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Riemannian Geometry by Eckhard Meinrenken
This PDF covers the following topics related to Riemannian Geometry, Manifolds, Examples of manifolds, Submanifolds, Tangent spaces, Tangent map, Tangent bundle, Vector fields as derivations, Flows of vector fields, Geometric interpretation of the Lie bracket, Lie groups and Lie algebras, Frobenius theorem, Riemannian metrics, Existence of Riemannian metrics, Length of curves, Connections and parallel transport, Geodesics, The HopfRinow Theorem, The curvature tensor, Connections on vector bundles.
Author(s): Eckhard Meinrenken, University of Toronto
58 Pages 
These lecture notes are based on the course in Riemannian geometry at the University of Illinois over a period of many years. This book contains below topics, Riemannian metrics, Minimizers, Connections, The Riemannian Connection, Calculus of variations, Riemannian curvature, Conjugate Points, Completeness, Curvature and topology.
Author(s): Richard L. Bishop
64 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
Riemannian Geometry an introductory Course
This note covers curves and surfaces in the plane and in space, Riemannian manifolds, Connections, Geodesics, The second fundamental form.
Author(s): E P Van Den Ban and E Looijenga
42 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