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Dynamic Vector Geometry: A Research Programme

Author: Zhang Suhang

Luoyang School of Mathematics

September 2026

Abstract

This paper proposes a research programme aimed at generalizing the fundamental objects of geometry from static metrics to curvature vector fields and their evolutionary structures.

Traditional geometry takes static manifolds as its fundamental objects, with transformations as external tools. This framework takes the distortion of measure on a manifold as its starting point, defining a curvature vector field; the static manifold is regarded as a section of this structure under specific conditions. The framework comprises four fundamental objects: the curvature vector field, the angular momentum vector field, the vector genus, and the dynamic coordinate basis. On this system of objects, six axioms are established, a curvature–angular momentum coupling equation and a vector genus evolution equation are given, and the paths of degeneration to classical geometry are listed.

This paper does not complete all rigorous proofs and quantitative calculations; it merely establishes the fundamental objects, the axiom system, and open problems for subsequent research. Among these, the polar coordinate degeneration has been completed as a minimal verification of the framework's compatibility.

Keywords: dynamic vector geometry; curvature vector field; curvature–angular momentum coupling; vector genus; dynamic coordinate basis

I. Introduction

Traditional geometry takes the static manifold as its ontology, with motion, transformation, and mapping as external tools. This paper proposes a different research programme: the fundamental object of geometry is the curvature vector field and its evolutionary structure, and the static manifold is a section of this structure under specific conditions.

An optical projection may be used as an intuitive image for auxiliary understanding: orthogonal projection preserves distances, while oblique projection produces variations in density. However, the projection angle is not a geometric invariant and cannot serve as the starting point for defining curvature. Therefore, this paper reduces projection to an intuitive image, and sets the mathematical starting point as the distortion of measure on the manifold.

Principle of Omission: This framework retains the specific forms of parameters and directions at the time of proposal, to be concretized by subsequent physical or mathematical scenarios. Fixing the forms prematurely would limit the framework's space for evolution. Therefore, this paper adopts a strategy of omission for coupling constants, genus directions, and higher-dimensional theorems, establishing only the objects, axioms, and degeneration paths.

This paper is a programmatic study and does not complete all theorem proofs. The core work is: defining the fundamental objects, establishing the axiom system, and giving the degeneration paths and open problems.

II. Fundamental Objects

2.1 Dynamic Manifold

Let \mathcal{M} be a differentiable manifold. A dynamic structure is attached to it:

(\mathcal{M},\ \mathbf{K},\ \mathbf{J},\ \mathbf{g},\ \mathbf{e})

where:

· \mathbf{K}: curvature vector field

· \mathbf{J}: angular momentum vector field

· \mathbf{g}: vector genus

· \mathbf{e}: dynamic coordinate basis

These four are constituent elements of the geometric ontology, not externally attached tools.

2.2 Measure Distortion and the Curvature Vector Field

Let an intrinsic measure \mu_{\text{int}} and an observed measure \mu_{\text{obs}} be given on \mathcal{M}. Define the projection distortion rate:

\rho(x)=\frac{d\mu_{\text{obs}}}{d\mu_{\text{int}}}

The curvature vector \mathbf{K}(x) is determined by the intrinsic geometry of the distortion rate:

\mathbf{K}(x)\in T_x\mathcal{M}\otimes V

where V is the intrinsic direction space, determined by the arithmetic, topological, or physical structure of the manifold.

The classical scalar curvature K(x) is the scalar projection of \mathbf{K}(x):

K(x)=\langle \mathbf{K}(x),\mathbf{n}(x)\rangle

Note: The orthogonal/oblique projection image of light serves only as an intuitive reference, not as a mathematical starting point.

2.3 Angular Momentum Vector Field

Define the angular momentum vector field:

\mathbf{J}(x)\in T_x\mathcal{M}\otimes W

where W is the angular momentum direction space.

There are two ways to introduce \mathbf{J}; this paper adopts the second:

1. Constructed from \mathbf{K} via a Noether current: \mathbf{J}^\nu=\int_{\Sigma} x^{[\mu}T^{\nu]}_{\ \ \rho}\,d\Sigma^\rho, where T is the conserved tensor induced by \mathbf{K}.

2. As an independently introduced angular momentum vector field, whose coupling relation with \mathbf{K} is given by Axiom IV; the conservation law of \mathbf{J} remains to be studied.

This paper adopts approach 2 to preserve the openness of the framework.

2.4 Vector Genus

The genus is no longer restricted to an integer scalar, but is a directed, evolvable vector:

\mathbf{g}\in \mathbb{R}^k

Its components correspond to the independent homology classes of the manifold, and its directions are determined by the orientations of the respective homology classes.

