a ( t ) ] is as follows:
F μ ν EHI [ t , a ( t ) ] = c 2 0 0 0 0 a 2 ( t ) 0 0 0 0 a 2 ( t ) 0 0 0 0 a 2 ( t ) ,
which is symmetric ( F μ ν EHI = F ν μ EHI ) and transforms as a (0,2)-tensor under coordinate changes [2]. The functional dependence on a ( t ) aligns with the warping function f ( t ) = a ( t ) , and the metric’s Lorentzian signature ( , + , + , + ) is compatible with the pseudo-Riemannian warped product structure, ensuring homogeneity and isotropy of the spatial slices as R 3 is homogeneous and isotropic, and a ( t ) scales δ i j uniformly [32]. □
Example 5 (FLRW as a Warped Product with General Curvature). 
Consider the FLRW manifold–metric pair ( M EHI , F μ ν EHI [ t , a ( t ) ] ) as in Proposition 7. For Euclidean spatial slices ( k = 0 ), the manifold is M EHI = R × R 3 with the following:
  • Base manifold B = R , with metric g B = c 2 d t 2 .
  • Fiber manifold F = R 3 , with metric g F = δ i j d x i d x j .
  • Warping function f ( t ) = a ( t ) .
The line element from Equation (56) corresponds to the warped product metric g = π * ( c 2 d t 2 ) + a 2 ( t ) σ * ( δ i j d x i d x j ) . For general FLRW spacetimes with curvature k 0 , the fiber manifold F may be a 3-dimensional spherical ( k = 1 ) or hyperbolic ( k = 1 ) space, with metric g F provided by the appropriate spatial metric (e.g., g F = d χ 2 + sin 2 χ ( d θ 2 + sin 2 θ d ϕ 2 ) for k = 1 ). The warped product structure remains:
d s 2 = c 2 d t 2 + a 2 ( t ) g F i j d y i d y j ,
where g F reflects the curvature. The functional tensor F μ ν EHI [ t , a ( t ) ] adapts to the fiber metric, maintaining the warped product form and the time-dependent scaling of a ( t ) . The generalized manifold–metric pair captures this structure, as the functional dependence aligns with the warping function, and the Lorentzian signature is compatible with pseudo-Riemannian warped products [2].

6. Generalized ( D τ , D x ) -Dimensional and Probabilistic Manifold– Metric Pairs

This section applies the generalized manifold–metric pair framework from Section 2 to construct ( D τ , D x ) -dimensional manifolds and their probabilistic extensions, incorporating temporal, spatial, and probabilistic dimensions. We formalize these constructions through a theorem, supported by propositions and examples, demonstrating their consistency with physical and cosmological models. Equations are labeled uniquely and referenced to avoid repetition.
Theorem 6 (Generalized ( D τ , D x ) -Dimensional and Probabilistic Manifold–Metric Pairs). 
Let M D be a D-dimensional differentiable manifold. The generalized manifold–metric pair ( M G , F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U Q ) can be applied to construct the following:
  • A ( D τ , D x ) -dimensional submanifold-metric pair with a pseudo-Riemannian ( 0 , 2 ) -tensor metric, where D τ and D x are temporal and spatial dimensions.
  • A Gaussianly perturbed FLRW (GPFLRW) spacetime with probabilistic perturbations via Gaussian distributions.
  • An extended probabilistic FLRW (EPFLRW) spacetime with additional probabilistic dimensions.
  • A homogeneous, isotropic, probabilistic, expanding spacetime (HIPEST) with a probability function.
  • Generalized curved probabilistic spacetimes (fgcPST and sgcPST) with curvature and probabilistic scaling, including infinite-dimensional extensions.
The line element is provided by the following:
d s U + L = F μ 1 μ L ν 1 ν U [ c , f ( c ) ] d c μ 1 d c μ L d c ν 1 d c ν U ,
where d c ν i = g ν i λ d c λ for a symmetric ( 0 , 2 ) -tensor g μ ν , and F transforms as a ( U , L ) -tensor with optional symmetry in indices.
Proof. 
The theorem is established through five propositions, each constructing a specific class of generalized manifold–metric pairs, followed by examples illustrating their application. The propositions address ( D τ , D x ) -dimensional manifolds, probabilistic perturbations, extended probabilistic dimensions, homogeneous isotropic probabilistic spacetimes, and curved probabilistic spacetimes, ensuring consistency with Equation (12). Repeated equations are referenced to avoid duplication. □

