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[5] viXra:2604.0105 [pdf] submitted on 2026-04-28 23:19:12
Authors: Abdelmajid Ben Hadj Salem
Comments: 11 Pages.
In this article, we present the equations of the geodesic lines of a surface in $R^3$ and then we determine the calculation of the geodesic lines of the ellipsoid of revolution with a numerical example.
Category: Geometry
[4] viXra:2604.0102 [pdf] submitted on 2026-04-27 16:54:02
Authors: Norm Cimon
Comments: 19 Pages.
The impetus for the work is this quote:"...as shown by Gel’fand’s approach, we can only abstract a unique manifold if our algebra is commutative."[1]Geometric algebra is non-commutative. Components of different grades can be staged on different manifolds. As operations on those elements proceed, they can effect the promotion and/or demotion of components to higher and/or lower grades, and thus to different manifolds. This paper includes imagery that visually displays bivector addition and rotation on a sphere.David Hestenes interpreted the vector product or rotor in two-dimensions:"as a directed arc of fixed length that can be rotated at will on the unit circle, just as we interpret a vectoras a directed line segment that can be translated at will without changing its length or directionu2026"[2]Rotors can be used to develop addition and multiplication of bivectors on a sphere. For those rotational dynamics, rotors of lengthare the basis elements. The geometric algebra of bivectors — Hamilton’s "pure quaternions" — is thus shown to transparently operate on a spherical manifold.This paper also explores the possible generalizations that emerge from the placement of the graded elements which make up a geometric algebra onto separate manifolds.
Category: Geometry
[3] viXra:2604.0078 [pdf] submitted on 2026-04-21 23:47:05
Authors: Jean-Yves Boulay
Comments: 12 Pages. (Note by viXra Admin: Please cite scientific references of other authors)
Grounded in a novel mathematical framework, this study demonstrates that any Euclidean triangle can be uniquely categorized into one of four canonical classes based on the intersection of isosceles and right-angled characteristics. We prove that these four classes form a complementary entanglement capable of saturating a rectangular space without voids. Furthermore, this geometric configuration is shown to be isomorphic with specific numerical assemblies in number theory, establishing a direct link between the fundamental sequence of whole numbers and the stability of geometric structures.
Category: Geometry
[2] viXra:2604.0039 [pdf] submitted on 2026-04-12 00:35:54
Authors: Daniel Henrique Pereira
Comments: 25 Pages.
We develop the complete Riemannian geometry of Victoria—Nash asymmetric equilibrium manifolds (VNAE) for $n$-player games. The metric (g_{ij} = iota_i iota_j delta_{ij} + varepsilon H_{ij}(V,iota)) yields explicit Levi-Civita connection (Gamma^k_{ij}), Riemann tensor (R^i_{,jkl}) with fourth-order $V$-derivative cancellation, Ricci tensor (R_{ij} approx kappabigl(iota_{i,j} iota_i - kappa partial_i^2 iota_ibigr) delta_{ij}), and scalar curvature (K_s approx sum_{i<j} iota_i iota_j det H_{ij}^s + O(varepsilon^2)). Positive/negative/zero signatures classify stability geometrically. The Lyapunov—Morse functional (mathcal{L}) satisfies (frac{d^2}{dt^2}mathcal{L}big|_{mathrm{VNAE}} approx -2 operatorname{Ric}(dot{s}^perp,dot{s}^perp)) along gradient flows, establishing Ricci curvature as the normal contraction rate. Classical Nash, von Neumann’s minimax theorem, and Lyapunov stability emerge as degenerate flat limits as (varepsilonto0).
Category: Geometry
[1] viXra:2604.0031 [pdf] submitted on 2026-04-11 01:03:35
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