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Semi Riemann Geometry and General Relativity
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.
Author(s): ShlomoSternberg
251 Pages 
Riemannian Geometry by Shiping Liu USTC
This PDF covers the following topics related to Riemannian Geometry : Introduction, Riemannian Metric, Geodesics, Connections, Curvatures, Space forms and Jacobi fields, Comparison Theorem, Candidates for Synthetic Curvature Conditions.
Author(s): Shiping Liu, Department of Mathematics, USTC
218 Pages
An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity
This book covers the following topics: Differentiable Manifolds, Differential Forms, Riemannian Manifolds, Curvature, Geometric Mechanics, Relativity.
Author(s): Leonor Godinho and Jose Natario
272 Pages
This note covers the following topics: What is a manifold, Analysis on Riemannian manifolds, Geodesics and curvature, The Bishop volume comparison theorem.
Author(s): F.E. Burstall
29 Pages
Riemannian manifolds with geometric structures
The main aim of this book is to get a way of union of various differential geometric structures on Riemannian manifolds in one scheme.
Author(s): Alexander A. Ermolitski
187 Pages