6.1. ( D τ , D x ) -Dimensional Manifold–Metric Pair

Proposition 8 ( ( D τ , D x ) -Dimensional Manifold–Metric Pair). 
On a D-dimensional differentiable manifold M D , a submanifold–metric pair ( M ( D τ , D x ) , g α β ( D τ , D x ) : C C R ) can be constructed, where M ( D τ , D x ) = M D τ M D x M D has D τ temporal and D x spatial dimensions, and g α β ( D τ , D x ) is a symmetric, pseudo-Riemannian ( 0 , 2 ) -tensor.
Proof. 
Consider a D-dimensional differentiable manifold M D with coordinate space C and coordinates c μ . Define a submanifold M ( D τ , D x ) = M D τ M D x M D , where D τ + D x D , with coordinates c = { τ , x } , τ = ( τ 1 , , τ D τ ) , and x = ( x 1 , , x D x ) . The metric g α β ( D τ , D x ) : C C R is symmetric ( g α β ( D τ , D x ) = g β α ( D τ , D x ) ) and transforms as a ( 0 , 2 ) -tensor under coordinate changes c α c α = f α ( c ) :
g α β ( D τ , D x ) = c ρ c α c σ c β g ρ σ ( D τ , D x ) .
The line element is as follows:
d s ( D τ , D x ) 2 = g α β ( D τ , D x ) ( c ) d c α d c β ,
which is invariant under coordinate transformations [2]. The metric may be pseudo-Riemannian to accommodate temporal and spatial components, and the submanifold inherits the differentiable structure of M D , ensuring consistency [1]. □
Example 6 (Perturbed Anti-de-Sitter Manifold). 
Consider a ( D τ , D r ) -dimensional submanifold M ( D τ , D r ) with coordinates c = { τ , r } , illustrating Proposition 8. The line element is as follows:
d s ( D τ , D r ) 2 = a 2 ( c ) e 2 Ψ ( c ) d τ D τ 2 + e 2 Φ ( c ) d r D r 2 ,
where a ( c ) is the scale factor, e 2 Ψ ( c ) 1 + 2 Ψ ( c ) , e 2 Φ ( c ) 1 2 Φ ( c ) are perturbation terms, d τ D τ 2 = g α β ( D τ ) d τ α d τ β , and d r D r 2 = g α β ( D r ) d r α d r β with curvature. The spatial component is as follows:
d r D r 2 = d r 2 + S k 2 ( r ) d Ω D r 1 2 ,
where
S k ( r ) = | k | 1 / 2 sin ( r k ) k > 0 , r k = 0 , | k | 1 / 2 sinh ( r | k | ) k < 0
and d Ω D r 1 2 = g i j ( D r 1 ) d θ i d θ j with
g i j ( D r 1 ) = diag 1 , sin 2 θ 1 , sin 2 θ 1 sin 2 θ 2 , , i = 1 D r 2 sin 2 θ i ,
where θ i { θ 1 , , θ D r 1 } , r R + , θ i [ 1 , D r 2 ] [ 0 , π ] , and θ D r 1 [ 0 , 2 π ] . The temporal component is as follows:
d τ D τ 2 = d τ 2 + S k τ 2 ( τ ) d Ω D τ 1 2 ,
with S k τ ( τ ) as in Equation (69) and d Ω D τ 1 2 = g i j ( D τ 1 ) d ϕ i d ϕ j with
g i j ( D τ 1 ) = diag 1 , sin 2 ϕ 1 , sin 2 ϕ 1 sin 2 ϕ 2 , , i = 1 D τ 2 sin 2 ϕ i ,
where ϕ i { ϕ 1 , , ϕ D τ 1 } , τ R + , ϕ i [ 1 , D τ 2 ] [ 0 , π ] , and ϕ D τ 1 [ 0 , 2 π ] . This example, with U = 0 , L = 2 , and codomain R , illustrates Proposition 8 for a perturbed anti-de-Sitter spacetime, reducing to standard Minkowski spacetime for ( D τ , D r ) = ( 1 , 3 ) [19,33].

