Riemannian Geometry: Definitions, Pictures, and Results

TL;DR

This paper provides a clear, visual, and accessible overview of Riemannian geometry tailored for physics, emphasizing concepts, relationships, and geometric intuition, with novel figures and a focus on manifolds with connections.

Contribution

It offers a visual and conceptual approach to Riemannian geometry, including novel figures and a focus on manifolds with connections, avoiding traditional proofs and calculations.

Findings

Detailed figures and geometric viewpoints enhance understanding.

Parallel development of coordinate-free, tensor, and Cartan formalisms.

Clarification of common confusions and alternative notations.

Abstract

A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel.