PRIMER ON SMOOTH MANIFOLDS, A

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.

Contents:

• 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

Readership: Undergraduate students in Mathematics and Physics, as textbook for a one-semester intro course on differential geometry; master, PhD students and researchers in physics and mathematics needing a quick introduction to smooth manifolds and calculus thereon.

Key Features:

• Unlike most available textbooks on Differential Geometry, these lecture notes introduce smooth manifolds and calculus on them in a quick and simple way that is suitable for undergraduate students in Mathematics and Physics
• The presentation is clear and concise, yet it does not sacrifice rigor
• The book covers exactly the material needed for a one semester course in Differential Geometry, so instructors need not to make any choice
• One of the peculiar aspects of the book is its special emphasis on symmetries and infinitesimal symmetries in Differential Geometry, and a thorough discussion of Cartan calculus on differential forms is also provided