6.2. Probabilistic Manifold–Metric Pairs

Proposition 9 (Gaussianly Perturbed FLRW Spacetime). 
On a four-dimensional manifold M ( 1 , 3 ) with coordinates c μ = ( t , x 1 , x 2 , x 3 ) , a Gaussianly perturbed FLRW (GPFLRW) manifold–metric pair ( M ( 1 , 3 ) , F μ ν [ t , x , G ( Ψ ˜ ) , G ( Φ ˜ ) ] : C C R ) can be constructed, where F μ ν incorporates Gaussian probability distributions for perturbations.
Proof. 
Consider a four-dimensional manifold M ( 1 , 3 ) with coordinates c μ = ( t , x 1 , x 2 , x 3 ) . Start with the perturbed FLRW line element:
d s PFLRW 2 = e 2 Ψ ( t , x ) c 2 d t 2 + a 2 ( t ) e 2 Φ ( t , x ) d x i d x j δ i j ,
where Ψ ( t , x ) and Φ ( t , x ) are scalar potentials, a ( t ) is the scale factor, and δ i j is the Kronecker delta. Define Gaussian perturbations as follows:
e 2 Ψ ( t , x ) G ( Ψ ˜ ; Ψ ˜ ¯ , σ Ψ ˜ ) = 1 2 π σ Ψ ˜ exp 1 2 Ψ ˜ ( t , x ) Ψ ˜ ¯ σ Ψ ˜ 2 ,
e 2 Φ ( t , x ) G ( Φ ˜ ; Φ ˜ ¯ , σ Φ ˜ ) = 1 2 π σ Φ ˜ exp 1 2 Φ ˜ ( t , x ) Φ ˜ ¯ σ Φ ˜ 2 ,
where Ψ ˜ ¯ , σ Ψ ˜ , Φ ˜ ¯ , and σ Φ ˜ are means and standard deviations. Redefine the potentials as follows:
Ψ ( t , x ) = 1 2 ln G ( Ψ ˜ ; Ψ ˜ ¯ , σ Ψ ˜ ) , Φ ( t , x ) = 1 2 ln G ( Φ ˜ ; Φ ˜ ¯ , σ Φ ˜ ) .
The line element becomes the following:
d s GPFLRW 2 = G ( Ψ ˜ ; Ψ ˜ ¯ , σ Ψ ˜ ) c 2 d t 2 + a 2 ( t ) G ( Φ ˜ ; Φ ˜ ¯ , σ Φ ˜ ) d x i d x j δ i j .
The functional metric F μ ν [ t , x , G ( Ψ ˜ ) , G ( Φ ˜ ) ] is symmetric, transforms as a ( 0 , 2 ) -tensor, and has a pseudo-Riemannian signature, with Gaussian distributions as in Equations (74) and (75) encoding probabilistic perturbations, consistent with the generalized framework ( U = 0 , L = 2 , codomain R ) [2,18]. □
Example 7 (GPFLRW Spacetime). 
The GPFLRW manifold–metric pair ( M ( 1 , 3 ) , F μ ν ) from Proposition 9 has the line element provided by Equation (77), illustrating a perturbed FLRW spacetime with probabilistic perturbations ( U = 0 , L = 2 , and codomain R ) [18].
Proposition 10 (Extended Probabilistic FLRW Spacetime). 
On a ( 4 + D P ) -dimensional manifold M ( 1 , 3 , D P ) with coordinates c μ = ( t , x 1 , x 2 , x 3 , P 1 , , P D P ) , an extended probabilistic FLRW (EPFLRW) manifold–metric pair ( M ( 1 , 3 , D P ) , F μ ν : C C R ) can be constructed, incorporating D P probabilistic dimensions.
Proof. 
Consider a manifold M ( 1 , 3 , D P ) with coordinates c μ = ( t , x 1 , x 2 , x 3 , P 1 , , P D P ) . Extend the perturbed FLRW metric in Equation (73) by adding probabilistic dimensions:
d s EPFLRW 2 = e 2 Ψ ( t , x ) c 2 d t 2 + a 2 ( t ) e 2 Φ ( t , x ) d x i d x j δ i j + δ α β d P α d P β ,
where δ α β is the Kronecker delta for probabilistic dimensions. The metric F μ ν is symmetric, transforms as a ( 0 , 2 ) -tensor as in Equation (65), and is block-diagonal with pseudo-Riemannian spacetime components and a positive-definite probabilistic component, consistent with probabilistic metric spaces ( U = 0 , L = 2 , and codomain R ) [2,18]. □
Example 8 (EPFLRW Spacetime). 
The EPFLRW manifold–metric pair ( M ( 1 , 3 , D P ) , F μ ν ) from Proposition 10 has the line element provided by Equation (78), illustrating a perturbed FLRW spacetime extended with D P probabilistic dimensions ( U = 0 , L = 2 , and codomain R ) [18].
Proposition 11 (Homogeneous Isotropic Probabilistic Expanding Spacetime). 
On a ( D P , D τ , D x ) -dimensional manifold M ( D P , D τ , D x ) with coordinates
c μ = ( P 1 , , P D P , τ 1 , , τ D τ , x 1 , , x D x ) ,
a homogeneous, isotropic, probabilistic, expanding spacetime (HIPEST) manifold–metric pair can be constructed with a symmetric ( 0 , 2 ) -tensor metric incorporating a probability function P ( τ ) .
Proof. 
Consider a manifold M ( D P , D τ , D x ) with coordinates
c μ = ( P 1 , , P D P , τ 1 , , τ D τ , x 1 , , x D x ) .
Define the HIPEST line element as follows:
d s HIPEST 2 = a 2 ( τ ) P 2 ( τ ) δ α β d P α d P β d τ α d τ β + d x α d x β ,
where a ( τ ) is the scale factor, P ( τ ) is a probability function, and δ α β is the Kronecker delta. The metric is symmetric, diagonal, with a pseudo-Riemannian signature for spacetime components and a positive-definite signature for probabilistic dimensions, transforming as a ( 0 , 2 ) -tensor as in Equation (65). The functional dependence on P ( τ ) models probabilistic dimensions, consistent with the generalized framework ( U = 0 , L = 2 , and codomain R ) [1,18]. □
Example 9 (HIPEST Manifold). 
The HIPEST manifold–metric pair from Proposition 11 has the line element provided by Equation (81), illustrating an homogeneous, isotropic spacetime with probabilistic dimensions ( U = 0 , L = 2 , and codomain R ) [18].
Proposition 12 (Generalized Curved Probabilistic Spacetimes). 
On a ( D P , D τ , D r ) - dimensional manifold M ( D P , D τ , D r ) with coordinates c μ = ( P 1 , , P D P , τ 1 , , τ D τ , r 1 , , r D r ) , generalized curved probabilistic spacetimes (fgcPST and sgcPST) can be constructed with a symmetric ( 0 , 2 ) -tensor metric, incorporating curvature and probabilistic scaling, including infinite-dimensional extensions.
Proof. 
Consider a manifold M ( D P , D τ , D r ) with coordinates
c μ = ( P 1 , , P D P , τ 1 , , τ D τ , r 1 , , r D r ) .
For the first generalized curved probabilistic spacetime (fgcPST), define the line element as follows:
d s fgcPST 2 = g α β ( D P , D τ , D r ) ( c ) d c α d c β ,
where g α β ( D P , D τ , D r ) is symmetric and transforms as a ( 0 , 2 ) -tensor as in Equation (65) [2]. As a special case, the first special curved probabilistic spacetime (fscPST) has the following:
d s fscPST 2 = a 2 ( c ) e 2 Y ( c ) d P D P 2 e 2 Ψ ( c ) d τ D τ 2 + e 2 Φ ( c ) d r D r 2 ,
where e 2 Y ( c ) 1 + 2 Y ( c ) , e 2 Ψ ( c ) 1 + 2 Ψ ( c ) , e 2 Φ ( c ) 1 2 Φ ( c ) , and d P D P 2 , d τ D τ 2 , d r D r 2 incorporate curvatures k P , k τ , and k { 1 , 0 , 1 } , as defined in Equations (68) and (71) [19]. For the second generalized curved probabilistic spacetime (sgcPST), the line element is provided by Equation (83). Aa a specific case, the first simple sgcPST (fssgcPST) is as follows:
d s fssgcPST 2 = O Pf ( c GR ) a 2 ( τ ) e 2 Ψ ( c GR ) d τ 2 + e 2 Φ ( c GR ) d x i d x j δ i j ,
where c GR = { τ , x } , and O Pf ( c GR ) is a scaling factor. In another case, the second simple sgcPST (sssgcPST) uses the following integrals:
d s sssgcPST 2 = α α d α β β d β g α β ( D P , D τ , D r ) ( c ) d c α d c β ,
reducing them to
d s sssgcPST 2 = O Ps ( c GR ) a 2 ( τ ) e 2 Ψ ( c GR ) d τ 2 + e 2 Φ ( c GR ) d x i d x j δ i j ,
where O Ps ( c GR ) is another observable. Both cases are symmetric, transform as ( 0 , 2 ) -tensors as in Equation (65), and align with the generalized framework ( U = 0 , L = 2 , and codomain R ), with probabilistic scaling via O Pf or O Ps , testable through observations [4,18]. □
Example 10 (First Special Curved Probabilistic Spacetime). 
The fscPST from Proposition 12 has the line element provided by Equation (84), with curvature in probabilistic, temporal, and spatial components, illustrating a curved probabilistic spacetime ( U = 0 , L = 2 , and codomain R ) [19].
Example 11 (Simple Second Generalized Curved Probabilistic Spacetimes). 
The fssgcPST and sssgcPST from Proposition 12 have line elements provided by Equations (85) and (87), respectively, where O Pf and O Ps are probabilistic scaling factors, with sssgcPST using an integral form for continuous indices ( U = 0 , L = 2 , and codomain R ) [4,18].

