Lecture notes on the Fundamental Structures of Differential Geometry

TL;DR

This paper introduces a sheaf-theoretic approach to fundamental differential geometry structures, emphasizing differential operators and horizontal subbundles, based on lecture notes from a specialized course.

Contribution

It presents a novel sheaf-based framework for understanding core differential geometry concepts, bridging algebraic and geometric perspectives.

Findings

Provides a new sheaf-theoretic formulation of differential geometry

Clarifies the role of differential operators and horizontal subbundles

Enhances understanding of geometric structures through algebraic methods

Abstract

This is the first part of the lecture notes that grew out of the special course given during the 2021-2022 academic year. In these lecture notes we present an approach to the fundamental structures of differential geometry that uses the vernacular of sheaves, differential operators and horizontal subbundles.