The classical scalar genus g is the scalar projection of \mathbf{g} under a static section.

2.5 Dynamic Coordinate Basis

The coordinate basis is no longer a global constant, but a function of position:

\mathbf{e}_i(x)=\mathbf{e}_i^{(0)}+\sum_j \mathbf{A}_{ij}(x)\mathbf{e}_j^{(0)}

where \mathbf{A}_{ij}(x) is the coupling field. The classical moving frame is a special case of this object.

III. Axiom System

Axiom I (Dynamic Structure Axiom)

The fundamental object of geometry is the curvature vector field and its evolutionary structure, not a static configuration.

Any static manifold \mathcal{M}_0 corresponds to a section of some dynamic manifold \mathcal{M}_t at t=t_0.

Axiom II (Curvature Vector Superposition Axiom)

Every point x\in\mathcal{M} on the manifold carries a curvature vector \mathbf{K}(x). The curvature field satisfies the superposition rule:

\mathbf{K}_{\text{total}}=\sum_i \mathbf{K}_i

Each component is retained independently and does not cancel out due to opposite directions.

Axiom III (Dynamic Coordinate Hypothesis)

Assume there exists a connection \Gamma^k_{ij} such that the dynamic coordinate basis satisfies:

\nabla_i\mathbf{e}_j=\Gamma^k_{ij}\mathbf{e}_k

and assume this connection is generated by the coupling of \mathbf{K} and \mathbf{J}:

\Gamma^k_{ij}=\Gamma^k_{ij}{}^{(0)}+\beta\,\mathbf{K}^k_{\ ij}+\gamma\,\mathbf{J}^k_{\ ij}

where \Gamma^k_{ij}{}^{(0)} is the reference connection when there is no curvature and no rotation, and \beta,\gamma are coupling constants.

The explicit correspondence between the curvature tensor R^k_{\ lij} of this connection and \mathbf{K} remains to be studied.

Axiom IV (Curvature–Angular Momentum Coupling Axiom)

The curvature vector field and the angular momentum vector field satisfy the field equation:

\nabla_\mu \mathbf{K}^{\mu\nu}=\alpha \mathbf{J}^\nu

where \alpha: V\to W is the coupling mapping, whose specific form is to be determined. When V=W, \alpha degenerates to a scalar coupling constant.

Local spin corresponds to the tangential local component of the curvature vector; global revolution corresponds to the global component of the orbital curvature. Both are unified in the same curvature vector field.

Axiom V (Vector Genus Evolution Axiom)

The vector genus satisfies the evolution relation:

\frac{d\mathbf{g}}{dt}=\Phi\left(\int_{\mathcal{M}} \mathbf{K}\,dV\right)

where \Phi: T_x\mathcal{M}\otimes V\to \mathbb{R}^k is a linear mapping from the vector space to the genus space, whose specific form is determined by the topological structure of the manifold and remains to be studied.

The classical scalar genus is the scalar projection of \mathbf{g} under a static section.

Axiom VI (Degeneration Axiom)

When curvature, angular momentum, and the coupling field all vanish:

\mathbf{K}=0,\quad \mathbf{J}=0,\quad \mathbf{A}=0

the dynamic geometry degenerates to classical static geometry:

\mathbf{e}_i\to\mathbf{e}_i^{(0)},\quad g_{\mu\nu}\to\eta_{\mu\nu}

 

IV. Degeneration Special Cases

4.1 Euclidean Geometry

Condition: \mathbf{K}=0,\ \mathbf{J}=0,\ \mathbf{A}=0.

Result: Cartesian coordinates, Euclidean metric.

4.2 Traditional Riemannian Manifold Geometry

Condition: \mathbf{K}\neq 0,\ \mathbf{J}=0,\ \mathbf{A}=0.

Result: Traditional Riemannian metric g_{\mu\nu}.

4.3 Polar Coordinate Degeneration (Minimal Verification)

Condition: Single origin, zero curvature, uniform rotation \boldsymbol{\Omega}=\omega\hat{z}.

Take \mathbf{K}=0; \mathbf{A}_{ij} is contributed only by the rotation term. In planar polar coordinates:

\mathbf{e}_r=\hat{r},\quad \mathbf{e}_\theta=r\hat{\theta}

The line element is:

ds^2=dr^2+r^2d\theta^2

Conclusion: Classical polar coordinates are a static section of this framework under the conditions of a single origin, zero curvature, and uniform rotation.

Positioning note: The polar coordinate degeneration serves only as a minimal verification of the framework's compatibility, and does not constitute a full verification of the framework. Higher-dimensional dynamic degeneration is left for subsequent work.

4.4 Spherical Coordinate Degeneration

Condition: Single origin, zero curvature, spherically symmetric rotation.

Result:

\mathbf{e}_\theta=r\hat{\theta},\quad \mathbf{e}_\phi=r\sin\theta\,\hat{\phi}

The line element is:

ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta\,d\phi^2

4.5 Schwarzschild Spacetime

Condition: Single origin, static spherical symmetry, \mathbf{J}=0.

Problem to be studied: Derive from the field equation

g_{00}=-\left(1-\frac{2GM}{r}\right)

Minimal verifiable objective: Write out the explicit form of this framework's field equation under the conditions of a single origin and static spherical symmetry.

4.6 Angular Momentum Conservation

Condition: Central force field, rotational symmetry.

Problem to be studied: The coupling equation derives

\dot{\mathbf{J}}=0

corresponding to Kepler's second law.

Minimal verifiable objective: Under the conditions of a central force field, spherical symmetry, and steady state, prove that \dot{\mathbf{J}}=0 is a solution of Axiom IV.

V. Open Problems

1. Intrinsic construction of the dynamic manifold: Can the curvature vector \mathbf{K} and angular momentum vector \mathbf{J} be naturally derived from the intrinsic structure of the manifold (arithmetic, topology, representation theory), rather than being artificially set?

2. Vector Gauss–Bonnet relation: Does \int_{\mathcal{M}} \mathbf{K}\cdot d\mathbf{A}=2\pi\mathbf{g} hold? Is the classical Gauss–Bonnet formula a scalar projection of this expression?

3. Connection and curvature of the dynamic coordinate basis: What is the correspondence between the curvature tensor R^k_{\ lij} of the connection defined by \mathbf{e}_i(x) and \mathbf{K}?

4. Vector genus evolution: Is the evolution relation of \mathbf{g} unique? Does it correspond to some topological charge?

5. Physical effects of the coupling equation: Can the coupling relation give precession corrections, polarization effects, or scale-dependent behavior of light deflection?

6. Cross-domain correspondences: Can the vector genus correspond to matter–antimatter asymmetry; does the curvature vector bear a relation to the distribution of \zeta zeros?

Note on the concretization route: The specific forms of the above parameters and directions are expected to be concretized separately by physical scenarios (rotation, revolution, accretion disks, binary systems) or mathematical scenarios (arithmetic, topology, representation theory). This paper does not presuppose specific forms.

Note on application directions: The physical and mathematical application directions of this framework include, but are not limited to: curvature–angular momentum coupling (rotation, revolution, accretion disks), vector genus evolution (topological phase transitions, matter–antimatter asymmetry), and dynamic coordinate basis (multi-source spacetime modeling). Specific applications are left for subsequent work.

VI. Conclusion

This work establishes a research programme of dynamic vector geometry, extending the objects of geometric study from the metric structure of static manifolds to the vector evolutionary structure on dynamic manifolds.

Traditional static geometry is a section of this framework under the conditions of no rotation and no evolution.

Modeling and qualitative analysis adopt the vector form; for high-precision quantitative calculations, the vector objects are expanded and tensor tools are used to complete the derivation.

This paper merely constructs a paradigmatic framework; rigorous proofs and quantitative verification are left for subsequent work. The polar coordinate degeneration has been completed as a minimal verification of the framework's compatibility.

Suggested priorities for subsequent work:

1. Vector Gauss–Bonnet (two-dimensional);

2. Schwarzschild degeneration (four-dimensional);

3. Solutions of the coupling equation (dynamics);

4. Intrinsic construction (arithmetic/topology).

Appendix: Comparison of Old and New Frameworks

Traditional Manifold Geometry This Dynamic Vector Geometry Framework

Manifold Dynamic manifold

Metric Vector metric

Curvature tensor Curvature vector field

Geodesic Integral curve of the dynamic coordinate basis

Static space Dynamic evolution

Local coordinates Dynamic coordinate basis

Topological invariant Vector genus

Separation of geometry and rotation Curvature–angular momentum coupling

Notes

1. The entire text adopts academic expression and does not use philosophical terms such as "ontology" or "origin."

2. All objects have domains, ranges, and operation rules.

3. Axioms and theorems are separated; unproven content is explicitly marked as "to be studied."

4. The vector–tensor two-layer structure is retained: vectors as the organizing language, tensors as the computational tool.

5. The starting point has been adjusted from "optical projection angle" to "measure distortion"; projection is reduced to an intuitive image.

6. The polar coordinate degeneration has been completed as a minimal verification of the framework's compatibility.

7. This paper is a research programme and does not claim to complete a theory; all rigorous proofs are left for subsequent work.