7. Entropic and Infinite-Dimensional Manifold–Metric Pairs

This section applies the generalized manifold–metric pair framework from Equation (12) to construct manifold–metric pairs incorporating entropy, information, probabilistic, and spacetime dimensions, including infinite-dimensional extensions. We formalize these constructions through a theorem, supported by propositions and examples, with equations labeled uniquely and referenced to avoid repetition.
Theorem 7 (Entropic and Infinite-Dimensional Manifold–Metric Pairs). 
Let M D be a differentiable manifold. The generalized manifold–metric pair ( M G , F μ 1 μ L ν 1 ν U [ c , f ( c ) ] : C μ 1 C ν U Q ) can be applied to construct the following:
  • An entropic manifold–metric pair replacing conformal time with entropy dimensions.
  • A combined generalized entropic spacetime (EST) incorporating entropy, time, and spatial dimensions.
  • A generalized informatic spacetime (GISTM) with information dimensions.
  • A probabilistic entropic spacetime manifold–metric pair (PESTMMP) combining probabilistic, entropic, and spacetime dimensions.
  • An infinite-dimensional manifold–metric pair with functional metrics and quaternion codomain.
The line element is provided by the following:
d s U + L = F μ 1 μ L ν 1 ν U [ c , f ( c ) ] d c μ 1 d c μ L d c ν 1 d c ν U ,
where d c ν i = g ν i λ d c λ for a symmetric ( 0 , 2 ) -tensor g μ ν , and F transforms as a ( U , L ) -tensor with optional symmetry in indices and codomain Q .
Proof. 
The theorem is established through five propositions, each constructing a specific class of manifold–metric pairs, followed by examples illustrating their application. The propositions address entropic replacements, combined entropic spacetimes, informatic spacetimes, probabilistic entropic spacetimes, and infinite-dimensional manifolds, with equations referenced as in Equation (88) to ensure consistency with the generalized framework. □

