Semi-Riemann Geometry and General Relativity
by Shlomo Sternberg
Number of pages: 251
This book represents course notes for a one semester course at the undergraduate level giving an introduction to Riemannian geometry and its principal physical application, Einstein’s theory of general relativity. The background assumed is a good grounding in linear algebra and in advanced calculus, preferably in the language of differential forms.
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by Bang-Yen Chen - arXiv
Submanifold theory is a very active vast research field which plays an important role in the development of modern differential geometry. In this book, the author provides a broad review of Riemannian submanifolds in differential geometry.
by Bertrand Eynard - arXiv.org
An introduction to the geometry of compact Riemann surfaces. Contents: Riemann surfaces; Functions and forms on Riemann surfaces; Abel map, Jacobian and Theta function; Riemann-Roch; Moduli spaces; Eigenvector bundles and solutions of Lax equations.
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These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It starts with the definition of Riemannian and semi-Riemannian structures on manifolds.
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The main purpose of these lecture notes is to introduce the beautiful theory of Riemannian Geometry. Of special interest are the classical Lie groups allowing concrete calculations of many of the abstract notions on the menu.