Lectures on Differential Geometry
by John Douglas Moore
Publisher: University of California 2009
Number of pages: 263
This course will describe the foundations of Riemannian geometry, including geodesics and curvature, as well as connections in vector bundles, and then go on to discuss the relationships between curvature and topology. Topology will presented in two dual contrasting forms, de Rham cohomology and Morse homology.
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by Adam Marsh - arXiv
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 and relevant theorems.
by David R. Wilkins - Trinity College, Dublin
From the table of contents: Smooth Manifolds; Tangent Spaces; Affine Connections on Smooth Manifolds; Riemannian Manifolds; Geometry of Surfaces in R3; Geodesics in Riemannian Manifolds; Complete Riemannian Manifolds; Jacobi Fields.
by Ilkka Holopainen, Tuomas Sahlsten
Based on the lecture notes on differential geometry. From the contents: Differentiable manifolds, a brief review; Riemannian metrics; Connections; Geodesics; Curvature; Jacobi fields; Curvature and topology; Comparison geometry; The sphere theorem.
by Andrew Clarke, Bianca Santoro - arXiv
The holonomy group is one of the fundamental analytical objects that one can define on a Riemannian manfold. These notes provide a first introduction to the main general ideas on the study of the holonomy groups of a Riemannian manifold.