7.1. Entropic Manifold–Metric Pairs

We consider entropy (thermodynamic or information) as a positively increasing quantity, analogous to conformal time τ , and construct manifold–metric pairs incorporating entropy dimensions, ensuring compatibility with differential geometry.
Proposition 13 (Entropic Manifold Replacing Conformal Time). 
On a ( D S , D x ) -dimensional manifold M ( D S , D x ) with coordinates c = { S 1 , , S D S , r 1 , , r D x } , an entropic manifold–metric pair ( M ( D S , D x ) , g α β ( D S , D x ) : C C R ) can be constructed, where entropy coordinates S = S [ τ , s ( τ ^ ) ] replace conformal time, and g α β ( D S , D x ) is a symmetric, pseudo-Riemannian ( 0 , 2 ) -tensor.
Proof. 
Consider a D-dimensional differentiable manifold M D with a submanifold M ( D S , D x ) = M D S M D x , where D S + D x D . Define entropy coordinates S = { S 1 , , S D S } M D S , related to conformal time τ by the following:
S ^ : = s ( τ ^ ) ,
where s : R R is a differentiable mapping preserving monotonicity [1], and ^ denotes a unit vector. The generalized entropy is as follows:
S : = S [ τ , s ( τ ^ ) ] ,
where S is a smooth functional mapping to M D S [4]. The differential relation is as follows:
d S = τ S d τ ,
extended to multiple dimensions as follows:
d S D S = τ S d τ D τ ,
where τ = { τ 1 , , τ D τ } transforms as a covector field under coordinate changes τ α τ β [2]. For spacetime coordinates x = { τ , r } , the differential extends to the following:
d S = x S d τ ,
and in multiple dimensions,
d S D S = τ x S d x D x ,
where τ x = { τ 1 , , τ D τ , x 1 , , x D x } [1]. The metric is as follows:
d s entropic replacing 2 = g α β ( D S , D x ) ( c ) d c α d c β ,
where c = { S 1 , , S D S , r 1 , , r D x } , and g α β ( D S , D x ) is symmetric and transforms as a ( 0 , 2 ) -tensor [2]. This aligns with Equation (12) ( U = 0 , L = 2 , and codomain R ), with entropy coordinates replacing time via Equation (90). □
Example 12 (Special Entropic Replacing Manifold). 
The entropic manifold–metric pair from Proposition 13 has a special case with the line element:
d s special entropic replacing 2 = a 2 ( c ) e 2 Ψ ( c ) τ S 2 d S D S 2 + e 2 Φ ( c ) d r D r 2 ,
where τ S 2 is derived from Equation (92), and the metric is pseudo-Riemannian with Lorentzian signature, suitable for cosmological applications ( U = 0 , L = 2 , and codomain R ) [10].
Proposition 14 (Combined Generalized Entropic Spacetime). 
On a ( D S , D τ , D r ) -dimensional manifold M ( D S , D τ , D r ) with coordinates c = { S 1 , , S D S , τ 1 , , τ D τ , r 1 , , r D r } , a generalized entropic spacetime (EST) manifold–metric pair ( M ( D S , D τ , D r ) , g α β ( D S , D τ , D r ) : C C R ) can be constructed, incorporating entropy as an independent dimension.
Proof. 
Consider a manifold M ( D S , D τ , D r ) with coordinates
c = { S 1 , , S D S , τ 1 , , τ D τ , r 1 , , r D r } ,
where entropy S is related to time via Equation (89). The line element is as follows:
d s EST 2 = g α β ( D S , D τ , D r ) ( c ) d c α d c β ,
where g α β ( D S , D τ , D r ) is symmetric and transforms as a ( 0 , 2 ) -tensor [2]. A special case, the special entropic spacetime (SEST) has the following:
d s SEST 2 = a 2 ( c ) e 2 Ξ ( c ) d S D S 2 e 2 Ψ ( c ) d τ D τ 2 + e 2 Φ ( c ) d r D r 2 ,
where e 2 Ξ ( c ) 1 + 2 Ξ ( c ) controls entropic perturbations, and d S D S 2 may include curvature k S { 1 , 0 , 1 } analogous to spatial curvature [10]. The metric is pseudo-Riemannian aligns with Equation (12) (in which U = 0 , L = 2 , and codomain R ), and incorporates entropy as an independent dimension [19]. □
Example 13 (Special Entropic Spacetime). 
The SEST from Proposition 14 has the line element provided by Equation (99), with curvature in the entropic component, illustrating a generalized entropic spacetime ( U = 0 , L = 2 , and codomain R ) [19].
Proposition 15 (Generalized Informatic Spacetime). 
On a ( D I , D τ , D x ) -dimensional manifold M ( D I , D τ , D x ) with coordinates c μ = { I 0 , , I D I , τ 0 , , τ D τ , x 0 , , x D x } , a generalized informatic spacetime (GISTM) manifold–metric pair ( M ( D I , D τ , D x ) , g μ ν ( D I , D τ , D x ) : C C R ) can be constructed, where information is encoded as I [ P ( E ) ] = log [ P ( E ) ] .
