Concept of Manifold
The entire lecture was devoted to motivation and examples of -smooth manifolds (submanifolds of , spheres, tori, projective spaces, matrix groups etc).
Slightly more detailed plan of the lecture is here.
If you want to read more (which is most highly welcome), here are a few recommendations:
- F. Warner, Foundations of Differentiable Manifolds and Lie Groups
- W. M. Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry
- N. J. Hicks, Notes on Differential Geometry, Van Nostrand
- B. A. Dubrovin, Differential Geometry. Notes from SISSA course (Trieste, Italy)
A few thoughts on how to use these books. The subject (calculus on manifolds) is difficult because it involves both complicated concepts and the new language describing these concepts, and there is no way to learn these things but in parallel. One possibility to practice in the new language is to read as many texts about familiar subjects, as possible. This is what I suggest: if you believe you understand certain things, try to read about them in different books and make sure that different notation adopted by different authors does not detract you from the core.
A little bit more specific note. A closely related beautiful subject, Algebraic Geometry, was born from studies of how subsets of real or complex Euclidean space may look like. For some time it developed using mostly geometric/analytic tools, but eventually it was realized that to avoid problems with singularities, “double points”, “points at infinity” etc., one should start with the algebra of polynomials in one and several variables, its ideals, the quotient algebras and schemes in general. This approach brought tremendous achievements.
In my attempt to present the basic constructions of Calculus on Manifolds and, more generally, Differential Geometry, I decided to make the first several steps in a similar spirit and build objects from the algebra of -smooth functions on a manifold. Of course, these algebras are very different from the algebras of polynomials (in particular, they are not Noetherian), which makes life some times easier, some times more difficult.
See you in a week.