Advanced Manifold–Metric Pairs
Abstract
1. Introduction
2. Advanced Manifold–Metric Space Pairs
Generalized Manifold–Metric Pair Theorem
3. Metrizability Conditions
3.1. Metrizability Conditions in Standard Manifold–Metric Pairs
- Hausdorff: For any two distinct points , there exist disjoint open neighborhoods and such that and .
- Second-Countable: The topology of M has a countable basis, i.e., there exists a countable collection of open sets such that every open set in M can be expressed as a union of elements of .
- Locally Euclidean: For a D-dimensional manifold, every point has a neighborhood homeomorphic to an open subset of .
- Paracompact: Every open cover of M has a locally finite open refinement. That is, for any open cover , there exists a locally finite open cover such that each for some .
3.2. Standard Distance Function for (0,2)-Rank Tensor
4. Metrizability for Generic (U,L)-Rank Tensors
- a (2,0)-rank tensor 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
- A (0,2)-Rank Tensor: A symmetric, positive-definite tensor that defines a Riemannian metric.
- Tensor Operations: Operations such as contractions or inversions to relate the (U,L)-rank tensor to or its inverse . For example,
- for a (2,0)-rank tensor , the additional structure is a (0,2)-rank tensor such that ;
- for a (1,1)-rank tensor , a (0,2)-rank tensor such that for some tensor , or T preserves as an isometry ();
- for a (0,3)-rank tensor , a vector field such that is symmetric and positive-definite.
- Symmetry and Positive-Definiteness: The resulting (0,2)-rank tensor must be symmetric () and positive-definite ( for all non-zero ).
- Smoothness: All tensors and auxiliary fields (e.g., and ) must vary smoothly over M.
4.2. Analysis of (0,3)-Rank Tensor Distance Construction
- Symmetry: The tensor must be symmetric () to ensure .
- Positive-Definiteness: The expression for all non-zero , with equality only at . In the 2D example, can be negative, so the absolute value is necessary.
- Topology Equivalence: The distance must induce the manifold’s topology, requiring to be compatible with local Euclidean charts.
- Smoothness: The tensor must be smooth.
4.3. Generalized Metrizability for (0,L)-Rank Tensors
- Symmetry: The tensor must be symmetric under all index permutations:
- Positive-Definiteness: For all non-zero vectors ,
- Topology Equivalence: The distance must induce the manifold’s topology, requiring to be compatible with local Euclidean charts.
- Smoothness: The tensor must be smooth over M.
- Auxiliary Structure (Optional): If needed, vector fields such that
- 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 .
4.4. Generalized Metrization Theorem
4.5. Proof of the Generalized Theorem
4.6. Generalized Metrizability for (U,L)-Rank Tensors
- Symmetry: The tensor must be symmetric under permutations of covariant indices and contravariant indices separately:
- Positive-Definiteness: For all non-zero vectors and covectors ,
- Topology Equivalence: The distance must induce the manifold’s topology, requiring to be compatible with local Euclidean charts.
- Smoothness: The tensor and any auxiliary fields (e.g., ) must be smooth over M.
- Auxiliary Structure: A (0,2)-rank tensor , symmetric and positive-definite, to relate covariant and contravariant components: . Optionally, fields (e.g., vector fields and covector fields ) such that
- 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 .
4.7. Generalized Metrization Theorem
4.8. Proof of the Generalized Metrization Theorem
4.9. Comparison of the Generalized Manifold–Metric Pair with a Hessian Structure
- Tensor Rank: Both and are (U,L)-rank tensors, making them structurally identical.
- Line Element: The line elements are identical, as , both of order .
- Functional Dependence: The Hessian depends solely on , while includes , which may be incorporated into or represent coordinate parameterization.
- Symmetry: Both tensors are symmetric in their covariant and contravariant indices due to the commutativity of partial derivatives in , satisfying the symmetry requirement for .
- Positive-Definiteness: Both require , ensured by the absolute value in the distance function and the auxiliary metric .
- Applications: The Hessian structure is used in information geometry and statistical mechanics, while supports cosmological models like FLRW, where (with , ) is unlikely to be a Hessian unless a suitable is constructed.
5. Generic Advanced Manifold–Metric Pair Applications
5.1. Applications of Generalized Manifold–Metric Pairs
- Higher-rank -tensor metrics with real codomain on Euclidean manifolds.
- -tensor metrics with complex or quaternionic codomains for quantum or field-theoretic applications.
- Functional -tensor metrics with time-dependent scaling for cosmological spacetimes, such as the Friedmann–Lemaître–Robertson–Walker (FLRW) model.
An Expanding Homogeneous and Isotropic Manifold and Metric Pair
5.2. Comparison of a FLRW Manifold–Metric Pair with a Warped Product Structure
- Base manifold , with coordinate t and metric .
- Fiber manifold , with coordinates and metric , where is the Kronecker delta.
- Warping function , i.e., the scale factor, depending only on the time coordinate.
- Base manifold , with metric .
- Fiber manifold , with metric .
- Warping function .
6. Generalized -Dimensional and Probabilistic Manifold– Metric Pairs
- A -dimensional submanifold-metric pair with a pseudo-Riemannian -tensor metric, where and 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.
6.1. -Dimensional Manifold–Metric Pair
6.2. Probabilistic Manifold–Metric Pairs
7. Entropic and Infinite-Dimensional Manifold–Metric Pairs
- 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.
7.1. Entropic Manifold–Metric Pairs
7.2. Infinite Dimensional Manifold–Metric Pair
7.3. Nick Early’s Combinatorics Argument
8. Conclusions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Ntelis, P. Advanced Manifold–Metric Pairs. Mathematics 2025, 13, 2510. https://doi.org/10.3390/math13152510
Ntelis P. Advanced Manifold–Metric Pairs. Mathematics. 2025; 13(15):2510. https://doi.org/10.3390/math13152510
Chicago/Turabian StyleNtelis, Pierros. 2025. "Advanced Manifold–Metric Pairs" Mathematics 13, no. 15: 2510. https://doi.org/10.3390/math13152510
APA StyleNtelis, P. (2025). Advanced Manifold–Metric Pairs. Mathematics, 13(15), 2510. https://doi.org/10.3390/math13152510