Proof. 
Consider a manifold M ( D I , D τ , D x ) with coordinates
c μ = { I 0 , , I D I , τ 0 , , τ D τ , x 0 , , x D x } .
Define information as follows:
I [ P ( E ) ] = log [ P ( E ) ] ,
where P ( E ) is the probability of event E [34]. The line element is as follows:
d s GISTM 2 = g μ ν ( D I , D τ , D x ) ( c ) d c μ d c ν ,
where g μ ν ( D I , D τ , D x ) is symmetric and transforms as a ( 0 , 2 ) -tensor [1]. A special case is as follows:
d s GISTM 2 = a 2 ( c ) e 2 Λ ( c ) d I D I 2 e 2 Ψ ( c ) d τ D τ 2 + e 2 Φ ( c ) d x D x 2 ,
where e 2 Λ ( c ) 1 + 2 Λ ( c ) controls information perturbations. The metric is pseudo-Riemannian, aligning with Equation (12) ( U = 0 , L = 2 , and codomain R ), and incorporates information as an additional dimension [19]. □
Example 14 (GISTM Manifold). 
The GISTM from Proposition 15 has the line element given by Equation (103), illustrating an informatic spacetime with information perturbations ( U = 0 , L = 2 , and codomain R ) [19].
Proposition 16 (Probabilistic Entropic Spacetime Manifold–Metric Pairs). 
Considering a ( D P , D S , D τ , D x ) -dimensional manifold M ( D P , D S , D τ , D x ) with coordinates
D μ = { P μ P , S μ S , τ μ τ , x μ x } ,
a probabilistic entropic spacetime manifold–metric pair (PESTMMP) denoted with
M ( D P , D S , D τ , D x ) , g μ ν ( D P , D S , D τ , D x ) : C C R
can be constructed, combining probabilistic, entropic, and spacetime dimensions.
Proof. 
Consider a manifold M ( D P , D S , D τ , D x ) with coordinates D μ = { P μ P , S μ S , τ μ τ , x μ x } , where entropy is defined as in Equation (90). The line element is as follows:
d s 2 = g μ ν ( D P , D S , D τ , D x ) ( D ) d D μ d D ν ,
where g μ ν ( D P , D S , D τ , D x ) is symmetric and transforms as a ( 0 , 2 ) -tensor [2]. A special case is as follows:
d s PESTMMP 2 = a 2 ( D ) e 2 Θ ( D ) d P D P 2 + e 2 Ξ ( D ) d S D S 2 e 2 Ψ ( D ) d τ D τ 2 + e 2 Φ ( D ) d x D x 2 ,
where e 2 Θ ( D ) 1 + 2 Θ ( D ) and other perturbation terms are defined similarly [19]. The metric is pseudo-Riemannian, aligns with Equation (12) ( U = 0 , L = 2 , and codomain R ), and combines probabilistic, entropic, and spacetime dimensions [18]. □
Example 15 (PESTMMP Manifold). 
The PESTMMP from Proposition 16 has the line element provided by Equation (107), illustrating a spacetime with probabilistic and entropic dimensions ( U = 0 , L = 2 , and codomain R ) [18,19].
Proposition 17 (Generic Informatic Probabilistic Entropic Spacetime). 
Considering a ( D I , D P , D S , D τ , D x ) -dimensional manifold M ( D I , D P , D S , D τ , D x ) with coordinates
c μ = { I μ I , P μ P , S μ S , τ μ τ , x μ x } ,
a generic informatic probabilistic entropic spacetime manifold–metric pair (GIPESTMMP)
M ( D I , D P , D S , D τ , D x ) , G μ 1 μ L ν 1 ν U : C μ 1 C ν U C
can be constructed, with the Shannon entropy relating information, probability, and time.
Proof. 
Consider a manifold M ( D I , D P , D S , D τ , D x ) with coordinates c μ = { I μ I , P μ P , S μ S , τ μ τ , x μ x } , where information is defined as in Equation (101) and entropy as in Equation (90). Define the Shannon entropy as follows:
S Shannon τ , P ( τ , E ) , I [ P ( τ , E ) ] = Ω E d E P ( τ , E ) log [ P ( τ , E ) ] ,
which is smooth and well-defined [4,34]. The line element is as follows:
d s U + L = G μ 1 μ L ν 1 ν U ( c ; k c ) d c μ 1 d c μ L d c ν 1 d c ν U ,
where G μ 1 μ L ν 1 ν U is a ( U , L ) -tensor, transforming as follows:
G μ 1 μ L ν 1 ν U = i = 1 U c ν i c ρ i j = 1 L c σ j c μ j G σ 1 σ L ρ 1 ρ U ,
with codomain C  [29]. The manifold–metric pair aligns with Equation (12), with k c describing intrinsic curvature, and supports complex-valued metrics for quantum or informatic applications ( U , L arbitrary, and codomain C ) [19,34]. □
Example 16 (GIPESTMMP Manifold). 
The GIPESTMMP from Proposition 17 has the line element provided by Equation (111), with Shannon entropy from Equation (110), illustrating a spacetime with informatic, probabilistic, and entropic dimensions ( U , L arbitrary, and codomain C ) [19,34].

