This book introduces the reader to the
concept of smooth manifold through abstract definitions and, more importantly,
through many we believe relevant examples. Topics covered includes: Manifolds,
Natural Constructions on Manifolds, Calculus on Manifolds, Riemannian Geometry,
Elements of the Calculus of Variations, The Fundamental group and Covering
Spaces, Cohomology, Characteristic classes, Classical Integral Geometry,
Elliptic Equations on Manifolds and Dirac Operators.

The purpose of these notes is to
introduce and study differentiable manifolds. Topics covered includes:
Manifolds in Euclidean space, Abstract manifolds, The tangent space, Topological
properties of manifolds, Vector fields and Lie algebras, Tensors, Differential
forms and Integration.