Generalized Differential Geometry

TL;DR

This paper introduces a generalized differential geometry framework using generalized functions and manifolds, enabling the treatment of singularities and nonlinearities in differential equations with potential implications for space-time theories.

Contribution

It develops a generalized manifold embedding that simplifies handling singularities and nonlinearities, linking it to existing global theories and exploring generalized space-time models.

Findings

Singularities vanish on the generalized manifold

Products of nonlinearities become well-defined

Potential impacts on classical space-time models

Abstract

Generalized Functions play a central role in the understanding of differential equations containing singularities and nonlinearities. Introducing infinitesimals and infinities to deal with these obstructions leads to controversies concerning the existence, rigor and the amount of non-standard analysis needed to understand these theories. Milieus constructed over the generalized reals sidestep them all. A Riemannian manifold M embeds discretely into a generalized manifold $M^*$ on which singularities vanish and products of nonlinearities make sense. Linking this to an already existing global theory provides an algebra embedding $\kappa :\hat{{\cal{G}}}(M)\longrightarrow {\cal{C}}^{\infty}(M^*,\widetilde{\mathbb{R}}_f)$. Generalized Space-Time is constructed and its possible effects on Classical Space-Time are examined.