Zero Divisor Manifolds

TL;DR

The paper develops a geometric framework over algebras with zero divisors by formulating zero divisor manifolds from the dual real number algebra $\mathcal{R}^{(2)}$. It introduces $\mathcal{R}^{(2)}$-modules, a $(\mathcal{R}^{(2)}, \mathcal{R})$-basis, and the finite dimension $(n, m)$, culminating in the construction of the real projective $\mathcal{R}^{(2)}$-space $\mathcal{R}^{(2)}\mathcal{P}^{n,m}$ with a smooth atlas. The work further defines dual real differentiability and $\mathcal{R}^{(2)}$-smooth maps, and initiates a symplectic analogue by presenting a Hu-Liu symplectic form and a canonical basis for finite-dimensional symplectic $\mathcal{R}^{(2)}$-modules. Overall, it lays foundational tools for generalized differential geometry and symplectic theory in the presence of zero divisors, with potential implications for extended geometric models.

Abstract

We develop the basic properties of $R^{(2)}$-modules, introduce the concept of zero divisor manifolds, construct projective $R^{(2)}$-space which generalizes the real projective space, and initiate the study of the counterpart of symplectic spaces

Theorems & Definitions (8)

• Definition 1.1
• Proposition 1.1
• Proposition 1.2
• Definition 2.1
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Proposition 3.1