Textbook in PDF format

Differential Geometry is one of the major branches of current Mathematics, and it is an unavoidable language in modern Physics. The main characters in Differential Geometry are smooth manifolds: a class of geometric objects that locally behave like the standard Euclidean space.The book provides a first introduction to smooth manifolds, aimed at undergraduate students in Mathematics and Physics. The only prerequisites are the Linear Algebra and Calculus typically covered in the first two years. The presentation is as simple as possible, but it does not sacrifice the rigor. The lecture notes are divided into 10 chapters, with gradually increasing difficulty. The first chapters cover basic material, while the last ones present more sophisticated topics. The definitions, propositions, and proofs are complemented by examples and exercises. The exercises, which include part of the proofs, are designed to help the reader learn the language of Differential Geometry and develop their problem-solving skills in the area. The exercises are also aimed at promoting an active learning process. Finally, the book contains pictures which are useful aids for the visualization of abstract geometric situations. The lecture notes can be used by instructors as teaching material in a one-semester course on smooth manifolds.
Preface.
About the Author.
Charts, Atlases, and Smooth Manifolds.
Smooth Maps and Submanifolds.
Tangent Vectors.
Full Rank Smooth Maps.
Vector Fields.
Flows and Symmetries.
Covectors and Differential 1-Forms.
Differential Forms and Cartan Calculus.
Vector Bundles.
Integration on Manifolds.
Bibliography.
Index