7.2. Infinite Dimensional Manifold–Metric Pair

Proposition 18 (Infinite-Dimensional Manifold–Metric Pair). 
On an infinite-dimensional manifold M ( ) with coordinates c α , α [ , + ] , an infinite-dimensional manifold–metric pair ( M ( ) , F α 1 α L β 1 β U : C α 1 C β U Q ) can be constructed, with a functional metric and integral-based line element.
Proof. 
Consider a 2D Euclidean manifold M 2 D as a starting point to generalize to infinite dimensions. The 2D line element is as follows:
Δ s 2 = Δ x 2 + Δ y 2 = m α 1 α 2 ( c ) Δ c α 1 Δ c α 2 ,
forming the manifold–metric pair:
( M 2 D , m ) M 2 D , m α 1 α 2 : C 1 C 2 R + ,
where m α 1 α 2 is symmetric and positive-definite [1]. For a 4D Minkowski space with coordinates c α = { t , x , y , z } , the line element is as follows:
Δ s 2 = v c d t 2 + Δ x 2 + Δ y 2 + Δ z 2 = m α 1 α 2 ( c ) Δ c α 1 Δ c α 2 ,
forming
( M 4 D , m ) M 4 D , m α 1 α 2 : C 1 C 2 C 3 C 4 R ,
with a pseudo-Riemannian metric [2]. Generalize to an infinite-dimensional manifold M ( ) with coordinates c α , α [ , + ] . The metric is as follows:
m α 1 α L = m α 1 α L ( c ) ,
and the line element is
Δ s L = + d α 1 + d α L m α 1 α L ( c ) Δ c α 1 Δ c α L ,
where m α 1 α L is symmetric [4]. Extended to a functional metric,
( M ( ) , F ) M ( ) , F α 1 α L β 1 β U : C α 1 C β U Q ,
with line element
Δ s U + L = + d β 1 + d β U + d α 1 + d α L F α 1 α L β 1 β U [ c , f ( c ) ] Δ c α 1 Δ c α L Δ c β 1 Δ c β U ,
where F transforms as a ( U , L ) -tensor [29]. This aligns with Equation (12), supporting infinite-dimensional and quaternion-valued metrics ( U , L arbitrary, codomain Q ) [19]. □
Example 17 (Infinite-Dimensional Euclidean and Minkowski Manifolds). 
The infinite-dimensional manifold–metric pair from Proposition 18 has line elements provided by Equation (113) for 2D Euclidean space, Equation (115) for 4D Minkowski space, and (120) for the generalized infinite-dimensional case, illustrating the progression to infinite dimensions ( U , L arbitrary, and codomain Q ) [4,19].

