Beginners are supposed to be familiar with fundamentals of Commutative Algebra,
Topology and Differential Geometry. For instance, all this can be found in the
M. F. Atiyah, I. G. MacDonald, - Introduction to Commutative Algebra, - Westview
Chapter 1: Rings and Ideals;
Chapter 2: Modules.
It is strongly suggested to solve exercises to these two chapters.
John M. Lee, -
Introduction to Smooth Manifolds, -
Springer-Verlag, Graduate Texts in Mathematics, Vol. 218, 2003.
Topology (p. 540);
• Continuity and convergency (pp. 541-543);
• Hausdorff spaces (pp. 543-544);
• Base and countability (pp. 544-545);
• Subspace, product spaces, and disjoint unions (pp.
• Quotient spaces and quotient maps (pp. 548-549);
• Open and closed maps (pp. 550-550);
• Connectedness (pp. 550-552);
• Compactness (pp. 552-553).
•Examples of Smooth Manifolds.
(Chapter 1 is also
available on the author's web page.)
•Smooth functions and smooth maps (pp. 31-37);
•Partitions of unity (pp. 49-57).
•Tangent vectors (pp. 61-65);
•Pushforwards (pp. 65-69);
•Computations in coordinates (pp. 69-75);
•Tangent vectors to a curves (pp. 75-77).
•The tangent bundle (pp. 81-82);
•Vector fields on manifold (pp. 82-89).
•Covectors (pp. 125-127);
•Tangent covectors on manifold (pp. 127-129);
•The cotangent bundle (pp. 129-132);
•The differential of a function (pp. 132-136);
•Pullbacks (pp. 136-138).
Jet Nestruev, - Smooth manifolds and Observables
Springer-Verlag, Graduate Texts in Mathematics, Vol. 220, 2002.
First chapters of this book will introduce you to the spirit of the school. People
who have read this book
and solved 70% of the exercises will be able follow the veteran courses.