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Riemannian Geometry
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 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
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
A Course in Riemannian Geometry(Wilkins D.R pdf)
This note covers the following topics, Smooth Manifolds, Tangent Spaces, Affine Connections on Smooth Manifolds, Riemannian Manifolds, Geometry of Surfaces in R3, Geodesics in Riemannian Manifolds, Complete Riemannian Manifolds and Jacobi Fields.
Author(s): David R. Wilkins
72 Pages
An Introduction to Riemannian Geometry
This note covers the following topics, Differentiable Manifolds, The Tangent Space, The Tangent Bundle, Riemannian Manifolds, The LeviCivita Connection, Geodesics, The Riemann Curvature Tensor, Curvature and Local Geometry.
Author(s): Sigmundur Gudmundsson
111 Pages