Abstract Geometric Algebra. Orthogonal and Symplectic Geometries

TL;DR

This paper explores the conditions defining orthogonal and symplectic abstract differential geometries, providing detailed sheaf-theoretic versions of key theorems like the symplectic Gram-Schmidt and Witt's theorems.

Contribution

It offers a detailed sheaf-theoretic framework for orthogonal and symplectic geometries, including proofs of fundamental theorems in this context.

Findings

Sheaf-theoretic formulation of symplectic Gram-Schmidt theorem

Sheaf-theoretic version of Witt's theorem

Characterization conditions for orthogonal and symplectic geometries

Abstract

Our main interest in this paper is chiefly concerned with the conditions characterizing \textit{orthogonal and symplectic abstract differential geometries}. A detailed account about the sheaf-theoretic version of the \textit{symplectic Gram-Schmidt theorem} and of the \textit{Witt's theorem} is also given.