Riemannian Geometry

TL;DR

This paper provides an in-depth exploration of Riemannian geometry using bundle frameworks, covering fundamental structures, geodesics, curvature, and advanced topics like volume comparison, with added problems and streamlined background.

Contribution

It introduces a comprehensive, problem-rich presentation of Riemannian geometry emphasizing bundle methods and advanced theorems, with unique focus on metric Taylor series and Riccati equations.

Findings

Taylor series for the metric in normal coordinates

Use of matrix Riccati equations in volume comparison

Extended coverage of geodesic and curvature properties

Abstract

These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian and semi-Riemannian structures on manifolds. Affine connections, geodesics, torsion and curvature, the exponential map, and the Riemannian connection follow quickly. The Taylor series for of the metric in normal coordinates is an unusual feature. The last third covers, first and second variation of energy, completeness, cut points, the Hadamard-Cartan theorem, and finishes with the use of matrix Riccati equations to prove the volume comparison theorem. There is a comprehensive index.