Preliminary Exploration of the Fusion Configuration as a Concrete Prototype of Fiber Bundles

Author: Zhang Suhang, Luoyang, Henan

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Abstract

Based on previous research results on spherical cap‑derived self‑similar geometry (SCD‑SG), this paper conducts an exploratory topological correlation analysis. By comparing the spherical base body and discrete spherical cap convex structures of SCD‑SG with the core elements of fiber bundles—total space, base space, fiber, projection, connection, etc.—one by one, we verify the correspondence in spatial construction logic between the two.

Taking the SCD‑SG configuration, which widely exists in nature and engineering, as a concrete real‑world prototype of abstract fiber bundles allows us both to intuitively explain topological concepts such as base space, fiber, and local triviality using physical structures, and to analyze the internal unit coupling, spatial continuity, and overall structural stability of SCD‑SG using the theoretical framework of fiber bundles. This paper is a qualitative exploratory study, clarifying the structural mapping relationships and applicable boundaries, thereby reserving space for further topological modeling of such geometries.

Keywords: spherical cap‑derived self‑similar geometry (SCD‑SG); fiber bundle; base space; fiber; projection map; local triviality; structural topological mapping

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1. Introduction

Two previous studies have successively completed the morphological definition, geometric independence determination, and analysis of energy optimality and steady‑state convergence mechanisms for spherical cap‑derived self‑similar geometry (SCD‑SG). SCD‑SG takes a closed spherical surface as its load‑bearing base, with regularly arranged spherical cap convex units of uniform morphology on the surface. The whole structure is smooth, continuous, and of finite hierarchy, commonly found in spherical enveloped viruses, spherical conformal antennas, and spherical sensor arrays.

From the perspective of differential topology, the “base carrier + attached units” composite architecture of SCD‑SG is highly consistent with the fundamental construction paradigm of fiber bundles. Classical fiber bundle theory mostly uses abstract mathematical models as examples and lacks intuitively observable, naturally existing physical references.

Taking the SCD‑SG geometric form as the object, this paper systematically introduces the basic concepts of base space, total space, fiber, projection map, local triviality, connection, etc., tentatively establishes a structural correspondence between physical geometry and fiber bundle theory, and carries out an exploratory cross‑disciplinary topological comparison.

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2. Basic Conceptual System of Fiber Bundles

The mathematical architecture of a standard fiber bundle includes a full set of basic topological elements. The core defining components are as follows:

1. Base space (base manifold): The underlying topological space that carries all attached structures; it is the supporting carrier of the bundle structure.

2. Fiber: An isomorphic topological subset attached to each point of the base space; all fibers in the same bundle are topologically equivalent.

3. Total space: The overall topological space formed by the base space together with all fibers.

4. Projection map: A continuous map from the total space to the base space, uniquely assigning each fiber element to its attachment point on the base space.

5. Local triviality: For any point in the base space, there exists a neighborhood such that the corresponding subset of the total space is topologically equivalent to the direct product of that neighborhood and the fiber.

6. Connection: A rule that defines the parallel transport of fibers along paths in the base space, ensuring smooth joining of the total space surface and continuous variation of the morphology.

The essence of a fiber bundle is to describe a spatial structural relationship in which the base space carries homogeneous fiber units that are locally decomposable and globally continuously coupled.

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3. Complete Correspondence Between SCD‑SG Geometry and Fiber Bundle Elements

Using the full set of topological concepts, we match and map the SCD‑SG morphological structure to the components of a fiber bundle point by point.

3.1 SCD‑SG Spherical Base Body – Base Space

The central closed smooth spherical surface of SCD‑SG is a two‑dimensional compact, boundaryless manifold. All spherical cap convexities are distributed on this spherical surface. This spherical surface exactly corresponds to the base space of a fiber bundle, serving as the fundamental load‑bearing space of the entire structure.

3.2 Single Spherical Cap Convex Unit – Fiber

Each independent spherical cap convexity on the SCD‑SG surface is morphologically and topologically identical to the others and is attached to a discrete base point on the spherical base surface. Each spherical cap can be regarded as a fiber, and the isomorphic units satisfy the basic requirement of fiber isomorphism.

3.3 Overall Composite Form – Total Space

The complete SCD‑SG geometry consists of the underlying spherical base body together with all spherical cap convexities. This corresponds to the total space defined in fiber bundles, encompassing both the carrier and all attached structures.

3.4 Attachment Relationship Between Base Point and Convexity – Projection Map

Any spherical‑cap fiber uniquely corresponds to an attachment anchor point on the base spherical surface. This one‑to‑one subordinate positioning relationship is equivalent to the projection map on a fiber bundle, realizing a continuous projection from elements of the total space to base points on the base space.

3.5 Small Local Region on the Spherical Surface – Local Triviality

Take any sufficiently small neighborhood on the base spherical surface of SCD‑SG. Within this neighborhood, the composite structure of the base body and convexities is simple and regular, without complex twisting or distortion. The local structure can be approximately decomposed as a direct product of the base region and the fiber unit, satisfying the core condition of local triviality.

3.6 Smooth Interface Between Spherical Surface and Spherical Cap – Connection

The curvature transitions smoothly between the base surface of SCD‑SG and the spherical cap convexities. The orientation of the fibers changes continuously with the curvature of the base surface, without abrupt discontinuities or fractures. This smooth joining constraint corresponds to the connection in fiber bundles, ensuring the global geometric continuity of the total space.

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4. Rationality of the Mapping Relationship and Applicable Boundaries

From the perspective of spatial construction logic, SCD‑SG geometry can fully map to all the basic fiber‑bundle concepts—base space, total space, fiber, projection map, local triviality, and connection. The structural coupling method conforms to the core construction rules of fiber bundles, qualifying SCD‑SG as a candidate concrete prototype of fiber bundles.

At the same time, we clarify the objective boundaries of this exploration:

A standard theoretical fiber bundle is fully covered by fibers over the entire base space. SCD‑SG, in contrast, exhibits a discrete sparse fiber distribution: fiber units exist only at limited anchor points, and it is not a strictly globally dense fiber bundle. The topological construction frameworks of the two are consistent, but there are objective differences in metric geometry, unit distribution density, etc. SCD‑SG is an approximate physical prototype.

The bidirectional application value is clear:

· Using the real SCD‑SG form as a reference, one can intuitively understand abstract topological concepts such as base space, fiber, and local triviality.
· Conversely, relying on the full theoretical toolbox of fiber bundles, one can analyze the arrangement rules, spatial constraints, structural deformations, and overall stability of SCD‑SG.

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5. Summary and Outlook

This exploration has systematically introduced the core concepts of base space, total space, fiber, projection, local triviality, connection, etc., and has demonstrated that the composite configuration of SCD‑SG spherical cap‑derived self‑similar geometry can form a complete element‑wise mapping with the fiber bundle system at the topological architecture level.

This natural geometric entity can serve as an intuitive physical example for fiber bundle theory, alleviating the difficulty of understanding purely abstract models. Subsequent research may further investigate topological invariant calculations for SCD‑SG, definitions of sparse fiber bundles, and structural topology optimization based on base space constraints and fiber distribution characteristics.

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References

[1] Steenrod N. The Topology of Fibre Bundles[M]. Princeton University Press, 1951.
[2] Zhang S. Morphological definition of spherical cap‑derived self‑similar geometry[R].
[3] Walls A C, et al. Structure of SARS‑CoV‑2 spike glycoprotein[J]. Cell, 2020.

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(End of the paper）