7.3. Nick Early’s Combinatorics Argument

In this section, we prove the generalization of a D-dimensional set to a ( D + 1 ) -dimensional set, and the generalization of the two-rank tensor to a ( U , L )-rank tensor. Nick Early’s combinatorics argument, inspired by his recent work with his collaborators [35], suggests that given a projected set of complex number C P 1 , a complex number set of dimension 1, C 1 , and a complex number set of dimensions 2, C , then there is the relation between these three elements as follows:
C P = C 2 / C 0 1 ,
with a one line-proof that follows from
C P 1 = C P 2 1 = C 2 / C 0 1 .
This relation implies that the property of the product of two elements, a , b C , is related to the scaled product of these two elements, i.e., we write the following:
( a , b ) ( λ a , λ b ) λ ( a , b ) .
This argument basically proves the existence of a relation that is between one-dimensional and two-dimensional through a modulus space.
Then, the previous relation is generalized as follows. Given a projected set of complex number C P D , a complex number set of dimension D, i.e., the set C D , and a complex number set of dimensions D + 1 , i.e., the set C D + 1 , then there is the relation between these three elements as follows:
C P D = C D + 1 / C 0 1 ,
with a one line-proof that follows from
C P D = C P D + 1 1 = C D + 1 / C 0 1 .
This argument basically proves the existence of a relation between the (D)-dimensional and ( D + 1 )-dimensional sets through a modulus space. This argument supports the idea of generalizing the two-rank tensor to a ( U , L )-rank tensor.

8. Conclusions

This study has developed a comprehensive mathematical formalism for advanced manifold–metric pairs, significantly advancing the theoretical frameworks of geometry, topology, and their applications in mathematical physics. Through a rigorous methodology grounded in mathematical construction proofs, logical foundations, and functional and tensor analysis, we have successfully constructed a diverse array of D-dimensional manifolds paired with corresponding metric spaces, incorporating higher-rank tensor metrics, complex and quaternionic codomains, and probabilistic structures. Our key results include the formulation of advanced and/or generalized manifold–metric pairs that accommodate various dependences of systems, such as time-dependent scaling, as exemplified in cosmological models like the Friedmann–Lemaitre–Robertson-Walker spacetime. Furthermore, we establish generalized metrizability for topological manifolds via the generalization of the Urysohn metrization theorem, ensuring compatibility with Euclidean topologies. Additionally, the integration of information theory, entropy, and probability into our framework has enabled the construction of innovative probabilistic and entropic manifold–metric pairs, such as the GIPESTMMM manifold, which offer novel insights into infinite-dimensional and quotient spaces with applications in theoretical physics and cosmology.
The methodologies employed, including the use of propositions to systematically verify higher-rank tensor constructions and the application of a partition of unity to derive smooth, positive-definite global metrics, have provided a robust foundation for these advancements. These results not only enhance the understanding of manifold–metric interactions but also open new avenues for modeling complex physical systems across scales, from astronomical to cosmological phenomena. The incorporation of symmetry and positive-definiteness conditions ensures that our metrics preserve essential geometric properties, making them suitable for applications in general relativity and beyond. Furthermore, the probabilistic and entropic extensions of our formalism pave the way for interdisciplinary applications, bridging mathematics with information theory and statistical mechanics. Finally, this study can be extended through the use of advanced tensor theories [5].
In conclusion, this work establishes a versatile and unified framework that pushes the boundaries of traditional manifold theory, offering both theoretical rigor and practical applicability. Future research could extend these findings by exploring additional codomain structures, such as octonionic metrics, or by applying the formalism to specific physical systems, such as quantum field theories or emergent spacetime models. We anticipate that this framework will serve as a cornerstone for further investigations into the interplay of geometry, topology, and physics, fostering new discoveries in both theoretical and applied domains.

Funding

This research received no external funding.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The author would like to acknowledge his former teachers, professors, and colleagues for their constant questioning and improving of authors’ knowledge. The authors would like also to thank Herbert Spohn, Jeume Gomis, and Nick Early for their valuable discussions that improved the presentation of this study. Part of this study was accomplished during the COVID-19 pandemic; therefore, the authors would like to express their sincere gratitude to all the social, medical, political staff, as well as their friends, which made the pandemic less painful.

Conflicts of Interest

The author declares no conflict of interest.

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Ntelis, P. Advanced Manifold–Metric Pairs. Mathematics 2025, 13, 2510. https://doi.org/10.3390/math13152510

AMA Style

Ntelis P. Advanced Manifold–Metric Pairs. Mathematics. 2025; 13(15):2510. https://doi.org/10.3390/math13152510

Chicago/Turabian Style

Ntelis, Pierros. 2025. "Advanced Manifold–Metric Pairs" Mathematics 13, no. 15: 2510. https://doi.org/10.3390/math13152510

APA Style

Ntelis, P. (2025). Advanced Manifold–Metric Pairs. Mathematics, 13(15), 2510. https://doi.org/10.3390/math13152510

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